15 on cross multiplication

In this section we look at the useful procedure of cross
multiplcation.
Cross Multiplication

multiplcation.

multiplcation.
Many procedures with two fractions utilize the operation of
cross– multiplication as shown below.

multiplcation.
a
b
c
d

multiplcation.
a
b
c
d
ad bc

multiplcation.
a
b
c
d
Take the denominators and multiply them diagonally across.
ad bc

multiplcation.
What we get are two numbers.
a
b
c
d
ad bc

multiplcation.
a
b
c
d
ad bc
Make sure that the denominators cross over and up so the
numerators stay put.

multiplcation.
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
c
d
adbc

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,

we say the ratio of sugar to flour is 3 to 2, and it’s written as
3 : 2 for sugar : flour.

3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is
2 : 3.

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.

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.

Example A.
Write 3/4 cup of sugar as and 2/3 cup of flour as3
4 S
2
3 F.

Example A.
4 S
2
3 F.
We have the ratio 3
4
S : 2
3
F

Example A.
4 S
2
3 F.
We have the ratio 3
4
S : 2
3
F cross multiply we’ve 9S : 8F.

Example A.
4 S
2
3 F.
We have the ratio 3
4
S : 2
3
Hence in integers, the ratio is 9 : 8 for sugar : flour.

Example A.
4 S
2
3 F.
We have the ratio 3
4
S : 2
3
Hence in integers, the ratio is 9 : 8 for sugar : flour.
Remark: A ratio such as 8 : 4 should be simplified to 2 : 1.

Cross–Multiplication Test for Comparing Two Fractions

When comparing two fractions to see which is larger and
which is smaller.

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

3
5
9
15
=45 45
we get

3
5
9
15
=45 45 so
3
5
9
15
=
we get

3
5
9
15
=45 45 so
3
5
9
15
=
we get
3
5
5
8

3
5
9
15
=45 45 so
3
5
9
15
=
we get
Cross– multiply 3
5
5
8

3
5
9
15
=45 45 so
3
5
9
15
=
we get
Cross– multiply 3
5
5
8
24 25
we get

3
5
9
15
=45 45 so
3
5
9
15
=
we get
Cross– multiply 3
5
5
8
24 25
we get
moreless

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
.

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.)

Cross–Multiplication for Addition or Subtraction

We may cross multiply to add or subtract two fractions

a
b
c
d
±

a
b
c
d± =
ad ±bc

a
b
c
d± =
ad ±bc
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

a
b
c
d
Afterwards we reduce if necessary for the simplified answer.
± =
ad ±bc
bd

a
b
c
d
Example B. Calculate
± =
ad ±bc
bd
3
5
5
6
–a.

a
b
c
d
± =
ad ±bc
bd
3
5
5
6
– =
5*5 – 6*3
6*5
a.

a
b
c
d
± =
ad ±bc
bd
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
=a.

a
b
c
d
± =
ad ±bc
bd
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
=a.
5
12
5
9
–b.

a
b
c
d
± =
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.

a
b
c
d
± =
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.

a
b
c
d
± =
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
=

a
b
c
d
± =
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
=
In a. the LCD = 30 = 6*5 so the crossing method is the same as
the Multiplier Method.

a
b
c
d
± =
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
=
the Multiplier Method. However in b. the crossing method
yielded an answer that needed to be reduced.

a
b
c
d
± =
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
=
the Multiplier Method. However in b. the crossing method
yielded an answer that needed to be reduced. we need both
methods.

Clearing Denominators–for Comparing Multiple Fractions
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

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
–
C. Use cross–multiplication to combine the fractions.

15 on cross multiplication

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