Reciprocals of Fractions and Mixed Numbers | Grade 7 Math | LevelKite

You already know that $\frac{1}{4}$ is one of $4$ equal parts of a whole. So $4$ copies of $\frac{1}{4}$ put the whole back together: $4 \times \frac{1}{4} = \frac{4}{4} = 1$ .

The numbers $4$ and $\frac{1}{4}$ are partners: multiply them and you get exactly $1$ . In this lesson you will learn the name for such partners and how to find the partner of a fraction such as $\frac{5}{8}$ or of a number such as $2\frac{3}{4}$ .

Now try $\frac{2}{3}$ . How many copies of $\frac{2}{3}$ make exactly $1$ whole?

Three copies make $\frac{2}{3} + \frac{2}{3} + \frac{2}{3} = \frac{6}{3} = 2$ . That is twice as much as you want, so you need half as many copies. Half of $3$ copies is $\frac{3}{2}$ copies, or one and a half copies. One full copy is $\frac{2}{3}$ and half a copy is $\frac{1}{3}$ , and together they make $\frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1$ .

You get the same result when you multiply the top numbers together and the bottom numbers together:

\frac{3}{2} \times \frac{2}{3} = \frac{3 \times 2}{2 \times 3} = \frac{6}{6} = 1

Compare the partners. The partner of $4$ , which equals $\frac{4}{1}$ , is $\frac{1}{4}$ . The partner of $\frac{2}{3}$ is $\frac{3}{2}$ . Each time, the top number and the bottom number trade places. That pattern gives you a fast way to find the partner of a number, as long as the number is not $0$ .

Key concept

A nonzero number times its reciprocal is 1

The reciprocal of a number is the number you multiply it by to get a product (the result of multiplying) of exactly $1$ . Only a nonzero number, meaning a number that is not $0$ , has a reciprocal, because $0$ times any number is $0$ , never $1$ .

Let $a$ and $b$ each stand for a number that is not $0$ . The fraction $\frac{a}{b}$ has numerator (top number) $a$ and denominator (bottom number) $b$ . Its reciprocal is $\frac{b}{a}$ : you swap the numerator and the denominator.

\frac{a}{b} \times \frac{b}{a} = \frac{a \times b}{b \times a} = 1 \quad \text{when } a \neq 0 \text{ and } b \neq 0

Why does the swap work? Multiply $\frac{a}{b}$ by $\frac{b}{a}$ . The numerators multiply to $a \times b$ , and the denominators multiply to $b \times a$ . Changing the order in a multiplication does not change the product, so the top and the bottom are the same nonzero number. A nonzero number divided by itself is $1$ .

You need both conditions. If $b$ were $0$ , then $\frac{a}{b}$ would have no value, because a denominator can never be $0$ . If $a$ were $0$ , then $\frac{a}{b}$ would equal $0$ , and the swapped fraction $\frac{b}{a}$ would have $0$ as its denominator. That agrees with the key idea: $0$ has no reciprocal.

Worked example

Reciprocal of a fraction

Find the reciprocal of $\frac{5}{8}$ .

Name the two parts of the fraction. The numerator is 5 and the denominator is 8. Neither one is zero, so the fraction is a nonzero number and it has a reciprocal.
$5 \neq 0 \quad \text{and} \quad 8 \neq 0$
Swap the numerator and the denominator. You can do this because the new denominator, 5, is not zero.
$\text{reciprocal of } \frac{5}{8} = \frac{8}{5}$
Check with the definition: a nonzero number times its reciprocal must equal 1. Multiply the numerators together and the denominators together. The top and the bottom are both 40, and a nonzero number divided by itself is 1.
$\frac{5}{8} \times \frac{8}{5} = \frac{5 \times 8}{8 \times 5} = \frac{40}{40} = 1$

Answer

The reciprocal of $\frac{5}{8}$ is $\frac{8}{5}$ , because $\frac{5}{8} \times \frac{8}{5} = 1$ .

A mixed number shows a whole number and a fraction between $0$ and $1$ written together, such as $2\frac{3}{4}$ . It means $2 + \frac{3}{4}$ , and it is one single quantity. Because $2\frac{3}{4}$ is greater than $1$ , it is not $0$ , so it has a reciprocal.

A mixed number does not show one numerator over one denominator, so you cannot swap yet. First rewrite all of it as an improper fraction, which is a fraction whose numerator is greater than or equal to its denominator. Then check that the numerator and the denominator are both nonzero, and swap them.

Worked example

Reciprocal of a mixed number

Find the reciprocal of $2\frac{3}{4}$ .

Write the mixed number as its two parts added together. It means 2 wholes plus 3 fourths of a whole.
$2\frac{3}{4} = 2 + \frac{3}{4}$
Rewrite the 2 wholes in fourths so that both parts count pieces of the same size. One whole is 4 fourths, so 2 wholes are 2 times 4, or 8, fourths. Add the numerators, because 8 fourths plus 3 fourths make 11 fourths.
$2 + \frac{3}{4} = \frac{2 \times 4}{4} + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$
Now the number has one numerator, 11, and one denominator, 4. Both are nonzero, so you can swap them.
$\text{reciprocal of } \frac{11}{4} = \frac{4}{11}$
Check with the definition: a nonzero number times its reciprocal must equal 1. Use the fraction form of the mixed number. The top and the bottom are both 44, and a nonzero number divided by itself is 1.
$\frac{11}{4} \times \frac{4}{11} = \frac{11 \times 4}{4 \times 11} = \frac{44}{44} = 1$
Check the size as well. The starting number is greater than 1, so less than one full copy of it makes 1. The reciprocal must be between 0 and 1, and it is, because 4 is less than 11.
$0 < \frac{4}{11} < 1$

Answer

The reciprocal of $2\frac{3}{4}$ is $\frac{4}{11}$ , because $\frac{11}{4} \times \frac{4}{11} = 1$ .

Common mistake: flipping only the fraction part. Someone might keep the $2$ and flip only the $\frac{3}{4}$ , turning $2\frac{3}{4}$ into $2\frac{4}{3}$ . Test that with the definition. $2\frac{4}{3}$ means $2 + \frac{4}{3} = \frac{6}{3} + \frac{4}{3} = \frac{10}{3}$ , and $\frac{11}{4} \times \frac{10}{3} = \frac{11 \times 10}{4 \times 3} = \frac{110}{12}$ , which is more than $9$ . The reciprocal of a nonzero number must give a product of exactly $1$ .

This fails because flipping $\frac{3}{4}$ gives the reciprocal of $\frac{3}{4}$ alone, not of $2 + \frac{3}{4}$ . The swap rule works on one fraction that stands for the entire number. So write $2\frac{3}{4} = \frac{11}{4}$ first, then swap to get $\frac{4}{11}$ .

Each nonzero number and its reciprocal multiply to 1, so each one is the reciprocal of the other.

Number	As one fraction	Reciprocal	Check
$\frac{1}{6}$	$\frac{1}{6}$	$\frac{6}{1} = 6$	$\frac{1}{6} \times \frac{6}{1} = \frac{1 \times 6}{6 \times 1} = \frac{6}{6} = 1$
$6$	$\frac{6}{1}$	$\frac{1}{6}$	$\frac{6}{1} \times \frac{1}{6} = \frac{6 \times 1}{1 \times 6} = \frac{6}{6} = 1$
$\frac{7}{3}$	$\frac{7}{3}$	$\frac{3}{7}$	$\frac{7}{3} \times \frac{3}{7} = \frac{7 \times 3}{3 \times 7} = \frac{21}{21} = 1$
$1\frac{2}{5}$	$\frac{5}{5} + \frac{2}{5} = \frac{7}{5}$	$\frac{5}{7}$	$\frac{7}{5} \times \frac{5}{7} = \frac{7 \times 5}{5 \times 7} = \frac{35}{35} = 1$
$3\frac{1}{3}$	$\frac{9}{3} + \frac{1}{3} = \frac{10}{3}$	$\frac{3}{10}$	$\frac{10}{3} \times \frac{3}{10} = \frac{10 \times 3}{3 \times 10} = \frac{30}{30} = 1$

Here is the idea to keep. A reciprocal is a multiplying partner: a nonzero number times its reciprocal is exactly $1$ . To find it, write the number as one fraction with a nonzero numerator and a nonzero denominator, then swap the two. For a mixed number, rewrite all of it as an improper fraction first. The reciprocal keeps the sign of the number, and $0$ has no reciprocal. To check any answer, multiply: the product must be $1$ .

Next, you will add and subtract fractions, and then multiply fractions and mixed numbers.