Basic Mathematics for College Students

 

 

There is another way to determine whether a proportion is true or false. Before

we can discuss it, we need to introduce some more vocabulary of proportions.

Each of the four numbers in a proportion is called a term.The first and fourth

terms are called the extremes,and the second and third terms are called the means.

In the proportion shown above, theproduct of the extremes is equal to the product

of the means.

These products can be found by multiplying diagonally in the proportion. We call

and cross products.

Cross products

Note that the cross products are equal. This example illustrates the following

property of proportions.

Cross-Products Property (Means-Extremes Property)

To determine whether a proportion is true or false, first multiply along one

diagonal, and then multiply along the other diagonal.

• If the cross products are equal,the proportion is true.


• If the cross products are not equal,the proportion is false.
  (If the product of the extremes is equalto the product of the means, the
  proportion is true. If the product of the extremes is not equalto the product of
  the means, the proportion is false.)

1

2



3

6

1  6  6 2  3  6

 

1  6 2  3

1 # 6  6 and 2 # 3  6

Third term (mean)

 Fourth term (extreme)

1 

2



3

 6

First term (extreme) 

Second term (mean)

430 Chapter 5 Ratio, Proportion, and Measurement

b.Neither ratio in the proportion is in simplest form. To simplify each

ratio, we proceed as follows:

Since the ratios on the left and right sides of the proportion are not equal ,

the proportion is false.

1152 ^1542

45

12



3

1

 15

3

1

 4



15

4

30

4



2

1

 15

2

1

 2



15

2

30

4 

45

12

EXAMPLE (^3) Determine whether each proportion is true or false.
a. b.
StrategyWe will check to see whether the cross products are equal (the product
of the extremes is equal to the product of the means).
WHYIf the cross products are equal, the proportion is true. If the cross products
are not equal, the proportion is false.
Solution
a.
Each cross product is 63.
Since the cross products are equal, the proportion is true.

3

7



9

21

3  21  63 7  9  63

8

3



13

5

3

7



9

21

Self Check 3

Determine whether the

proportion

is true or false.

Now TryProblem 25

6

13



18

39