1 f7 on cross-multiplication

Cross Multiplication
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The importance of the LCD is that the LCD is the smallest,
hence the easiest number to “multiply” all the fractions in
question into whole numbers.

Example A. a. Find the LCD
of the following list of fractions,
{ 2/3, 5/8, 7/12, 3/4}

{ 2/3, 5/8, 7/12, 3/4}
List the multiples of 12
(from 7/12): 12, 24, 36,..
we find that the LCD is 24.

{ 2/3, 5/8, 7/12, 3/4}
b. Multiply the LCD to each
fraction to change the list
into a whole-number list.
(from 7/12): 12, 24, 36,..

{ 2/3, 5/8, 7/12, 3/4}
(from 7/12): 12, 24, 36,..
{ 2/3, 5/8, 7/12, 3/4} x 24

{ 2/3, 5/8, 7/12, 3/4}
(from 7/12): 12, 24, 36,..
{ 2/3, 5/8, 7/12, 3/4} x 24
8
{ 2/3, 5/8, 7/12, 3/4} x 24

{ 2/3, 5/8, 7/12, 3/4}
(from 7/12): 12, 24, 36,..
{ 2/3, 5/8, 7/12, 3/4} x 24
8
{ 2/3, 5/8, 7/12, 3/4} x 24
{ 16,

{ 2/3, 5/8, 7/12, 3/4}
(from 7/12): 12, 24, 36,..
{ 2/3, 5/8, 7/12, 3/4} x 24
8 3
{ 2/3, 5/8, 7/12, 3/4} x 24
{ 16,

{ 2/3, 5/8, 7/12, 3/4}
(from 7/12): 12, 24, 36,..
{ 2/3, 5/8, 7/12, 3/4} x 24
8 3
{ 2/3, 5/8, 7/12, 3/4} x 24
{ 16, 15,

{ 2/3, 5/8, 7/12, 3/4}
(from 7/12): 12, 24, 36,..
{ 2/3, 5/8, 7/12, 3/4} x 24
8 3 62
{ 2/3, 5/8, 7/12, 3/4} x 24
{ 16, 15, 14, 18}

{ 2/3, 5/8, 7/12, 3/4}
(from 7/12): 12, 24, 36,..
{ 2/3, 5/8, 7/12, 3/4} x 24
8 3 62
{ 16, 15, 14, 18}
c. List the fractions from
the largest to the smallest.
From the whole numbers,
listing the fractions from the
largest to the smallest:
{ 2/3, 5/8, 7/12, 3/4} x 24

{ 2/3, 5/8, 7/12, 3/4}
(from 7/12): 12, 24, 36,..
{ 2/3, 5/8, 7/12, 3/4} x 24
8 3 62
{ 16, 15, 14, 18}
c. List the fractions from
the largest to the smallest.
From the whole numbers,
listing the fractions from the
largest to the smallest: 3/4, 2/3, 5/8, 7/12
1418 16 15
{ 2/3, 5/8, 7/12, 3/4} x 24

With two fractions, here is an easier and useful procedure
for clearing their denominators.

a
b
c
d
Given two fractions as shown below,

a
b
c
d
Given two fractions as shown below, multiplying to each fraction
by the common denominator bd (which may not be the LCD),
to clear the denominators,

a
b
c
d
bdbd

a
b
c
d
bdbd
ad

a
b
c
d
bdbd
ad bc

to clear the denominators, yield the same outcome as:
taking the denominators and multiply them diagonally across.
a
b
c
d
bdbd
ad bc

a
b
c
d
ad bc
(cross–multiplication)

a
b
c
d
ad bc
Be sure the denominators cross over and up
so the outcomes correspond to the fractions.

a
b
c
d
ad bc
Do not cross downward as shown here!!
a
b
c
d
adbc

a
b
c
d
ad bc
Do not cross downward as shown here!!
a
b
c
d
adbc
(The results are out of order!)

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

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

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

Example B.
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 B.
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 C. 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.

Comparing Multiple Fractions
The reason that Cross–multiply them, the side with the larger
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.)

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.

problem two different ways. If both methods yielded the same
answer then the answer is likely to be correct.

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.

to cross check an answer. For example, in part b. above,
we obtain an answer via the crossing method.

we obtain an answer via the crossing method. Let’s cross
check the first answer using the Multiplier Method.
5
12
5
9
–

Since the LCD = 36, we multiply and divide by 36.
5
12
5
9
–

5
12
5
9
–( ( *36 / 36

5
12
5
9
–( ( *36 / 36
34

5
12
5
9
–( ( *36 / 36
= (5*4 – 5*3) / 36 = 5/36
34

5
12
5
9
–( ( *36 / 36
= (5*4 – 5*3) / 36 = 5/36
34
This is the same as before hence it’s very likely to be correct.

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.

Comments
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.
Comments
locate the mistake.

small number of fractions. These two methods generalize to
addition and subtraction of fractional (rational) formulas in
later topics.
Comments
locate the mistake.

later topics. Each method leads to various ways of handling
various fractional algebra problems where each way has its
own advantages and disadvantage.
Comments
locate the mistake.

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.
Comments
locate the mistake.

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.

1 f7 on cross-multiplication

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