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- 1. MATH: COMPUTATIONAL SKILLS Section I DePaul Math Placement Test
- 2. <ul><li>There are different types of numbering systems that you must be familiar with in order to understand different properties of mathematical functions </li></ul><ul><li>Whole Numbers : the set of counting numbers, including zero {0, 1, 2, 3, ..} . </li></ul><ul><li>Natural Numbers : the set of all whole numbers except zero {1, 2, 3, 4, . . .}. </li></ul><ul><li>Integers : the set of all positive and negative whole numbers, including zero. </li></ul><ul><li>Fractions and decimals are not included {. . . , –3, –2, –1, 0, 1, 2, 3, . . .}. </li></ul><ul><li>Rational Numbers : the set of all numbers that can be expressed as a </li></ul><ul><li>quotient of integers. Any set of rational numbers includes all integers, and </li></ul><ul><li>all fractions that contain integers in the numerator and denominator(m/n). </li></ul><ul><li>Real Numbers : every number on the number line. The set of real numbers </li></ul><ul><li>includes all rational and irrational numbers. </li></ul>Different Types of Numbers
- 3. <ul><li>The following rules define how positive and negative numbers operate under various operations. </li></ul><ul><li>Addition and Subtraction : Positive and negative numbers </li></ul><ul><li>Multiplication : </li></ul><ul><li>positive * positive = positive ( Example: 2*3=6) </li></ul><ul><li>negative * negative = positive ( Example: - 2*-3=6) </li></ul><ul><li>positive * negative = negative ( Example: 4*-5=-20) </li></ul><ul><li>negative * positive = negative ( Example: -4*2=-8) </li></ul><ul><li>Division : </li></ul><ul><li>positive / positive = positive ( Example: 9/3=3) </li></ul><ul><li>negative / negative = positive ( Example: -12/-3=4 ) </li></ul><ul><li>positive / negative = negative ( Example: 6/-3=-2) </li></ul><ul><li>negative / positive = negative ( Example: -6/3=-2) </li></ul><ul><li>The rules for multiplication and division are exactly the same since any division operation can be written as a form of multiplication: a ÷b = a / b = a X 1 / b . </li></ul>Operations of Positive and Negative Numbers
- 4. The most basic form of mathematical expressions involving several mathematical operations can only be solved by using the order of PEMDAS . This catchy acronym stands for: Parentheses : first, perform the operations in the innermost parentheses. A set of parentheses supercedes any other operation. Exponents : before you do any other operation, raise all the required bases to the prescribed exponent. Exponents include roots, since root operations are the equivalent of raising a base to the 1 / n, where ‘n’ is any integer. Multiplication and Division : perform multiplication and division. Addition and Subtraction : perform addition and subtraction. Order of Operations
- 5. Here are two examples illustrating the usage of PEMDAS . Let’s work through a few examples to see how order of operations and PEMDAS work. 3 X 2 3 + 6÷4. Since nothing is enclosed in parentheses, the first operation we carry out is exponentiation: 3 X 2 3 + 6÷4 = 38+6÷4 Next, we do all the necessary multiplication and division: 3 X 8+6÷4 = 24÷1.5 Lastly, we perform the required addition and subtraction. Our final answer is: 24÷1.5= 25.5 Here a few question to try yourself. Evaluate : 6 (2 3 2(5-3)) Hint: Start solving from the innermost parenthesis first (Final Answer is 12). Here’s another question. Evaluate: 5-2 2 ⁄6+4. Hint: Solve the numerator and denominator separately. Order of Operations: Examples
- 6. <ul><li>An exponent defines the number of times a number is to be multiplied by itself. For example, in a b , where a is the base and b the exponent, a is multiplied by itself b times. In a numerical example, 2 5 = 2 x 2 x 2 x 2 x 2 . An exponent can also be referred to as a power. </li></ul><ul><li>The following are other terms related to exponents with which you should be familiar: </li></ul><ul><ul><ul><li>Base . The base refers to the 3 in 3 5 . It is the number that is being multiplied by itself however many times specified by the exponent. </li></ul></ul></ul><ul><ul><ul><li>Exponent . The exponent is the 5 in 3 5 . It indicates the number of times the base is to be multiplied with itself. </li></ul></ul></ul><ul><ul><ul><li>Square . Saying that a number is squared means that the number has been raised to the second power, i.e., that it has an exponent of 2. In the expression 6 2 , 6 has been squared. </li></ul></ul></ul><ul><ul><ul><li>Cube . Saying that a number is cubed means that it has been raised to the third power, i.e., that it has an exponent of 3. In the expression 4 3 , 4 has been cubed. </li></ul></ul></ul><ul><ul><ul><ul><li>Exponents </li></ul></ul></ul></ul>
- 7. In order to add or subtract numbers with exponents, you must first find the value of each power, then add the two numbers . For example, to add 3 3 + 4 2 , you must expand the exponents to get (3x 3 x 3) + (4 x 4), and then, 27 + 16 = 43. However, algebraic expressions that have the same bases and exponents, such a s 3 x 4 and 5 x 4 , c an be added and subtracted. For exampl e, 3 x 4 + 5 x 4 = 8 x 4 . Adding and Subtracting Numbers with Exponents
- 8. To multiply exponential numbers raised to the same exponen t , raise their product to that exponent: a n x b n = (ax b) n = (ab) n 4 3 x 5 3 =(4x5) 3 = 20 3 To divide exponential numbers raised to the same exponen t , raise their quotient to that exponent: a n /b n = (a/b) n 12 5 / 3 5 = (12/3) 5 = 4 3 To multiply exponential numbers or terms that have the same base , add the exponents together: a m b n = (ab) (m+n) 3 6 x3 2 = 3 (6+2) = 3 8 To divide two same-base exponential numbers or terms , just subtract the exponents: a m /b n = (a/b) (m-n) 3 6 /3 2 = 3 (6-2) = 3 4 When an exponent is raised to another exponent in cases, (3 2 ) 4 and (x 4 ) 3 . In such cases, multiply the exponents: (a m ) n = a (mn) (3 2 ) 4 = 3 (2x 4) = 3 8 Multiplying and Dividing Numbers with Exponents
- 9. <ul><li>To summarize what was said earlier, when you multiply to negative numbers you get a positive number. And, when you multiply a negative number by a positive number, you get a negative number. These rules affect how negative numbers function in reference to exponents. </li></ul><ul><li>When you raise a negative number to an even number exponent , you get a positive number. </li></ul><ul><ul><li>For example, (–2) 4 = 16. </li></ul></ul><ul><ul><li>Let’s break it down. (-2) 4 means, –2 X –2 X –2 X–2. When you multiply the first two –2’s together, you get 4 because you are multiplying two negative numbers. Then, when you multiply the 4 by the next –2, you get –8 , since you are multiplying a positive number by a negative number. Finally, you multiply the –8 by the last –2 and get 16, since you’re once again multiplying two negative numbers. </li></ul></ul><ul><li>When you raise a negative number to an odd power , you get a negative number. To see why, refer to the example above and stop the process at –8, which equals (–2) 3 . </li></ul>Negative Numbers and Exponents
- 10. <ul><li>There are a few special properties of certain exponents that you also should know. </li></ul><ul><li> </li></ul><ul><ul><li>Zero </li></ul></ul><ul><ul><li>Any base raised to the power of zero is equal to 1. If you see any exponent of the form x 0 , you should know that its value is 1. Note, however, that 0 0 is undefined. </li></ul></ul><ul><li> </li></ul><ul><ul><li>One </li></ul></ul><ul><ul><li>Any base raised to the power of one is equal to itself. For example , 2 1 = 2, (–67) 1 = –67, and x 1 = x. This can be helpful when you’re attempting an operation on exponential terms with the same base. </li></ul></ul><ul><ul><li>For example: </li></ul></ul><ul><ul><li> (3x 6 ) X (x 4 )= 3x (6+4) = 3x 7 </li></ul></ul>Special Exponents
- 11. Fractional Exponents Exponents can be fractions, too. When a number or term is raised to a fractional power, it is called taking the root of that number or term. This expression can be converted into a more convenient form: x (a/b) = b For example, 2 13 ⁄ 5 is equal to the fifth root of 2 to the thirteenth power: 13 = 6.063 The symbol is also known as the radical, and anything under the radical (in this case 2 13 ) is called the radicand. For a more familiar example, look at 9 1⁄2 , which is the same as : 1 = = 2
- 12. Negative Exponents Seeing a negative number as a power may be a little strange the first time around. But the principle at work is simple. Any number or term raised to a negative power is equal to the reciprocal of that base raised to the opposite power. For example: x -5 = (1/ x 5 ) Or a slightly more complicated example: (2/3) -3 =(1/(2/3)) -3 = (3/2) 3 = 27/8 Now You’ve got the four rules of special exponents. Here are some examples to firm up your knowledge: x (1/8) = 1 = 4 2/3 x 4 8/5 = 4 (2/3 +8/5) = 4 (34/15) = 34 (3 -2 ) x = 3 -2 x = 1/3 2x 3(xy) 0 = 3 b -1 = 1/b x - 2/3 x z -2/3 = (xz) -2/3 = 1/( 2 ) 1 2x/3w4 = 1 ( one raised to any power is still one)
- 13. Roots and Radicals We just saw that roots express fractional exponents. But it is often easier to work with roots in a different format. When a number or term is raised to a fractional power, the expression can be converted into one involving a root in the following way: x 5/3 = 5 with the √sign as the radical sign and x a as the radicand . Roots are like exponents, only backward. For example, to square the number 3 is to multiple 3 by itself two times: 3 2 = 3 x 3 = 9. The root of 9, is 3. In other words, the square root of a number is the number that, when squared, is equal to the given number.
- 14. Square Roots and Cube Roots Square roots are the most commonly used roots, but there are also Cube roots (numbers raised to 1 / 3 ), fourth roots, fifth roots, and so on. Each root is represented by a radical sign with the appropriate number next to it (a radical without any superscript denotes a square root). cube roots are shown a s and fourth roots as . These roots of higher degrees operate the same way square roots do. Because 3 3 = 27, it follows that the cube root of 27 is 3. Here are a few examples: = 4 because 4 2 =16 = ½ because (1/2) 2 = ¼ I f x n = y, then = x The same rules that apply to multiplying and dividing exponential terms with the same exponent apply to roots as well. Consider these examples: X = = 4 X = Just be sure that the roots are of the same degree (i.e., you are multiplying or dividing all square roots or all roots of the fifth power).
- 15. <ul><li>Scientific notation is a convention used to express large numbers. A number written in scientific notation has two parts: </li></ul><ul><li>A number between 1 and 10. </li></ul><ul><li>The power of 10 by which you must multiply the first number in order to obtain the large number that is being represented. </li></ul><ul><li>The following examples express numbers in scientific notation: </li></ul><ul><li>3,000,000 = 3.0 x 10 6 </li></ul><ul><li> 4,123,273,823 = 4.123273823 x 10 9 </li></ul><ul><li>Finding the product would be pretty nasty—even if you’re using a calculator. Approximating each number using scientific notation makes the problem a lot easier: </li></ul><ul><li>13, 234, 836, 436 X 555, 317, 897, 542, 222, 010 1.32 X10 13 X 5.55 X 10 17 </li></ul><ul><li> = (1.32 X5.55) X(10 13 X 10 17 ) = 7.326 X 10 30 </li></ul><ul><li>Also, note the way in which we combined the terms in the last example to make the multiplication a little simpler: </li></ul><ul><li>1.32 x 10 13 x 5.55 x 10 17 = (1.32 x 5.55) x (10 13 x 10 17 ) </li></ul>Scientific Notation
- 16. <ul><li>A fraction describes a part of a whole. It is composed of two expressions, a numerator and a denominator. The numerator of a fraction is the quantity above the fraction bar, and the denominator is the quantity below the fraction bar. For example, in the fraction 1 / 2 , 1 is the numerator and 2 is the denominator. </li></ul><ul><li> </li></ul><ul><li>Equivalent Fractions </li></ul><ul><li>Two fractions are equivalent if they describe equal parts of the same whole. To determine if two fractions are equivalent, multiply the denominator and numerator of one fraction so that the denominators of the two fractions are equal. For example, 1 / 2 = 3 / 6 because if you multiply the numerator and denominator of 1 / 2 by 3, you get </li></ul><ul><li> 1 x 3 = 3 2 x 3 6 </li></ul><ul><li> As long as you multiply or divide both the numerator and denominator of a fraction by the same nonzero number, you will not change the overall value of the fraction. Fractions represent a part of a whole, so if you increase both the part and whole by the same multiple, you will not change their fundamental relationship. </li></ul>Fractions
- 17. Reducing fractions makes life with fractions much simpler. It makes unwieldy fractions, such as 450 / 600 , smaller and easier to work with . To reduce a fraction to lowest terms, divide the numerator and denominator by their greatest common factor. For example, for 450 / 600 , the GCF of 450 and 600 i s 150. The fraction reduces to 3 / 4 . A fraction is in reduced form if its numerator and denominator are relatively prime (their GCF is 1). Therefore, it makes sense that the equivalent fractions we studied in the previous section all reduce to the same fraction. For example , the equivalent fractions 4 / 6 and 8 / 12 both reduce to 2 / 3 . Reducing Fractions
- 18. When dealing with integers, large positive numbers with a lot of digits, like 5,000,000 , are greater than numbers with fewer digits, such as 5. 200 / 20,000 might seem like an impressive fraction, bu t 2 / 3 is actually larger because 2 is a much bigger part o f 3 than 200 is of 20,000. Cross-multiplication: While dealing with two fractions that have different numerators and denominators, such as 200 / 20,000 and 2 / 3 ., a n easy way to compare these two fractions is to use cross multiplication. Simply multiply the numerator of each fraction by the denominator of the other. then write the product of each multiplication next to the numerator you used to get it. We’ll cross-multiply 200 / 20,000 and 2 / 3 : 600 = 200 2 = 40,000 20,000 3 Since 40,000 > 600, 2 / 3 is the greater fraction. Comparing Fractions
- 19. On the math placement test you will need to know how to add and subtract two different types of fractions. The fractions will either have the same or different denominators. Fractions with the Same Denominators Fractions are extremely easy to add and subtract if they have the same denominator. In addition problems, all you have to do is add up the numerators: 1 + 3 + 13 = 17 20 20 20 20 Subtraction works similarly. If the denominators of the fractions are equal, then you simply subtract one numerator from the other: 13 – 2 = 11 20 20 20 Adding and Subtracting Fractions
- 20. <ul><li>If the fractions do not have equal denominators , the process of manipulating them becomes somewhat difficult. </li></ul><ul><li>To make the denominators the same and then to subtract as described above. The best way to do this is to find the least common denominator (LCD), which is simply the least common multiple of the two denominators. For example, the LCD of 1 / 2 and 2 / 3 i s 6, s ince 6 is the LCM o f 2 and 3. </li></ul><ul><li>After you’ve equalized the denominators of the two fractions, is to multiply each numerator by the same value as their respective denominator. Let’s take a look at how to do this for our example , 1 / 2 + 2 / 3 . </li></ul><ul><li>For 1 / 2 : </li></ul><ul><li>Numerator = 1 x 3 = 3 </li></ul><ul><li>Denominator = 2 x 3 = 6 </li></ul><ul><li>So, the new fraction is 3 / 6 . The same process is repeated for the second fraction, 2 / 3 : </li></ul><ul><ul><li>Numerator = 2 x 2 = 4 </li></ul></ul><ul><ul><li> Denominator = 3 x 2 = 6 </li></ul></ul><ul><ul><li>The new fraction is 4 / 6 . The final step is to perform the addition or subtraction. In this case , 3 / 6 + 4 / 6 = 7 / 6 . </li></ul></ul><ul><li> </li></ul><ul><li> </li></ul>Fractions with Different Denominators
- 21. <ul><li>Multiplying Fractions </li></ul><ul><li>Multiplying fractions is quite simple. The product of two fractions is the product of their numerators over the product of their denominators. This can be represented as: </li></ul><ul><li> a x c = ac b d bd </li></ul><ul><li>Here’s a numerical example, 3 x 2 = 3 x 2 = 6 </li></ul><ul><li>7 5 7 x 5 35 </li></ul><ul><li>Dividing Fractions </li></ul><ul><li>Multiplication and division are inverse operations. It makes sense, then, that to perform division with fractions, you need to flip the second fraction over, which is also called taking its reciprocal, and then multiply: </li></ul><ul><li> a ÷ c = a x d = ad b d b c bc </li></ul><ul><li> </li></ul><ul><li>Here’s a numerical example: 1 ÷ 4 = 1 x 5 = 5 </li></ul><ul><li> 2 5 2 4 8 </li></ul><ul><li> </li></ul><ul><li> </li></ul>Multiplying and dividing Fractions
- 22. A mixed number is an integer followed by a fraction, like 1 1 / 2 . It is another form of an improper fraction, which is a fraction greater than one. But any operation such as addition, subtraction, multiplication, or division can be performed only on the improper fraction form, so you need to know how to convert between the two. Let’s convert the mixed number 1 1 / 2 into an improper fraction. You multiply the integer portion of the mixed number by the denominator and add that product to the numerator. So 1 x 2 + 1 = 3, making 3 the numerator of the improper fraction. Pu t 3 over the original denominator , 2, and you have your converted fraction, 3 / 2 . Here’s another example: 3 2 / 13 = (3x13) +2 = 41 13 13 Mixed Numbers
- 23. The least common multiple (LCM) of two integers is the smallest multiple that the two numbers have in common. Like the GCF, the least common multiple of two numbers is useful when manipulating fractions. To find the LCM of two integers, you must first find the integers’ prime factorizations. The least common multiple is the smallest prime factorization that contains every prime number in each of the two original prime factorizations. If the same prime factor appears in the prime factorizations of both integers, multiply the factor by the greatest number of times it appears in the factorization of either number. For example, what is the least common multiple of 4 and 6? We must first find their prime factorizations. 4 = 2 x 2 Least Common Multiple
- 24. Let’s try a harder question. Q. What is the LCM of 14 and 38 ? A. we start by finding the prime factorizations of both numbers: 14= 2 x 7 38= 2 x 19 Here 2 appears in both prime factorizations, but not more than once in each, so we only need to use one 2. Therefore, the LCM of 7 and 38 is 2x 7 x 18 =266 For practice, find the LCM of the following pairs of integers: 12 and 32 15 and 26 34 and 40 Compare your answers to the solutions: 12 = 2 3 X3. 32 = 2 5 . The LCM is 2 5 X 3 = 96. 15 = 3 X 5. 26 = 2 X 13. The LCM is 2 X 3 X 5 X 13 = 390. 34 = 2 X 17. 40 = 2 3 X 5. The LCM is 2 3 X 5 X 17 = 680. Least Common Multiple
- 25. Decimals are just another way to express fractions. To produce a decimal, divide the numerator of a fraction by the denominator. For example, 1 / 2 = 1÷2 = .5. Comparing Decimals As with fractions, comparing decimals can be a bit deceptive. As a general rule, when comparing two decimals such as .3 with .003, the decimal with more leading zeros is smaller. But if asked to compare . 003 with .0009, however, you might overlook the additional zero and, because 9 is the larger integer, choose . 0009 as the larger decimal. That, of course, would be wrong. Take care to avoid such mistakes. One way is to line up the decimal points of the two decimals: .0009 is smaller than . 0030 Similarly, . 000900 is smaller than .000925 Decimals
- 26. Let’s convert .3875 into a fraction and see how this First, we eliminate the decimal point and make 3875 the numerator: .3875 = 3875 ? Since .3875 has four digits after the decimal point, we put four zeros in the denominator: .3875 = 3875 1000 Then, by finding the greatest common factor of 3875 and 10000, 125, we can reduce the fraction: 3875 = 31 1000 80 To convert from fractions back to decimals is a cinch. Simply carry out the necessary division on your calculator, such as for 31 / 80 : 3 = 3÷ 5= 0.6 5 Converting Decimal Fractions
- 27. Percents A percent is another way to describe a part of a whole (which means that percents are also another way to talk about fractions or decimals). Percent literally means “of 100” in Latin, so when you attend school 25 percent of the time, that means you only go to school 25 / 100 (or .25) of the time. Take a look at this example : 3 is what percent of 15? This question presents you with a whole, 15, and then asks you to determine how much of that whole 3 represents in percentage form. Since a percent is “of 100,” to solve the question, you have to set the fraction 3 / 15 equal to x / 100 : 3 = x 15 100 You then cross-multiply .07 X 1100 = 7% of 1100=77 .97 X 13 = 97% of 13 =12.61
- 28. Writing Equations <ul><li>Here’s another problem : </li></ul><ul><li>Q Gus needs to paint his house, which has a surface area of x square feet. The brand of paint he buys (at a cost of p dollars a can) comes in cans that cover y square feet each. Gus also needs to buy ten pairs of new jeans .They cost d dollars a pair. If Gus makes these purchases, what is the difference between the cost of the paint and the cost of the jeans? </li></ul><ul><li>A. Assume he doesn’t buy any excess paint—that is, the required amount is not a </li></ul><ul><li>fraction of a can. </li></ul><ul><li>This word problem is long and complicated, but you need to carry out </li></ul><ul><li>just four steps to solve it: </li></ul><ul><li>Gus must buy x /y cans of paint to cover his house. </li></ul><ul><li>This will cost him xp /y dollars. </li></ul><ul><li>The jeans Gus buys cost 10 d dollars. </li></ul><ul><li>Thus, the difference, in dollars, between the cost of the paint and the cost of </li></ul><ul><li>the jeans is xp /y – 10 d . </li></ul><ul><li>Now, test your knowledge of the topics discussed by clicking on the sample </li></ul><ul><li>test links given below. </li></ul>

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