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 
ask Dr. Math!

- Doctor Reno, The Math Forum
  http://mathforum.org/dr.math/