This 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 ratio in whole integers. An example demonstrates taking a ratio of 3/4 to 2/3 and rewriting it as 9:8 using cross multiplication. The document also notes that cross multiplication can be used to compare two fractions, with the fraction corresponding to the larger product being the larger fraction.
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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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.
2. In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
3. In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication
Cross Multiplication
4. 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.
Cross Multiplication
5. In this section we look at the useful procedure of cross
multiplcation.
a
b
c
d
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Cross Multiplication
6. In this section we look at the useful procedure of cross
multiplcation.
a
b
c
d
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Cross Multiplication
7. In this section we look at the useful procedure of cross
multiplcation.
a
b
c
d
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
ad bc
Cross Multiplication
8. In this section we look at the useful procedure of cross
multiplcation.
a
b
c
d
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
ad bc
Cross Multiplication
9. In this section we look at the useful procedure of cross
multiplcation.
What we get are two numbers.
a
b
c
d
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
ad bc
Cross Multiplication
10. In this section we look at the useful procedure of cross
multiplcation.
What we get are two numbers.
a
b
c
d
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
ad bc
Make sure that the denominators cross over and up so the
numerators stay put.
Cross Multiplication
11. In this section we look at the useful procedure of cross
multiplcation.
What we get are two numbers.
a
b
c
d
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
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
c
d
adbc
Cross Multiplication
12. In this section we look at the useful procedure of cross
multiplcation.
What we get are two numbers.
a
b
c
d
Cross Multiplication
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.
Take the denominators and multiply them diagonally across.
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
c
d
adbc
Cross Multiplication
13. Here are some operations where we may cross multiply.
Cross Multiplication
14. Here are some operations where we may cross multiply.
Rephrasing Fractional Ratios
Cross Multiplication
15. 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,
Cross Multiplication
16. 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.
Cross Multiplication
17. 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.
Cross Multiplication
18. 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.
Cross Multiplication
19. 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.
Cross Multiplication
20. 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.
Cross Multiplication
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 as3
4 S
2
3 F.
Cross Multiplication
22. 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 as3
4 S
2
3 F.
We have the ratio 3
4
S : 2
3
F
Cross Multiplication
23. 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 as3
4 S
2
3 F.
We have the ratio 3
4
S : 2
3
F cross multiply we’ve 9S : 8F.
Cross Multiplication
24. 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 as3
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.
Cross Multiplication
25. 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 as3
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.
Cross Multiplication
Remark: A ratio such as 8 : 4 should be simplified to 2 : 1.
27. Cross–Multiplication Test for Comparing Two Fractions
Cross Multiplication
When comparing two fractions to see which is larger and
which is smaller.
28. Cross–Multiplication Test for Comparing Two Fractions
Cross Multiplication
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 Test for Comparing Two Fractions
Cross Multiplication
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 Test for Comparing Two Fractions
Hence cross– multiply
Cross Multiplication
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.
3
5
9
15
31. Cross–Multiplication Test for Comparing Two Fractions
Hence cross– multiply
Cross Multiplication
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.
3
5
9
15
=45 45
we get
32. Cross–Multiplication Test for Comparing Two Fractions
Hence cross– multiply
Cross Multiplication
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.
3
5
9
15
=45 45 so
3
5
9
15
=
we get
33. Cross–Multiplication Test for Comparing Two Fractions
Hence cross– multiply
Cross Multiplication
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.
3
5
9
15
=45 45 so
3
5
9
15
=
we get
3
5
5
8
34. Cross–Multiplication Test for Comparing Two Fractions
Hence cross– multiply
Cross Multiplication
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.
3
5
9
15
=45 45 so
3
5
9
15
=
we get
Cross– multiply 3
5
5
8
35. Cross–Multiplication Test for Comparing Two Fractions
Hence cross– multiply
Cross Multiplication
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.
3
5
9
15
=45 45 so
3
5
9
15
=
we get
Cross– multiply 3
5
5
8
24 25
we get
36. Cross–Multiplication Test for Comparing Two Fractions
Hence cross– multiply
Cross Multiplication
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.
3
5
9
15
=45 45 so
3
5
9
15
=
we get
Cross– multiply 3
5
5
8
24 25
we get
moreless
37. Cross–Multiplication Test for Comparing Two Fractions
Hence cross– multiply
Cross Multiplication
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.
3
5
9
15
=45 45 so
3
5
9
15
=
we get
Cross– multiply 3
5
5
8
24 25
Hence 3
5
5
8
is less than
we get
moreless
.
38. Cross–Multiplication Test for Comparing Two Fractions
Hence cross– multiply
Cross Multiplication
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.
3
5
9
15
=45 45 so
3
5
9
15
=
we get
Cross– multiply 3
5
5
8
24 25
Hence 3
5
5
8
is less than
we get
moreless
.
(Which is more
7
11
9
14
or ? Do it by inspection.)
41. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions
a
b
c
d
±
Cross Multiplication
42. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions
a
b
c
d± =
ad ±bc
Cross Multiplication
43. Cross–Multiplication for Addition or Subtraction
a
b
c
d± =
ad ±bc
Cross Multiplication
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
44. Cross–Multiplication for Addition or Subtraction
a
b
c
d± =
ad ±bc
bd
Cross Multiplication
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
45. 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
Afterwards we reduce if necessary for the simplified answer.
± =
ad ±bc
bd
Cross Multiplication
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
b
c
d
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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
b
c
d
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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
b
c
d
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
3
5
5
6
– =
5*5 – 6*3
6*5
a.
Cross Multiplication
49. 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
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
=a.
Cross Multiplication
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
b
c
d
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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
b
c
d
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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
b
c
d
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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 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
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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.
Cross Multiplication
54. 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
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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. 5
36
=
Cross Multiplication
55. 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
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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. 5
36
=
Cross Multiplication
In a. the LCD = 30 = 6*5 so the crossing method is the same as
the Multiplier Method.
56. 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
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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. 5
36
=
Cross Multiplication
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 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
Afterwards we reduce if necessary for the simplified answer.
Example B. Calculate
± =
ad ±bc
bd
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. 5
36
=
Cross Multiplication
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.
58. Clearing Denominators–for Comparing Multiple Fractions
Cross Multiplication
Cross multiplication of two fractions clears the denominators out
of the picture when comparing them. To compare three or more
fractions, we use their LCD to accomplish the same goal.
(Multiplier Method for Comparing Fractions)
To compare three or more fractions, multiply each fraction by the
LCD first then compare their products.
2
3
Example C.
Arrange the fractions
from the smallest to the largest.
,
3
5 ,
4
7
The LCD is 3(5)(7) = 105, multiply it
to the fractions. Comparing the
products, we see that from the
smallest to the largest, they are
2
3,
3
5 ,
4
7
x 105
60,63, 70
2
3 ,
3
5,
4
7
the largestthe 2nd largest
the smallest
59. Ex. Restate the following ratios in integers.
9. In a market, ¾ of an apple may be traded with ½ a pear.
Restate this using integers.
1
2
1
3
:1. 2. 3. 4.2
3
1
2:
3
4
1
3
:
2
3
3
4
:
3
5
1
2:5. 6. 7. 8.1
6
1
7
:
3
5
4
7:
5
2
7
4:
Determine which fraction is more and which is less.
2
3
3
4
,10. 11. 12. 13.4
5
3
4
,
4
7
3
5
,
5
6
4
5
,
5
9
4
7
,14. 15. 16. 17.7
10
2
3
,
5
12
3
7
,
13
8
8
5
,
1
2
1
3
+18. 19. 20. 21.1
2
1
3
–
2
3
3
2
+
3
4
2
5
+
5
6
4
7
–22. 23. 24. 25.7
10
2
5
–
5
11
3
4
+
5
9
7
15
–
Cross Multiplication
C. Use cross–multiplication to combine the fractions.