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Fraction Shortcuts and Word Problems Study Guide

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Updated on Oct 4, 2011

Introduction to Fraction Shortcuts and Word Problems

I have hardly ever known a mathematician who was capable of reasoning.

—PLATO, classical Greek philosopher (427 b.c.e. –347 b.c.e.)

The final fraction lesson is devoted to arithmetic shortcuts with fractions (addition, subtraction, and division) and to word problems.

The first part of this lesson shows you some shortcuts for doing arithmetic with fractions. The rest of the lesson reviews all of the fraction lessons by presenting you with word problems. Fraction word problems are especially important because they come up so frequently in everyday living, as you'll see from the familiar situations presented in the word problems.

Shortcut for Addition and Subtraction

Instead of wasting time looking for the least common denominator (LCD) when adding or subtracting, try this "cross multiplication" trick to quickly add or subtract two fractions:

Example:

  1. Top number: "Cross multiply" 5 × 8 and 6 × 3; then add:      
  2. Bottom number: Multiply 6 × 8, the two bottom numbers:
  3. Reduce:   

When using the shortcut for subtraction, you must be careful about the order of subtraction: Begin the "cross multiply" step with the top number of the first fraction. (The "hook" to help you remember where to begin is to think about how you read. You begin at the top left—where you'll find number that starts the process, the top number of the first fraction.)

Example:

  1. Top number: Cross multiply 5 × 4 and subtract 3 × 6:                
  2. Bottom number: Multiply 6 × 4, the two bottom numbers:
  3. Reduce:                                                                                         

Shortcut for Division: Extremes over Means

Extremes over means is a fast way to divide fractions. This concept is also best explained by example, say . But first, let's rewrite the example as and provide two definitions:

Here's how to do it:

1. Multiply the extremes to get the top number of the answer:
2. Multiply the means to get the bottom number of the answer:

You can even use extremes over means when one of the numbers is a whole number or a mixed number. First change the whole number or mixed number into a fraction, and then use the shortcut.

Example:  

1. Change the 2 into a fraction and rewrite the division:
2. Multiply the extremes to get the top number of the answer:
3. Multiply the means to get the bottom number of the answer:
4. Optional: Change the improper fraction to a mixed number:

Tip

Many fraction word problems involve measurement or time. Since feet are based on units of 12 inches, multiply a fraction of a foot by 12 to change it to inches. Since hours are based on units of 60 minutes, multiply a fraction of an hour by 60 to change it to minutes.

Example: What is of an hour? × 60 = 40 minutes Example: What is of a foot? × 12 = 4.5 inches

When given mixed units that contain feet and inches, the number of inches is never just added on as a decimal. In other words, 6 feet 4 inches6.4 feet. It's best to convert 6 feet 4 inches to inches by multiplying the feet by 12 and adding on the inches: 6 × 12 + 4 = 76 inches.

Tip

Throughout the day, look around to find things you can use to make into word problems. The word problem has to involve fractions, so look for groups or portions of a whole. You could use the number of pencils and pens that make up your whole writing instrument supply or the number of CDs and MP3s that make up your music collection. Write down a word problem and solve it using the word problems in this lesson to guide you.

Fraction Shortcuts and Word Problems Sample Questions

Solutions to Sample Questions

Question 1

1. Cross multiply 2 × 5 and subtract 3 × 3:
2. Multiply 3 × 5, the two bottom numbers:

Question 2

1. Change each mixed number into an improper fraction and rewrite the division problem:
2. Multiply the extremes to get the top number of the answer:
3. Multiply the means to get the bottom number of the answer:
4. Reduce:

Find practice problems and solutions for these concepts at Fraction Shortcuts and Word Problems Practice Questions.

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