The document discusses finding the least common multiple (LCM) of numbers. It defines the LCM as the smallest number that is a multiple of all the given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from prime factorizations. The preferred method when the LCM is large is to construct it by fully factorizing each number into prime factors and taking the highest power of each prime factor.
The document defines the least common multiple (LCM) as the smallest positive number that is a multiple of two or more given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from the prime factorizations. When the numbers are large, constructing the LCM is easier than listing multiples. The process of construction involves factorizing each number into prime factors and taking the highest power of each prime factor. The product of these highest powers gives the LCM. The LCM of denominators of fractions is also defined as the least common denominator (LCD).
3 lcm and lcd, addition and subtraction of fractionselem-alg-sample
The document defines the least common multiple (LCM) as the smallest positive number that is a multiple of two or more given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from prime factorizations. The constructing method involves factorizing each number, identifying the highest power of each prime factor, and taking their product. This is described as easier than listing when the LCM is large. The least common denominator (LCD) of fractions is also defined as the LCM of the denominators.
The document discusses methods for finding the least common multiple (LCM) of two or more numbers. It defines the LCM as the smallest number that is a multiple of all the numbers. Two methods are described: listing the multiples and finding their common multiples, and constructing the LCM by factorizing each number and taking the highest power of each prime factor. An example uses the constructing method to find the LCM of 8, 15, and 18 as 360.
The document discusses the least common multiple (LCM) of numbers. The LCM of two or more numbers is the smallest positive number that is a multiple of each number. Two methods are presented for finding the LCM: 1) listing the multiples of each number and finding their common multiples, and 2) factorizing each number and taking the highest power of each prime factor. Examples are provided to illustrate both methods. The least common denominator (LCD) of fractions is also introduced as being the LCM of the denominators.
The document discusses using the least common multiple (LCD) to convert fractions to equivalent whole numbers. It provides an example of finding the LCD of fractions {2/3, 5/8, 7/12, 3/4} which is 24. Multiplying each fraction by the LCD converts them to the whole number list {16, 15, 14, 18}. The fractions are then listed from largest to smallest as 3/4, 2/3, 5/8, 7/12.
14 lcm, addition and subtraction of fractionsalg1testreview
The document discusses methods for finding the least common multiple (LCM) of two or more numbers. It defines the LCM as the smallest number that is a multiple of all the numbers. Two methods are described: listing the multiples and finding the smallest common one, and constructing the LCM by factorizing each number and taking the highest power of each prime factor. An example using each method is provided to find the LCM of different sets of numbers.
The document discusses finding the least common multiple (LCM) and least common denominator (LCD) of numbers. It provides examples of:
1) Finding the LCM of numbers by listing their common multiples or constructing it from their prime factorizations.
2) Using the LCM, also called the LCD, of denominators to divide a pizza evenly between people who each want a fractional amount.
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
The document defines the least common multiple (LCM) as the smallest positive number that is a multiple of two or more given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from the prime factorizations. When the numbers are large, constructing the LCM is easier than listing multiples. The process of construction involves factorizing each number into prime factors and taking the highest power of each prime factor. The product of these highest powers gives the LCM. The LCM of denominators of fractions is also defined as the least common denominator (LCD).
3 lcm and lcd, addition and subtraction of fractionselem-alg-sample
The document defines the least common multiple (LCM) as the smallest positive number that is a multiple of two or more given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from prime factorizations. The constructing method involves factorizing each number, identifying the highest power of each prime factor, and taking their product. This is described as easier than listing when the LCM is large. The least common denominator (LCD) of fractions is also defined as the LCM of the denominators.
The document discusses methods for finding the least common multiple (LCM) of two or more numbers. It defines the LCM as the smallest number that is a multiple of all the numbers. Two methods are described: listing the multiples and finding their common multiples, and constructing the LCM by factorizing each number and taking the highest power of each prime factor. An example uses the constructing method to find the LCM of 8, 15, and 18 as 360.
The document discusses the least common multiple (LCM) of numbers. The LCM of two or more numbers is the smallest positive number that is a multiple of each number. Two methods are presented for finding the LCM: 1) listing the multiples of each number and finding their common multiples, and 2) factorizing each number and taking the highest power of each prime factor. Examples are provided to illustrate both methods. The least common denominator (LCD) of fractions is also introduced as being the LCM of the denominators.
The document discusses using the least common multiple (LCD) to convert fractions to equivalent whole numbers. It provides an example of finding the LCD of fractions {2/3, 5/8, 7/12, 3/4} which is 24. Multiplying each fraction by the LCD converts them to the whole number list {16, 15, 14, 18}. The fractions are then listed from largest to smallest as 3/4, 2/3, 5/8, 7/12.
14 lcm, addition and subtraction of fractionsalg1testreview
The document discusses methods for finding the least common multiple (LCM) of two or more numbers. It defines the LCM as the smallest number that is a multiple of all the numbers. Two methods are described: listing the multiples and finding the smallest common one, and constructing the LCM by factorizing each number and taking the highest power of each prime factor. An example using each method is provided to find the LCM of different sets of numbers.
The document discusses finding the least common multiple (LCM) and least common denominator (LCD) of numbers. It provides examples of:
1) Finding the LCM of numbers by listing their common multiples or constructing it from their prime factorizations.
2) Using the LCM, also called the LCD, of denominators to divide a pizza evenly between people who each want a fractional amount.
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
1 5 multiplication and division of rational expressionsmath123b
The document discusses methods for finding the least common multiple (LCM) of numbers. It describes the searching method, which involves finding the smallest number that is a multiple of all the given numbers. An example finds the LCM of 18, 24, 16 to be 144. The document also introduces the construction method, which builds the minimum coverage needed to fulfill all requirements.
14 the lcm and the multiplier method for addition and subtraction of rational...elem-alg-sample
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required for each subject across different college requirements.
2 the least common multiple and clearing the denominatorsmath123b
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required across different college applications in each subject area.
This document discusses the key concepts from several units in mathematics including integers, groups, finite groups, subgroups, and groups in coding theory. It then provides details on specific topics within these units, including equivalence relations, congruence relations, equivalence class partitions, the division algorithm, greatest common divisors (GCD) using division, and Euclid's lemma. The document aims to provide students with fundamental mathematical principles, methods, and tools to model, solve, and interpret a variety of problems. It also discusses enhancing students' development, problem solving skills, communication, and attitude towards mathematics.
Common Multiples and Least Common MultipleBrooke Young
The document explains how to find the least common multiple (LCM) of two numbers. It defines key terms like product, multiple, and common multiple. It then provides examples of finding the LCM of 3 and 6, 9 and 12, and 4 and 6. For each example, it lists the multiples of each number, circles the common multiples, and identifies the smallest common multiple as the LCM. The overall process is to list multiples, identify common multiples, and select the smallest value from the common multiples as the LCM.
This document discusses different types of means - arithmetic mean, geometric mean, and harmonic mean. It provides definitions and formulas for calculating each type of mean between two numbers. It also presents examples of calculating means and solving word problems involving arithmetic progressions and geometric/harmonic progressions. The key information covered includes definitions of arithmetic, geometric, and harmonic means; formulas for calculating each mean; and examples of applying the concepts to word problems.
This document discusses different methods for finding the least common multiple (LCM) of two or more numbers. It explains that the LCM is the smallest number that the given numbers will all divide into evenly. Then it describes four methods: multiplication, prime factorization, Venn diagrams, and using the greatest common factor (GCF). For each method it provides an example of finding the LCM of 24 and 18, and the answer is consistently 72.
The document discusses methods for finding the least common multiple (LCM) of numbers. It describes the searching method, which involves finding the smallest number that is a multiple of all the given numbers. An example finds the LCM of 18, 24, 16 to be 144. The document also introduces the construction method, which builds the minimum coverage needed to fulfill all requirements.
The document introduces the concept of the least common multiple (LCM) and provides examples of finding the LCM of numbers using their prime factorizations or a Venn diagram. It explains that the LCM is the smallest number that is a multiple of both numbers. Students are given practice problems to find the LCM of number pairs and expressions using these methods.
The document discusses finding the least common multiple (LCM) of numbers. It provides examples of finding the LCM of pairs of numbers by listing their multiples and finding the smallest number that is a multiple of both. It also introduces finding the LCM using prime factorizations and Venn diagrams, showing how to find the greatest power of each prime factor and multiply them to get the LCM. Students are given practice problems to find the LCM of various number pairs using these methods.
This document discusses finding the least common multiple (LCM) of sets of numbers. It begins with examples of finding the LCM of various number pairs and sets. These examples illustrate finding the prime factors of each number and identifying the smallest number that is a multiple of all numbers as their LCM. The document then provides practice problems for readers to identify the LCM of additional number sets. It concludes by recapping that the LCM is the smallest number that is divisible by all numbers in the set.
Finds the common multiples and the least common demo teachrosalio baybayan jr
The document outlines a lesson plan on finding common multiples and least common multiples (LCM) using different methods. It explains key terms like LCM, common, and multiple. Examples are given to demonstrate finding the LCM through listing common multiples, factor trees, or continuous division. Practice problems are provided for students to find the LCM of
This document provides instruction on finding the least common multiple (LCM) of two or more numbers using different strategies. It explains that the LCM is the smallest number that is a multiple of both/all numbers. Three methods are described: 1) listing the multiples of each number and finding their common multiples, 2) using prime factorizations and factor trees, and 3) the Limon CAKE method which involves setting up the numbers in a cake-shape and multiplying the left and bottom parts. Examples of finding the LCM of various number pairs are provided and worked through step-by-step using the different methods.
This document provides instruction on finding the least common multiple (LCM) of two or more numbers using different strategies. It explains that the LCM is the smallest number that is a multiple of both/all numbers. Three methods are described: 1) listing the multiples of each number and finding their common multiples, 2) using prime factorization and factor trees, and 3) using the Lemon CAKE method which involves setting up a division problem to systematically find the LCM. Examples of each method are worked through.
What is HCF? What is LCM? How you calculate HCF & LCM
of numbers & fractions quickly?
Find out in this short presentation by https://allexammocktest.in
The document provides lesson details for identifying multiples and finding the least common multiple (LCM) of numbers using different methods like continuous division. It includes examples of finding the LCM of pairs of numbers through continuous division steps. Groups of students are assigned to find the LCM of number sets using the listing method or continuous division and present their solutions. The document reviews LCM concepts and methods through examples and assessments.
Adding Fractions With Unlike DenominatorsSarah Hallum
To add or subtract fractions with unlike denominators:
1. Find the least common multiple (LCM) of the denominators.
2. Write the fractions with this LCM as the new denominator by multiplying the numerators and denominators.
3. Add or subtract the new numerators and put over the common denominator.
4. Simplify the final fraction if possible by dividing the numerator and denominator by common factors.
This document discusses least common multiples (LCM) and greatest common factors (GCF). It provides examples of finding the LCM and GCF of various number pairs and sets. It also discusses applications of LCM and GCF in fractions, measurements, and problems involving multiples. Some key points include how to list multiples to find the first number common to both lists for LCM, and finding the largest number that is a factor of both numbers for GCF. Word problems demonstrate using LCM and GCF to determine things like lengths, times, or amounts that are exactly divisible.
The document discusses finding the least common multiple (LCM) of two numbers. It defines a multiple as the result of multiplying a number by the counting numbers. The LCM is the smallest number that is common between the lists of multiples of two given numbers. Examples are provided to demonstrate finding the LCM by listing out the multiples of each number and identifying the first number that appears in both lists.
The document contains instructions and examples for comparing fractions using the least common denominator (LCD) method. It explains how to find the LCD and write fractions with a common denominator to allow for direct comparison. Examples are provided for comparing proper, improper and mixed fractions. Students are given practice problems to order fractions from least to greatest.
The document discusses finding the least common multiple (LCM) of different numbers. It provides examples of finding the LCM of 4 and 6, 9 and 15, and 8, 10, and 40. For each example, it lists the multiples of the given numbers and identifies the highest number that is a common multiple of all numbers as their LCM.
1 5 multiplication and division of rational expressionsmath123b
The document discusses methods for finding the least common multiple (LCM) of numbers. It describes the searching method, which involves finding the smallest number that is a multiple of all the given numbers. An example finds the LCM of 18, 24, 16 to be 144. The document also introduces the construction method, which builds the minimum coverage needed to fulfill all requirements.
14 the lcm and the multiplier method for addition and subtraction of rational...elem-alg-sample
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required for each subject across different college requirements.
2 the least common multiple and clearing the denominatorsmath123b
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required across different college applications in each subject area.
This document discusses the key concepts from several units in mathematics including integers, groups, finite groups, subgroups, and groups in coding theory. It then provides details on specific topics within these units, including equivalence relations, congruence relations, equivalence class partitions, the division algorithm, greatest common divisors (GCD) using division, and Euclid's lemma. The document aims to provide students with fundamental mathematical principles, methods, and tools to model, solve, and interpret a variety of problems. It also discusses enhancing students' development, problem solving skills, communication, and attitude towards mathematics.
Common Multiples and Least Common MultipleBrooke Young
The document explains how to find the least common multiple (LCM) of two numbers. It defines key terms like product, multiple, and common multiple. It then provides examples of finding the LCM of 3 and 6, 9 and 12, and 4 and 6. For each example, it lists the multiples of each number, circles the common multiples, and identifies the smallest common multiple as the LCM. The overall process is to list multiples, identify common multiples, and select the smallest value from the common multiples as the LCM.
This document discusses different types of means - arithmetic mean, geometric mean, and harmonic mean. It provides definitions and formulas for calculating each type of mean between two numbers. It also presents examples of calculating means and solving word problems involving arithmetic progressions and geometric/harmonic progressions. The key information covered includes definitions of arithmetic, geometric, and harmonic means; formulas for calculating each mean; and examples of applying the concepts to word problems.
This document discusses different methods for finding the least common multiple (LCM) of two or more numbers. It explains that the LCM is the smallest number that the given numbers will all divide into evenly. Then it describes four methods: multiplication, prime factorization, Venn diagrams, and using the greatest common factor (GCF). For each method it provides an example of finding the LCM of 24 and 18, and the answer is consistently 72.
The document discusses methods for finding the least common multiple (LCM) of numbers. It describes the searching method, which involves finding the smallest number that is a multiple of all the given numbers. An example finds the LCM of 18, 24, 16 to be 144. The document also introduces the construction method, which builds the minimum coverage needed to fulfill all requirements.
The document introduces the concept of the least common multiple (LCM) and provides examples of finding the LCM of numbers using their prime factorizations or a Venn diagram. It explains that the LCM is the smallest number that is a multiple of both numbers. Students are given practice problems to find the LCM of number pairs and expressions using these methods.
The document discusses finding the least common multiple (LCM) of numbers. It provides examples of finding the LCM of pairs of numbers by listing their multiples and finding the smallest number that is a multiple of both. It also introduces finding the LCM using prime factorizations and Venn diagrams, showing how to find the greatest power of each prime factor and multiply them to get the LCM. Students are given practice problems to find the LCM of various number pairs using these methods.
This document discusses finding the least common multiple (LCM) of sets of numbers. It begins with examples of finding the LCM of various number pairs and sets. These examples illustrate finding the prime factors of each number and identifying the smallest number that is a multiple of all numbers as their LCM. The document then provides practice problems for readers to identify the LCM of additional number sets. It concludes by recapping that the LCM is the smallest number that is divisible by all numbers in the set.
Finds the common multiples and the least common demo teachrosalio baybayan jr
The document outlines a lesson plan on finding common multiples and least common multiples (LCM) using different methods. It explains key terms like LCM, common, and multiple. Examples are given to demonstrate finding the LCM through listing common multiples, factor trees, or continuous division. Practice problems are provided for students to find the LCM of
This document provides instruction on finding the least common multiple (LCM) of two or more numbers using different strategies. It explains that the LCM is the smallest number that is a multiple of both/all numbers. Three methods are described: 1) listing the multiples of each number and finding their common multiples, 2) using prime factorizations and factor trees, and 3) the Limon CAKE method which involves setting up the numbers in a cake-shape and multiplying the left and bottom parts. Examples of finding the LCM of various number pairs are provided and worked through step-by-step using the different methods.
This document provides instruction on finding the least common multiple (LCM) of two or more numbers using different strategies. It explains that the LCM is the smallest number that is a multiple of both/all numbers. Three methods are described: 1) listing the multiples of each number and finding their common multiples, 2) using prime factorization and factor trees, and 3) using the Lemon CAKE method which involves setting up a division problem to systematically find the LCM. Examples of each method are worked through.
What is HCF? What is LCM? How you calculate HCF & LCM
of numbers & fractions quickly?
Find out in this short presentation by https://allexammocktest.in
The document provides lesson details for identifying multiples and finding the least common multiple (LCM) of numbers using different methods like continuous division. It includes examples of finding the LCM of pairs of numbers through continuous division steps. Groups of students are assigned to find the LCM of number sets using the listing method or continuous division and present their solutions. The document reviews LCM concepts and methods through examples and assessments.
Adding Fractions With Unlike DenominatorsSarah Hallum
To add or subtract fractions with unlike denominators:
1. Find the least common multiple (LCM) of the denominators.
2. Write the fractions with this LCM as the new denominator by multiplying the numerators and denominators.
3. Add or subtract the new numerators and put over the common denominator.
4. Simplify the final fraction if possible by dividing the numerator and denominator by common factors.
This document discusses least common multiples (LCM) and greatest common factors (GCF). It provides examples of finding the LCM and GCF of various number pairs and sets. It also discusses applications of LCM and GCF in fractions, measurements, and problems involving multiples. Some key points include how to list multiples to find the first number common to both lists for LCM, and finding the largest number that is a factor of both numbers for GCF. Word problems demonstrate using LCM and GCF to determine things like lengths, times, or amounts that are exactly divisible.
The document discusses finding the least common multiple (LCM) of two numbers. It defines a multiple as the result of multiplying a number by the counting numbers. The LCM is the smallest number that is common between the lists of multiples of two given numbers. Examples are provided to demonstrate finding the LCM by listing out the multiples of each number and identifying the first number that appears in both lists.
The document contains instructions and examples for comparing fractions using the least common denominator (LCD) method. It explains how to find the LCD and write fractions with a common denominator to allow for direct comparison. Examples are provided for comparing proper, improper and mixed fractions. Students are given practice problems to order fractions from least to greatest.
The document discusses finding the least common multiple (LCM) of different numbers. It provides examples of finding the LCM of 4 and 6, 9 and 15, and 8, 10, and 40. For each example, it lists the multiples of the given numbers and identifies the highest number that is a common multiple of all numbers as their LCM.
The document discusses finding the least common multiple (LCM) of different numbers. It provides examples of finding the LCM of 4 and 6, 9 and 15, and 8, 10, and 40. For each example, it lists the multiples of the given numbers and identifies the highest number that is a common multiple of all numbers as their LCM.
This document contains a lesson on finding the least common multiple (LCM) and least common denominator (LCD) of numbers. It defines key terms like multiple, LCM, LCD and provides examples of using the list method and prime factorization to calculate the LCM and LCD of various number sets. Examples include finding the LCM of 6 and 4, the LCD of fractions with denominators of 4 and 2, and determining the minimum number of packs of hats and shirts needed for a boy scout troop.
A prime number is a whole number greater than 1 that is only divisible by 1 and itself. Examples of prime numbers are 2, 3, 5. A composite number is a whole number that has more than two factors. Examples of composite numbers are 4, 6, 9.
The divisibility rules for 3, 4, 6, 9, 10 are: a number is divisible by 3 if the sum of its digits is divisible by 3; a number is divisible by 4 if the last two digits are divisible by 4; a number is divisible by 6 if it is divisible by both 2 and 3; a number is divisible by 9 if the sum of its digits is divisible by 9; a number is divisible by 10 if
Study Guide in Greatest Common Factor.pptxKennedyTabon
This document discusses finding the greatest common factor (GCF) and least common multiple (LCM) of numbers. It defines GCF as the highest number that divides all given numbers and LCM as the least number that is a multiple of all given numbers. Examples are provided of finding GCF and LCM using listing factors/multiples and continuous division methods. Properties of GCF and LCM are outlined. Practice problems are given to find the GCF of number sets using any method.
Adding & Subtracting Fractions - Part 1DACCaanderson
This document discusses how to add and subtract fractions that have different denominators, or uncommon denominators. It explains that to add or subtract fractions, they must first be converted to equivalent fractions that have a common denominator. This allows the fractions to represent equal sizes of the whole. It introduces the concept of the least common denominator (LCD) as the smallest number that both denominators divide into. The LCD is used to determine the common denominator when adding or subtracting fractions with uncommon denominators. Examples are provided to demonstrate finding the LCD and converting fractions to equivalent fractions with a common denominator.
The document discusses the concept of the least common multiple (LCM). It defines the LCM as the lowest number that is a multiple of two or more numbers. It provides examples of finding the LCM of different pairs of numbers by listing their multiples and circling the first number that is common to both lists. The document also discusses how the LCM can be used to find patterns involving multiples and to add or subtract fractions by finding a common denominator.
Similar to 15 addition and subtraction of fractions (20)
This document contains a sample algebra test with 24 multiple choice questions and solutions. Each question is followed by an answer option and a link to the solution. The solutions provide step-by-step workings to arrive at the correct answer. The test covers topics such as fractions, operations with fractions, percentages, word problems involving percentages, and solving simple equations. An interactive PowerPoint file is also included but viewing the slides is not interactive when using Slideshare.
The rectangular coordinate system represents points in a plane using perpendicular axes (x-axis and y-axis) that intersect at the origin (0,0). Each point is assigned an ordered pair (x,y) where x is the distance from the origin on the x-axis and y is the distance from the origin on the y-axis. The system divides the plane into four quadrants based on whether the x and y values are positive or negative. The rectangular coordinate system allows any point in the plane to be uniquely addressed using its x and y coordinates.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to the right of 0 and negative numbers to the left. Any number to the right of another number on the number line is defined as greater than the number to its left. Intervals on the number line, denoted by a < x < b, represent all the numbers between a and b. Examples are provided to illustrate drawing intervals on the number line.
The document discusses right triangles and the Pythagorean theorem. It defines a right triangle as one with a 90-degree angle, and labels the sides as the hypotenuse (the side opposite the right angle) and the two legs. It presents the Pythagorean theorem, which states that for any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. The document provides an example of using the theorem to calculate the height of a wall given the length of a leaning ladder. It also defines the square root and how it relates to finding the length of a side of a triangle based on the Pythagorean theorem.
The document discusses basic geometric shapes and formulas for calculating their perimeters. It defines a loop or polygon as a shape formed by connecting line segments end to end. Triangles have three sides and their perimeter is calculated as the sum of the three side lengths. Specific types of triangles like equilateral triangles are discussed. Rectangles are four-sided polygons with right angles, and squares are rectangles with four equal sides. Formulas are provided for calculating the perimeters of squares and rectangles based on their side lengths. Some example problems demonstrate applying these concepts and formulas to calculate perimeters of fenced or roped areas composed of multiple shapes.
The document discusses ratios, proportions, and how to solve proportional equations. It defines a ratio as two related quantities stated side by side, and gives the example of a 3:4 ratio of eggs to flour in a recipe. Proportions are defined as equal ratios. The key steps to solve proportional equations are: 1) write the ratios as fractions set equal to each other, 2) use cross-multiplication to convert the proportions into regular equations, and 3) solve the resulting equation using algebraic techniques. An example problem demonstrates these steps to solve a proportional equation for the variable x.
The document discusses exponents and rules for working with them. It defines exponents as the number of times a base is used as a factor in a repetitive multiplication. The main rules covered are:
- The multiply-add rule, which states that ANAK = AN+K
- The divide-subtract rule, which states that AN/AK = AN-K
Examples are provided to demonstrate calculating exponents and applying the rules.
The document discusses solving linear equations using examples of ordering pizzas. It explains that a linear equation contains linear expressions on both sides, such as 3x + 10 = 34, and can be solved by manipulating the equation through steps like subtraction to find the value of x that makes both sides equal. For example, in the equation 3x + 10 = 34, subtracting 10 from both sides and dividing both sides by 3 reveals that x = 8 is the solution.
The document discusses mathematical expressions and how to combine them. It defines an expression as a calculation procedure written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms are combined in the same way numbers are, while unlike terms cannot be combined. The simplest expressions are linear expressions of the form ax + b.
34 conversion between decimals, fractions and percentagesalg-ready-review
The document discusses the conversion between decimals, fractions, and percentages. It states that fractions, decimals, and percentages are different ways to express quantities. Fractions provide instructions to divide a whole into parts, while decimals standardize fractions to powers of 10 to make addition and subtraction easier. Percentages express a quantity as a ratio out of 100. The document then provides examples and steps for converting between these representations.
The document defines percentages as expressing "how many out of 100" and can be written as a fraction with the percentage symbol or as a decimal. It provides examples of common percentages like 1% = 1/100, 5% = 1/20, 10% = 1/10, 25% = 1/4, and 50% = 1/2. The document also works through examples of calculating percentages of a total amount, like finding 3/4 of $100 is $75, or that 45% of 60 pieces of candy is 27 pieces. Finally, it lists some important percentages that relate to coins: 5% = 1/20 for nickels, 10% = 1/10 for dimes, and 25%
The document discusses how to multiply multi-digit decimal numbers. It explains that multiplication of multi-digit numbers is done by multiplying the digits in place value, starting from the ones place. The results are recorded and carried over as needed. It provides a step-by-step example of 47 x 6, showing how each digit is multiplied and the results carried to the next place value. It notes that the same process is followed for decimals, but the decimal point is placed in the final product so that the total number of decimal places is correct.
31 decimals, addition and subtraction of decimalsalg-ready-review
This document introduces decimals by using an analogy of a cash register holding coins of various values. It explains that decimals allow tracking of smaller quantities by including base-10 fractions in the number system. It assumes the US Treasury makes fictional smaller value coins like "itties" and "bitties", then demonstrates writing decimal numbers as representations of coins in different slots of a cash register. Finally, it provides steps for comparing decimal values by lining up numbers at the decimal point and determining the largest by the digits from left to right.
The document discusses variables, expressions, and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values can change depending on the situation. Expressions are made using variables and mathematical operations, and evaluation involves replacing the variables in an expression with input values and calculating the output. The input values replace the variables within parentheses, and the process of evaluation finds the output. Several examples are provided to demonstrate evaluating different expressions by replacing variables with given input values.
The document discusses order of operations and how to correctly evaluate mathematical expressions. It provides examples of evaluating expressions involving multiplication, division, addition, subtraction, grouping symbols and exponents. The key steps are to perform operations within grouping symbols from the innermost out, then multiplication and division from left to right, followed by addition and subtraction from left to right. Setting clear rules for order of operations ensures the correct solution is obtained.
23 multiplication and division of signed numbersalg-ready-review
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use the following rules for the sign of the product: two numbers with the same sign yield a positive product; two numbers with opposite signs yield a negative product. It also discusses how multiplication is implied in algebra without an explicit operation symbol between terms.
The document discusses the Pythagorean theorem and square roots. It defines a right triangle as having one 90 degree angle. The Pythagorean theorem states that for a right triangle with sides a, b, c, where c is the hypotenuse opposite the right angle, a^2 + b^2 = c^2. An example uses the theorem to calculate the height of a wall given the length of a ladder leaning against it. Square roots are then introduced, with the square root of a number x defined as the positive number that produces x when squared.
The document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. This is done by writing the fractions as ratios, then multiplying the denominators diagonally to obtain two new numbers. The ratio between these new numbers represents the original fractional ratio using whole numbers. An example demonstrates taking a ratio of 3/4 cups sugar to 2/3 cups flour and rewriting it as 9:8 cups sugar to flour using cross multiplication. The document also notes cross multiplication can be used to compare two fractions, with the larger product corresponding to the larger fraction.
The document discusses rules for multiplying fractions. It states that to multiply fractions, one should multiply the numerators and multiply the denominators, canceling terms when possible. It then provides examples, such as multiplying 12/25 * 15/8, simplifying to 9/10. It also notes that word problems involving fractions of a quantity can often be solved by translating them into fraction multiplications.
The document defines fractions as numbers of the form p/q where p and q are whole numbers not equal to 0. Fractions represent parts of a whole, for example 3/6 of a pizza. The top number is the numerator and represents the number of parts, while the bottom number is the denominator and represents the total number of equal parts the whole was divided into. Calculations with fractions involve dividing the whole into the number of parts in the denominator and taking the number of parts indicated by the numerator. Whole numbers can be viewed as having a denominator of 1. Dividing by 0 is undefined in mathematics.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
2. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
LCM and LCD
3. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
LCM and LCD
4. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
LCM and LCD
5. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
LCM and LCD
6. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12,
LCM and LCD
7. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD
8. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD
We may improve the above listing-method for finding the LCM.
9. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
10. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
list its multiples in order until we find the one that can divide all
the other numbers.
11. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD
Example B. Find the LCM of 8, 9, and 12.
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
list its multiples in order until we find the one that can divide all
the other numbers.
12. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD
Example B. Find the LCM of 8, 9, and 12.
The largest number is 12 and the multiples of 12 are 12, 24,
36, 48, 60, 72, 84 …
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
list its multiples in order until we find the one that can divide all
the other numbers.
13. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD
Example B. Find the LCM of 8, 9, and 12.
The largest number is 12 and the multiples of 12 are 12, 24,
36, 48, 60, 72, 84 … The first number that is also a multiple
of 8 and 9 is 72.
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
list its multiples in order until we find the one that can divide all
the other numbers.
14. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD
Example B. Find the LCM of 8, 9, and 12.
The largest number is 12 and the multiples of 12 are 12, 24,
36, 48, 60, 72, 84 … The first number that is also a multiple
of 8 and 9 is 72. Hence LCM{8, 9, 12} = 72.
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
list its multiples in order until we find the one that can divide all
the other numbers.
15. But when the LCM is large, the listing method is cumbersome.
LCM and LCD
16. But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
17. To construct the LCM:
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
18. To construct the LCM:
a. Factor each number completely
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
19. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
20. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
21. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
22. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
23. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
24. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
25. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
26. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor:
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
27. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor:
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
28. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
29. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23, 32, 5,
30. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.
31. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.
32. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
The LCM of the denominators of a list of fractions is called the
least common denominator (LCD).
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.
33. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM and LCD
The LCM of the denominators of a list of fractions is called the
least common denominator (LCD). Following is an
application of the LCM.
34. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6.
LCM and LCD
35. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6.
LCM and LCD
Mary Chuck
In picture:
Joe
36. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
LCM and LCD
Mary Chuck
In picture:
Joe
37. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
LCM and LCD
Mary Chuck
In picture:
Joe
38. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching.
Mary Chuck
In picture:
Joe
39. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, …
Mary Chuck
In picture:
Joe
40. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM.
Mary Chuck
In picture:
Joe
41. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices
Mary Chuck
In picture:
Joe
42. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
Joe gets 12*
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3
43. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
Joe gets 12* = 4 slices
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3
44. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
Joe gets 12* = 4 slices 1
4
Mary gets 12*
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3
45. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3
46. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12*
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3
47. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12* = 2 slices
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3
48. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12* = 2 slices
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3
In total, that is 4 + 2 + 3 = 9 slices,
49. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12* = 2 slices
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3
In total, that is 4 + 2 + 3 = 9 slices, or of the pizza.
9
12
50. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12* = 2 slices
LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3
In total, that is 4 + 2 + 3 = 9 slices, or of the pizza.
9
12
=
3
4
51. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
52. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12
53. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
54. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
55. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
56. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
9
16
57. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
The new denominator is 48,
9
16
58. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
The new denominator is 48, then the new numerator is
48*
9
16
9
16
59. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
The new denominator is 48, then the new numerator is
48*
9
16
9
16
3
60. Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
The new denominator is 48, then the new numerator is
48* = 27 so =
9
16
9
16
3
9
16
27
48 .
61. Suppose a pizza is cut into 4 equal slices
Addition and Subtraction of Fractions
62. Suppose a pizza is cut into 4 equal slices
Addition and Subtraction of Fractions
63. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza,
Addition and Subtraction of Fractions
1
4
64. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza,
Addition and Subtraction of Fractions
1
4
2
4
65. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza, altogether they take
+
Addition and Subtraction of Fractions
1
4
2
4
1
4
2
4
66. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza, altogether they take
+
Addition and Subtraction of Fractions
1
4
2
4
=
3
4
of the entire pizza.
1
4
2
4
67. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza, altogether they take
+
Addition and Subtraction of Fractions
1
4
2
4
=
3
4
of the entire pizza. In picture:
+ =
1
4
2
4
3
4
68. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza, altogether they take
+
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Same
Denominator
1
4
2
4
=
3
4
of the entire pizza. In picture:
+ =
1
4
2
4
3
4
69. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza, altogether they take
+
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Same
Denominator
To add or subtract fractions of the same denominator, keep
the same denominator, add or subtract the numerators
1
4
2
4
=
3
4
of the entire pizza. In picture:
+ =
1
4
2
4
3
4
70. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza, altogether they take
+
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Same
Denominator
To add or subtract fractions of the same denominator, keep
the same denominator, add or subtract the numerators
1
4
2
4
=
3
4
of the entire pizza. In picture:
±
a
d
b
d = a ± b
d
+ =
1
4
2
4
3
4
71. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza, altogether they take
+
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Same
Denominator
To add or subtract fractions of the same denominator, keep
the same denominator, add or subtract the numerators
,then simplify the result.
1
4
2
4
=
3
4
of the entire pizza. In picture:
±
a
d
b
d = a ± b
d
+ =
1
4
2
4
3
4
81. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match.
82. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
1
2
83. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
+
1
2
1
3
84. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
+
1
2
1
3
=
?
?
85. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
+
1
2
1
3
=
?
?
86. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza into 6 slices.
+
1
2
1
3
=
?
?
87. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza into 6 slices. Both fractions may be
converted to have the denominator 6.
+
1
2
1
3
=
?
?
88. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza into 6 slices. Both fractions may be
converted to have the denominator 6. Specifically,
1
2
= 3
6
1
3
= 2
6
+
1
2
1
3
=
?
?
89. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza into 6 slices. Both fractions may be
converted to have the denominator 6. Specifically,
1
2
= 3
6
1
3
= 2
6
+
1
2
1
3
=
?
?
90. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza into 6 slices. Both fractions may be
converted to have the denominator 6. Specifically,
+
1
2
= 3
6
1
3
= 2
6
3
6
1
3
=
?
?
91. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza into 6 slices. Both fractions may be
converted to have the denominator 6. Specifically,
+
1
2
= 3
6
1
3
= 2
6
3
6
1
3
=
?
?
92. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza into 6 slices. Both fractions may be
converted to have the denominator 6. Specifically,
+
1
2
= 3
6
1
3
= 2
6
3
6
2
6
=
?
?
93. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza into 6 slices. Both fractions may be
converted to have the denominator 6. Specifically,
+
3
6
2
6
=
1
2
= 3
6
1
3
= 2
6
Hence, 1
2
+ 1
3
= 3
6
+ 2
6
= 5
6
5
6
94. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
Addition and Subtraction of Fractions
95. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator.
Addition and Subtraction of Fractions
96. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
97. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
98. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD
99. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD
2. Convert all the different-denominator-fractions to the have
the LCD as the denominator.
100. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD
2. Convert all the different-denominator-fractions to the have
the LCD as the denominator.
3. Add and subtract the adjusted fractions then simplify the
result.
101. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD
2. Convert all the different-denominator-fractions to the have
the LCD as the denominator.
3. Add and subtract the adjusted fractions then simplify the
result.
Example B:
5
6
3
8
+a.
102. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD
2. Convert all the different-denominator-fractions to the have
the LCD as the denominator.
3. Add and subtract the adjusted fractions then simplify the
result.
Example B:
5
6
3
8
+a.
Step 1: To find the LCD, list the multiples of 8
103. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD
2. Convert all the different-denominator-fractions to the have
the LCD as the denominator.
3. Add and subtract the adjusted fractions then simplify the
result.
Example B:
5
6
3
8
+a.
Step 1: To find the LCD, list the multiples of 8 which are
8, 16, 24, ..
104. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD
2. Convert all the different-denominator-fractions to the have
the LCD as the denominator.
3. Add and subtract the adjusted fractions then simplify the
result.
Example B:
5
6
3
8
+a.
Step 1: To find the LCD, list the multiples of 8 which are
8, 16, 24, .. we see that the LCD is 24.
105. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
106. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20,
5
6
5
6
107. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
108. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9,
3
8
3
8
109. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
110. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
111. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
112. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
113. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ –
16
9
114. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.–
16
9
115. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.–
16
9
Convert:
116. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 *
117. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
=
28
48
118. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
=
28
48
= 30
5
8
48 *
119. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
=
28
48
= 30
5
8
48 * so
5
8
=
30
48
120. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
=
28
48
= 30
5
8
48 * so
5
8
=
30
48
= 27
9
16
48 *
121. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
=
28
48
= 30
5
8
48 * so
5
8
=
30
48
= 27
9
16
48 * so
9
16
=
27
48
122. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28=
–
16
9
Convert:
7
12
48 * so
7
12
=
28
48
= 30
5
8
48 * so
5
8
=
30
48
= 27
9
16
48 * so
9
16
=
27
48
28
48
+
30
48
–
27
48
123. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28=
–
16
9
48
28 + 30 – 27
Convert:
7
12
48 * so
7
12
=
28
48
= 30
5
8
48 * so
5
8
=
30
48
= 27
9
16
48 * so
9
16
=
27
48
28
48
+
30
48
–
27
48
=
124. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28=
–
16
9
48
=
28 + 30 – 27
Convert:
7
12
48 * so
7
12
=
28
48
= 30
5
8
48 * so
5
8
=
30
48
= 27
9
16
48 * so
9
16
=
27
48
28
48
+
30
48
–
27
48
=
31
48
125. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28=
–
16
9
48
=
28 + 30 – 27
Convert:
7
12
48 * so
7
12
=
28
48
= 30
5
8
48 * so
5
8
=
30
48
= 27
9
16
48 * so
9
16
=
27
48
28
48
+
30
48
–
27
48
=
31
48
126. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
Addition and Subtraction of Fractions
127. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
Addition and Subtraction of Fractions
128. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5
Addition and Subtraction of Fractions
129. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2,
Addition and Subtraction of Fractions
130. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 =
Addition and Subtraction of Fractions
131. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
132. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
133. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
134. Example C. a.
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
135. Example C. a.
The LCD is 24.
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
136. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
137. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
138. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
4
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
139. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
4 3
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
140. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24 = (4*5 + 3*3) / 24
4 3
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
141. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24 = (4*5 + 3*3) / 24 = 29/24 =
4 3 29
24
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
143. The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
144. ( ) * 48 / 48
7
12
5
8+ – 16
9
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
145. ( ) * 48 / 48
7
12
5
8+ – 16
94
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
146. ( ) * 48 / 48
67
12
5
8+ – 16
94
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
147. ( ) * 48 / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
148. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
149. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
150. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
48
31
151. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
48
31
152. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8+ –
16
9
48
31
153. LCM and LCD
Exercise A. Find the LCM.
1. a.{6, 8} b. {6, 9} c. {3, 4}
d. {4, 10}
2. a.{5, 6, 8} b. {4, 6, 9} c. {3, 4, 5}
d. {4, 6, 10}
3. a.{6, 8, 9} b. {6, 9, 10} c. {4, 9, 10}
d. {6, 8, 10}
4. a.{4, 8, 15} b. {8, 9, 12} c. {6, 9, 15}
5. a.{6, 8, 15} b. {8, 9, 15} c. {6, 9, 16}
6. a.{8, 12, 15} b. { 9, 12, 15} c. { 9, 12, 16}
7. a.{8, 12, 18} b. {8, 12, 20} c. { 12, 15, 16}
8. a.{8, 12, 15, 18} b. {8, 12, 16, 20}
9. a.{8, 15, 18, 20} b. {9, 16, 20, 24}
154. B. Convert the fractions to fractions with the given
denominators.
10. Convert to denominator 12.
11. Convert to denominator 24.
12. Convert to denominator 36.
13. Convert to denominator 60.
2
3 ,
3
4 ,
5
6 ,
7
4
1
6 ,
3
4 ,
5
6 ,
3
8
7
12 ,
5
4 ,
8
9 ,
11
6
9
10 ,
7
12 ,
13
5 ,
11
15
LCM and LCD