Skill Builder : Comparing Fractions

Let's say we were given the following question on the SHSAT.

Example 1

Compare 3⁄7 and 4⁄9. Which fraction is greater?

One method of comparing the values of the two given fractions is to convert each fraction into decimal form. Doing so requires doing division. We would divide 3 by 7, and then repeat the process by dividing 4 by 9.

Converting fractions to decimal form by doing division would lead us to the answer, but doing so would require computation and time.

We would be better served by using a quicker method that requires just multiplication.

Cross-Multiplication Method for Comparing Fractions

Example 1

Compare 3⁄7 and 4⁄9. Which fraction is greater?

Step 1: Cross multiply. a) (Multiply the denominator of the first fraction (7) by the numerator of the second fraction (4). Write that product (7 x 4 = 28) above the second fraction. b) Repeat with the remaining denominator of the second fraction (9) and the numerator of the first fraction (3). 3 × 9 = 27

27 28

3⁄7 < 4⁄9

Step 2: Compare the products

We can see tjhat 27 is less 28. This lets us know that 3⁄7 is less than 4⁄9.

The cross-multiplication method of comparing the value of fractions works when comparing any two fractions, can be done quickly, and requires no conversion to decimal form.

Let's apply this technique to an actual SHSAT practice question.

EXAMPLE : 2020 SHSAT PRACTICE TEST B/ Question 102.

102. The Barnes family and the Ramirez family each have a pizza of equal size. The first pizza is cut into 18 equal slices, and the second pizza is cut into 15 equal slices. If the Barnes family eats 11 slices from the first pizza, what is the greatest number of whole slices from the second pizza that the Ramirez family can eat without eating a greater percentage of a pizza than the Barnes family ate?

     
     

     E. 6

     F. 7

     G. 8

     H. 9

Solution: The Barnes family ate 11⁄18 of the pizza. We are looking for the fraction of the second same-size pizza that the Ramirez family can eat that is smaller in value than 11⁄18 .

We can test the choices to see which choice will make the condition set out in the question true. Since we are looking for the greatest number of slices that the Ramirez family can eat, it's best to check Choice H, which is the largest number among the choices, first.

If the Ramirez family eats 9 slices, they will have eaten 9⁄15 of the entire pizza. Is 9⁄15 less than 11⁄18 ? If it is less, then Choice H is our answer.

We can see now that this lengthy word problem boils down to a simple question asking us to compare two fractions: 9⁄15 and 11⁄18.

We can use the cross-multiplication method to compare the two fractions.

18 x 9 = 162 15 x 11 =165

9⁄15 < 11⁄18

Notice that 162 is less than 165. This means, 9⁄15 is also less than 11⁄18.

So, when applied to our original question, this means the Ramirez family can eat 9 slices of their pizza and know that they will have eaten less total pizza than the Barnes Family. Choice H) 9 is our answer.

Here are five more practice questions to practice the cross-multiplication method of comparing fractions:

Compare the following fractions using cross multiplication. Which is greater?

1. 5/8  □  7/11

   
   

2. 6/13  □  4/9

   
   

3. 7/10  □  8/13

   
   

4. 9/14  □  5/8

   
   

5. 8/11  □  9/13

Whenever possible, try to use the cross-multiplication method to compare the relative values of two fractions on SHSAT questions. Doing so saves time and lessens computation. It will also ensure that you have ample time to apply to more difficult questions, which you'll be sure to see on the SHSAT.

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