Math Help: Understanding Reciprocals
You'll use reciprocals when you learn to divide fractions by whole numbers and other fractions. This skill is usually taught in 5th grade. Keep reading for help!
Reciprocals in Math
Reciprocals of Whole Numbers
A number's reciprocal is defined as one over that number (1/n). For example, the reciprocal of three is 1/3, and the reciprocal of 17 is 1/17.
Another way to think of reciprocals is that they are 'upsidedown' fractions. Whole numbers, like 3, 17 and 345, can all be written as fractions by putting them over one. For instance, 2 = 2/1, 17 = 17/1 and 345 = 345/1. The reciprocal of each number is the reverse of each fraction. The reciprocal of two (2/1) is 1/2, and the reciprocal of 345 (345/1) is 1/345.
Reciprocals of Fractions
Similarly, the reciprocal of a fraction is just that fraction reversed. For example, the reciprocal of 4/7 is 7/4, and the reciprocal of 13/8 is 8/13.
Division with Reciprocals
Reciprocals are mainly used when you're dividing by a fraction. To divide by fractions, you need to find the reciprocal of the fraction you're dividing by and then multiply the fractions. For example, if you're dividing five by 2/3, you'd write five as a fraction (5/1) and multiply it by the reciprocal of 2/3, which is 3/2. Since 5/1 x 3/2 = 15/2, that's your answer.
Reciprocal Practice Problems
Solve each of the following problems, and then check your answers using the key below. Don't forget to use a reciprocal for the division problems.
 1. What is the reciprocal of 11?
 2. Find the reciprocal of 7/8.
 3. What is 1/6 the reciprocal of?
 4. What fraction is 15/4 the reciprocal of?
 5. Solve 2/7 ÷ 1/7. Don't forget to simplify your answer.
 6. Find the quotient of 1/10 ÷ 2.
Answer Key
 1. Since the reciprocal of a number (n) is 1/n, the reciprocal of 11 is 1/11.
 2. To find the reciprocal of 7/8, you can simply switch the numerator with the denominator to get 8/7.
 3. Since 1/n is the reciprocal of a whole number, 1/6 is the reciprocal of six or 6/1.
 4. The fraction 15/4 is the reciprocal of 4/15.
 5. To solve 2/7 ÷ 1/7, start by finding the reciprocal of 1/7, which is 7/1. Then, multiply 2/7 x 7/1 to get 14/7. You can simplify 14/7 to two.
 6. To find the quotient of 1/10 ÷ 2, you'll need to write two as a fraction (2/1) and then find its reciprocal (1/2). Next, multiply 1/10 x 1/2 to get 1/20.
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