4 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.)

Multiplying by the LCD

Cross–multiplication is a shortcut for clearing denominators for
two fractions.

two fractions. When there are more than two fractions,
we use their LCD to clear their denominators when needed.

Example B.
A cookie recipe that calls for 1/4 cup of butter, 2/3 cup of flour
and 5/6 cup of sugar. Phrase this in terms of whole number.

Example B.
5
6
S
We have the three–way ratio:
1
4
B:
2
3 F:

Example B.
5
6
S
1
4
B:
2
3 F:( )12
We have the three–way ratio: multiply the list by the LCD 12.

Example B.
5
6
S
1
4
B:
2
3 F:( )12
3 4 2
3B: 8F: 10S

Example B.
5
6
S
1
4
B:
2
3 F:( )12
3 4 2
3B: 8F: 10S
Hence recipe calls for 3: 8: 10
for butter: flour: sugar.

Example B.
5
6
S
1
4
B:
2
3 F:( )12
3 4 2
3B: 8F: 10S
Example C.
Arrange 3/5, 2/3 and 7/12 from the smallest to the largest.

Example B.
5
6
S
1
4
B:
2
3 F:( )12
3 4 2
3B: 8F: 10S
Example C.
7
12
3
5
: 2
3 :Multiply the list by the LCD = 60.

Example B.
5
6
S
1
4
B:
2
3 F:( )12
3 4 2
3B: 8F: 10S
Example C.
7
12
3
5
: 2
3 :( )60Multiply the list by the LCD = 60.

Example B.
5
6
S
1
4
B:
2
3 F:( )12
3 4 2
3B: 8F: 10S
Example C.
7
12
3
5
: 2
3 :( )60
36: 40: 35
12 520
Multiply the list by the LCD = 60.

Example B.
5
6
S
1
4
B:
2
3 F:( )12
3 4 2
3B: 8F: 10S
Example C.
Multiply the list by the LCD = 60. 7
12
3
5
: 2
3 :( )60
36: 40: 35
7
12
3
5
2
3
Hence <<
12 520

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 D. 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=
In b. the crossing method gives an answer that needs to be
reduced because the denominator 9*12 =108 is not the LCD.

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.

4 cross multiplication

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4 cross multiplication