Associated Topics || Dr. Math Home || Search Dr. Math

### Cross Products

```
Date: 02/28/99 at 10:32:16
From: Sarah Harvard
Subject: Cross-products

I am having trouble understanding cross-products. Will you please
explain them to me?
```

```
Date: 02/28/99 at 12:47:52
From: Doctor Reno
Subject: Re: Cross-products

Hi, Sarah!

Cross-products can be used for three purposes: to compare fractions,
to determine whether a proportion is true, and to solve a proportion.

Fractions that represent the same quantity are called equivalent
fractions. For example:

3/6, 4/8, and 5/10 are all equivalent fractions for 1/2.

You can use cross-products as a shortcut method to find whether two
fractions are equivalent. If the cross-products are equal, then the
fractions are equivalent; if the cross-products are not equal, then
the fractions are not equivalent.

Are 3/10 and 15/50 equivalent?

3 x 50 = 150
10 x 15 = 150, so the fractions are equivalent.

What about 7/14 and 5/8? Are they equivalent?

7 x 8 = 56
14 x 5 = 70, so the fractions are NOT equivalent.

A proportion is a statement that two ratios are equal. You can
determine whether a proportion is true by using cross-products. A
proportion is true if the two fractions are equivalent and the
cross-products are equal.

Is the proportion 3/12 = 8/32 true or false? (We read 3/12 = 8/32 like
this: "three is to twelve as eight is to thirty-two.")

3 / 3      1
------  =  ---
12 / 3      4

8 / 8      1
------- =  ---
32 / 8      4

So we know the fractions are equivalent.

Are the cross-products equal? Does 3 x 32 = 8 x 12?

3 x 32 = 96
8 x 12 = 96
So YES, the proportion is TRUE.

Cross-products can also be used to solve proportions when one of the
numbers is unknown:

5      n
---- = ----   What is n?
20     48

We know from what we have learned above that 5 x 48 must equal 20 x n:

20 x n = 5 x 48
20 x n = 240
If you don't know how to solve for n yet, you can think: what
number multiplied by 20 gives me 240?
n = 12

We check this value for n by substituting it back into the proportion:

5     12
--- = ----
20     48

Are the two fractions equivalent?

5 / 5     1
------  = ---
20 / 5     4

12 / 12    1
------- = ---
48 / 12    4

So the proportion is true; our value of 12 for n was correct.

I hope this has helped, Sarah. If you have any more questions, please

- Doctor Reno, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Fractions
Middle School Fractions
Middle School Ratio and Proportion

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search