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.

1. Cross Multiplication
Back to Algebra–Ready Review Content.

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.

26. Cross–Multiplication Test for Comparing Two Fractions
Cross Multiplication

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.)

39. Cross–Multiplication for Addition or Subtraction
Cross Multiplication

40. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions
Cross Multiplication

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. 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.