The document discusses the mathematical procedure of cross multiplication. It explains that cross multiplication can be used to compare two fractions, with the fraction having the larger product being the greater value. It also describes how cross multiplication allows rewriting fractional ratios in whole numbers. An example shows converting a ratio of 3/4 cups of sugar to 2/3 cups of flour into a whole number ratio of 9:8.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
3. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
4. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
5. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
a
b
c
d
6. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
a
b
c
d
7. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
a
b
c
d
ad bc
8. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
a
b
c
d
ad bc
9. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
What we get are two numbers.
a
b
c
d
ad bc
10. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
What we get are two numbers.
a
b
c
d
ad bc
Make sure that the denominators cross over and up so the
numerators stay put.
11. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
What we get are two numbers.
a
b
c
d
ad bc
Make sure that the denominators cross over and up so the
numerators stay put. Do not cross downward as shown
here. a
b
cd
bc ad
12. Cross Multiplication
In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
What we get are two numbers.
a
b
c
d
ad bc
Make sure that the denominators cross over and up so the
numerators stay put. Do not cross downward as shown
here. a
b
cd
bc ad
14. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
15. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2,
16. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour.
17. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3.
18. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3. For most people a recipe that calls for the fractional ratio
of 3/4 cup sugar to 2/3 cup of flour is confusing.
19. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3. For most people a recipe that calls for the fractional ratio
of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to
cross multiply to rewrite this ratio in whole numbers.
20. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3. For most people a recipe that calls for the fractional ratio
of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to
cross multiply to rewrite this ratio in whole numbers.
Example A.
rewrite a recipe that calls for the fractional ratio of 3/4 cup
sugar to 2/3 cup of flour into ratio of whole numbers.
21. Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3. For most people a recipe that calls for the fractional ratio
of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to
cross multiply to rewrite this ratio in whole numbers.
Example A.
rewrite a recipe that calls for the fractional ratio of 3/4 cup
sugar to 2/3 cup of flour into ratio of whole numbers.
Write 3/4 cup of sugar as and 2/3 cup of flour as
3
4
S
2
3
F.
Cross Multiplication
22. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3. For most people a recipe that calls for the fractional ratio
of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to
cross multiply to rewrite this ratio in whole numbers.
Example A.
rewrite a recipe that calls for the fractional ratio of 3/4 cup
sugar to 2/3 cup of flour into ratio of whole numbers.
Write 3/4 cup of sugar as and 2/3 cup of flour as
3
4
S
2
3
F.
We have the ratio
3
4
S :
2
3
F
23. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3. For most people a recipe that calls for the fractional ratio
of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to
cross multiply to rewrite this ratio in whole numbers.
Example A.
rewrite a recipe that calls for the fractional ratio of 3/4 cup
sugar to 2/3 cup of flour into ratio of whole numbers.
Write 3/4 cup of sugar as and 2/3 cup of flour as
3
4
S
2
3
F.
We have the ratio
3
4
S :
2
3
F cross multiply we’ve 9S : 8F.
24. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3. For most people a recipe that calls for the fractional ratio
of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to
cross multiply to rewrite this ratio in whole numbers.
Example A.
rewrite a recipe that calls for the fractional ratio of 3/4 cup
sugar to 2/3 cup of flour into ratio of whole numbers.
Write 3/4 cup of sugar as and 2/3 cup of flour as
3
4
S
2
3
F.
We have the ratio
3
4
S :
2
3
F cross multiply we’ve 9S : 8F.
Hence in integers, the ratio is 9 : 8 for sugar : flour.
25. Cross Multiplication
Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
If a cookie recipe calls for 3 cups of sugar and 2 cups of flour,
we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3. For most people a recipe that calls for the fractional ratio
of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to
cross multiply to rewrite this ratio in whole numbers.
Example A.
rewrite a recipe that calls for the fractional ratio of 3/4 cup
sugar to 2/3 cup of flour into ratio of whole numbers.
Write 3/4 cup of sugar as and 2/3 cup of flour as
3
4
S
2
3
F.
We have the ratio
3
4
S :
2
3
F cross multiply we’ve 9S : 8F.
Hence in integers, the ratio is 9 : 8 for sugar : flour.
Remark: A ratio such as 8 : 4 should be simplified to 2 : 1.
28. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
29. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
30. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
Hence cross– multiply
3
5
9
15
31. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
Hence cross– multiply
3
5
9
15
45 = 45
we get
32. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
Hence cross– multiply
3
5
9
15
45 = 45 so
3
9
=
5
15
we get
33. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
Hence cross– multiply
3
5
9
15
45 = 45 so
3
9
=
5
15
we get
3
5
5
8
34. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
Hence cross– multiply
3
5
9
15
45 = 45 so
3
9
=
5
15
we get
Cross– multiply
3
5
5
8
35. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
Hence cross– multiply
3
5
9
15
45 = 45 so
3
9
=
5
15
we get
Cross– multiply
3
5
5
8
24 25
we get
36. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
Hence cross– multiply
3
5
9
15
45 = 45 so
3
9
=
5
15
we get
Cross– multiply
3
5
5
8
24 25
we get
less more
37. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
Hence cross– multiply
3
5
9
15
45 = 45 so
3
9
=
5
15
we get
Cross– multiply
3
5
5
8
24 25
Hence
3
5
5
8
is less than
we get
less more
.
38. Cross Multiplication
Cross–Multiplication Test for Comparing Two Fractions
When comparing two fractions to see which is larger and
which is smaller. Cross–multiply them, the side with the larger
product corresponds to the larger fraction.
In particular, if the cross multiplication products are the same
then the fraction are the same.
Hence cross– multiply
3
5
9
15
45 = 45 so
3
9
=
5
15
we get
Cross– multiply
3
5
5
8
24 25
Hence
3
5
5
8
is less than
we get
less more
.
(Which is more
7
11
9
14
or ? Do it by inspection.)
43. Cross Multiplication
Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
b
c
d
± =
ad ±bc
44. Cross Multiplication
Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
b
c
d
± =
ad ±bc
bd
45. Cross Multiplication
Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
46. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
a. –
Cross Multiplication
47. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
a. –
Cross Multiplication
48. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
a.
Cross Multiplication
49. Cross Multiplication
Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
a. =
50. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
a. =
5
12
5
9
b. –
Cross Multiplication
51. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
a. =
5
12
5
9
b. –
Cross Multiplication
52. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
a. =
5
12
5
9
– =
5*12 – 9*5
9*12
b.
Cross Multiplication
53. Cross Multiplication
Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
a. =
5
12
5
9
– =
5*12 – 9*5
9*12
15
108
b. =
54. Cross Multiplication
Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
a. =
b. = 5
5
12
5
9
– =
5*12 – 9*5
9*12
15
108
36
=
55. Cross Multiplication
Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
a. =
b. = 5
5
12
5
9
– =
5*12 – 9*5
9*12
15
108
36
=
In a. the LCD = 30 = 6*5 so the crossing method is the same as
the Multiplier Method.
56. Cross Multiplication
Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
a. =
b. = 5
5
12
5
9
– =
5*12 – 9*5
9*12
15
108
36
=
In a. the LCD = 30 = 6*5 so the crossing method is the same as
the Multiplier Method. However in b. the crossing method
yielded an answer that needed to be reduced.
57. Cross Multiplication
Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
± c
ad ±bc
=
b
d
bd
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
a. =
b. = 5
5
12
5
9
– =
5*12 – 9*5
9*12
15
108
36
=
In a. the LCD = 30 = 6*5 so the crossing method is the same as
the Multiplier Method. However in b. the crossing method
yielded an answer that needed to be reduced. we need both
methods.
59. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
60. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways.
61. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct.
62. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct. If two answers
are different then we have to clarify the mistake.
63. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct. If two answers
are different then we have to clarify the mistake.
When we + or – fractions, we can use the above two methods
to cross check an answer.
64. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct. If two answers
are different then we have to clarify the mistake.
When we + or – fractions, we can use the above two methods
to cross check an answer. For example, in part b. above,
we obtain an answer via the crossing method.
65. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct. If two answers
are different then we have to clarify the mistake.
When we + or – fractions, we can use the above two methods
to cross check an answer. For example, in part b. above,
we obtain an answer via the crossing method. Let’s cross
check the first answer using the Multiplier Method.
5
12
5
9
–
66. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct. If two answers
are different then we have to clarify the mistake.
When we + or – fractions, we can use the above two methods
to cross check an answer. For example, in part b. above,
we obtain an answer via the crossing method. Let’s cross
check the first answer using the Multiplier Method.
Since the LCD = 36, we multiply and divide by 36.
5
12
5
9
–
67. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct. If two answers
are different then we have to clarify the mistake.
When we + or – fractions, we can use the above two methods
to cross check an answer. For example, in part b. above,
we obtain an answer via the crossing method. Let’s cross
check the first answer using the Multiplier Method.
Since the LCD = 36, we multiply and divide by 36.
5
12
5
9
( –
(
*36 / 36
68. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct. If two answers
are different then we have to clarify the mistake.
When we + or – fractions, we can use the above two methods
to cross check an answer. For example, in part b. above,
we obtain an answer via the crossing method. Let’s cross
check the first answer using the Multiplier Method.
Since the LCD = 36, we multiply and divide by 36.
5
12
5
9
( –
(
*36 / 36
4 3
69. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct. If two answers
are different then we have to clarify the mistake.
When we + or – fractions, we can use the above two methods
to cross check an answer. For example, in part b. above,
we obtain an answer via the crossing method. Let’s cross
check the first answer using the Multiplier Method.
Since the LCD = 36, we multiply and divide by 36.
4 3
5
12
5
9
( –
(
*36 / 36
= (5*4 – 5*3) / 36 = 5/36
70. Cross Multiplication
The Double Check Strategy
One of the most difficult thing to do in mathematics is to know
that a mistake had taken place and to locate the mistake.
The Double Check is to cross check an answer by doing a
problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct. If two answers
are different then we have to clarify the mistake.
When we + or – fractions, we can use the above two methods
to cross check an answer. For example, in part b. above,
we obtain an answer via the crossing method. Let’s cross
check the first answer using the Multiplier Method.
Since the LCD = 36, we multiply and divide by 36.
4 3
5
12
5
9
( –
(
*36 / 36
= (5*4 – 5*3) / 36 = 5/36
This is the same as before hence it’s very likely to be correct.
71. Cross Multiplication
Comments
* The Double Check Strategy is an important tool for learning.
It reassures us if we’re heading in the right direction. It warns
us that a mistake had occurred so we should back track and
locate the mistake.
72. Cross Multiplication
Comments
* The Double Check Strategy is an important tool for learning.
It reassures us if we’re heading in the right direction. It warns
us that a mistake had occurred so we should back track and
locate the mistake.
Use this Double Check Strategy for learning!
73. Cross Multiplication
Comments
* The Double Check Strategy is an important tool for learning.
It reassures us if we’re heading in the right direction. It warns
us that a mistake had occurred so we should back track and
locate the mistake.
Use this Double Check Strategy for learning!
* The Multiplier Method and the Cross Multiplication Method
are two methods to double check addition and subtraction of
small number of fractions.
74. Cross Multiplication
Comments
* The Double Check Strategy is an important tool for learning.
It reassures us if we’re heading in the right direction. It warns
us that a mistake had occurred so we should back track and
locate the mistake.
Use this Double Check Strategy for learning!
* The Multiplier Method and the Cross Multiplication Method
are two methods to double check addition and subtraction of
small number of fractions. These two methods generalize to
addition and subtraction of fractional (rational) formulas in
later topics.
75. Cross Multiplication
Comments
* The Double Check Strategy is an important tool for learning.
It reassures us if we’re heading in the right direction. It warns
us that a mistake had occurred so we should back track and
locate the mistake.
Use this Double Check Strategy for learning!
* The Multiplier Method and the Cross Multiplication Method
are two methods to double check addition and subtraction of
small number of fractions. These two methods generalize to
addition and subtraction of fractional (rational) formulas in
later topics. Each method leads to various ways of handling
various fractional algebra problems where each way has its
own advantages and disadvantage.
76. Cross Multiplication
Comments
* The Double Check Strategy is an important tool for learning.
It reassures us if we’re heading in the right direction. It warns
us that a mistake had occurred so we should back track and
locate the mistake.
Use this Double Check Strategy for learning!
* The Multiplier Method and the Cross Multiplication Method
are two methods to double check addition and subtraction of
small number of fractions. These two methods generalize to
addition and subtraction of fractional (rational) formulas in
later topics. Each method leads to various ways of handling
various fractional algebra problems where each way has its
own advantages and disadvantage.
We use both methods through out this database.
77. Ex. Restate the following ratios in integers.
1
2
1
3
2
3
1
2
3
4
1
3
1. : 2. 3. 4.
:
:
2
3
3
4
:
3
5
1
2
1
6
1
7
3
5
4
7
5. : 6. 7. 8.
:
:
5
2
7
4
:
9. In a market, ¾ of an apple may be traded with ½ a pear.
Restate this using integers.
Determine which fraction is more and which is less.
2
3
3
4
4
5
3
4
10. , 11. 12. 13.
,
4
7
3
5
,
5
6
4
5
,
5
9
4
7
7
10
2
3
14. , 15. 16. 17.
,
5
12
3
7
,
13
8
8
5
,
1
2
1
3
1
2
1
3
18. + 19. 20. 21.
–
2
3
3
2
+
3
4
2
5
+
5
6
4
7
7
10
2
5
22. – 23. 24. 25.
–
5
11
3
4
+
5
9
7
15
–
Cross Multiplication
C. Use cross–multiplication to combine the fractions.