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!

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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 'upside-down' 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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