Reciprocals and Dividing Fractions | Grade 6 Math | LevelKite

Multiplication and division undo each other, as long as you never divide by zero. For example, $12 \div 4 = 3$ because $3 \times 4 = 12$ . In that division, $12$ is the dividend (the number being divided), $4$ is the divisor (the number you divide by), and $3$ is the quotient (the result).

The same link holds when the numbers are fractions. In this lesson you will use it to divide with fractions by multiplying. First you need one new tool: the reciprocal.

Two numbers are reciprocals of each other when their product (the result of multiplying them) is $1$ . For example, $\frac{2}{3}$ and $\frac{3}{2}$ are reciprocals, because $\frac{2}{3} \times \frac{3}{2} = \frac{2 \times 3}{3 \times 2} = \frac{6}{6} = 1$ .

To find the reciprocal of a fraction that is not zero, swap its numerator (the top number) and its denominator (the bottom number). This works because the product then has the same two numbers multiplied on the top and on the bottom, so it equals $1$ .

Some numbers need one step before you swap:

A whole number that is not zero. Write it over $1$ first. Since $5 = \frac{5}{1}$ , the reciprocal of $5$ is $\frac{1}{5}$ .
A mixed number (a whole number plus a fraction, such as $2\frac{1}{3}$ ). Rewrite it as one fraction first, then swap.

One number has no reciprocal at all: zero. The product of $0$ and any number is $0$ , never $1$ .

Reciprocals of four kinds of numbers

Kind of number	Number	Reciprocal	Check: the product is 1
A fraction less than 1	$\frac{2}{3}$	$\frac{3}{2}$	$\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1$
A fraction greater than 1	$\frac{9}{4}$	$\frac{4}{9}$	$\frac{9}{4} \times \frac{4}{9} = \frac{36}{36} = 1$
A whole number that is not zero	$5 = \frac{5}{1}$	$\frac{1}{5}$	$\frac{5}{1} \times \frac{1}{5} = \frac{5}{5} = 1$
A fraction with numerator 1	$\frac{1}{8}$	$\frac{8}{1} = 8$	$\frac{1}{8} \times \frac{8}{1} = \frac{8}{8} = 1$

Worked example

Reciprocal of a mixed number

Find the reciprocal of $2\frac{1}{3}$ .

Rewrite the mixed number as one fraction, because the swap only works on a number written as a single fraction. Each whole is 3 thirds, so 2 wholes are 6 thirds. Then add the 1 third.
$2\frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}$
Swap the numerator and the denominator. The number is not zero, so it has a reciprocal.
$\text{reciprocal of } \frac{7}{3} = \frac{3}{7}$
Check with the meaning of reciprocal: the product of the two numbers must be 1.
$\frac{7}{3} \times \frac{3}{7} = \frac{7 \times 3}{3 \times 7} = \frac{21}{21} = 1$

Answer

The reciprocal of $2\frac{1}{3}$ is $\frac{3}{7}$ .

Now you are ready to divide. Each fraction in this lesson is measured against the same whole, the number $1$ .

A unit fraction is a fraction whose numerator is $1$ , such as $\frac{1}{5}$ . Its reciprocal is a whole number. For example, $\frac{1}{5}$ and $5$ are reciprocals, because $5 \times \frac{1}{5} = \frac{5}{5} = 1$ .

A division can ask one of two questions, depending on the situation:

How many groups? $6 \div \frac{1}{5}$ can ask how many groups of size $\frac{1}{5}$ fit in $6$ . Then the quotient is a count of groups.
How big is one share? $\frac{1}{2} \div 3$ can ask how big one share is when $\frac{1}{2}$ is split into $3$ equal shares. Then the quotient is the size of one share.

The next two examples answer one question each.

Worked example

A whole number divided by a unit fraction

Find $6 \div \frac{1}{5}$ . Read it as: how many groups of size $\frac{1}{5}$ fit in $6$ ?

Count the groups in one whole first. It takes 5 groups of one fifth to make 1 whole. That is why 5 is the reciprocal of one fifth.
$5 \times \frac{1}{5} = \frac{5}{5} = 1$
The dividend is 6 wholes, and each whole holds 5 groups of one fifth, so multiply 6 by 5. This multiplies the dividend by the reciprocal of the divisor.
$6 \div \frac{1}{5} = 6 \times 5 = 30$
Check with multiplication. The divisor is not zero and the quotient is exact (not rounded), so the quotient times the divisor must give back the dividend.
$30 \times \frac{1}{5} = \frac{30}{5} = 6$

Answer

The quotient is $30$ : exactly $30$ groups of size $\frac{1}{5}$ fit in $6$ . Here the quotient is a count of groups.

Worked example

A unit fraction divided by a whole number

Find $\frac{1}{2} \div 3$ . Read it as: $\frac{1}{2}$ is split into $3$ equal shares. How big is one share?

Rename the dividend so that it splits evenly into 3 shares. Multiply its numerator and its denominator by 3. The value stays the same, because that is the same as multiplying by 1.
$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$
Split the 3 sixths into 3 equal shares. Each share gets 1 sixth.
$\frac{3}{6} \div 3 = \frac{1}{6}$
Compare this with multiplying the dividend by the reciprocal of the divisor. The divisor 3 is 3 over 1, so its reciprocal is one third. The product is the same number.
$\frac{1}{2} \times \frac{1}{3} = \frac{1 \times 1}{2 \times 3} = \frac{1}{6}$
Check as before. The divisor is not zero and the quotient is exact, so the quotient times the divisor must give back the dividend.
$\frac{1}{6} \times 3 = \frac{3}{6} = \frac{1}{2}$

Answer

The quotient is $\frac{1}{6}$ : each of the $3$ equal shares has size $\frac{1}{6}$ . Here the quotient is the size of one share, not a count of groups.

In the last two examples, dividing gave the same result as multiplying the dividend by the reciprocal of the divisor. That is not luck, and it does not depend on unit fractions. Here is the reason.

Multiplication and division are inverse operations, which means each one undoes the other. So when the divisor is not zero, the quotient is the one number that gives back the dividend when you multiply it by the divisor. You used this in each check: $30 \times \frac{1}{5} = 6$ and $\frac{1}{6} \times 3 = \frac{1}{2}$ .

Use that on $\frac{3}{4} \div \frac{2}{5}$ . Let $q$ stand for the quotient. Then $q$ must make $q \times \frac{2}{5} = \frac{3}{4}$ true. The reciprocal of the divisor $\frac{2}{5}$ is $\frac{5}{2}$ . Test $q = \frac{3}{4} \times \frac{5}{2}$ : multiply it by the divisor and see what comes back. You may regroup the factors (the numbers being multiplied), because regrouping does not change a product.

\left(\frac{3}{4} \times \frac{5}{2}\right) \times \frac{2}{5} = \frac{3}{4} \times \left(\frac{5}{2} \times \frac{2}{5}\right) = \frac{3}{4} \times 1 = \frac{3}{4}

The dividend came back. The reciprocal $\frac{5}{2}$ and the divisor $\frac{2}{5}$ multiplied to $1$ , and multiplying by $1$ left $\frac{3}{4}$ unchanged. So $\frac{3}{4} \times \frac{5}{2}$ is the quotient:

$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8}$

The same reasoning works for any dividend and any divisor that is not zero, because every number other than zero has a reciprocal.

Key concept

Divide by multiplying by the reciprocal

Dividing by a number that is not zero gives the same result as multiplying by its reciprocal. Keep the dividend. Change only the divisor to its reciprocal. Then multiply.

It works because the divisor times its reciprocal is $1$ , so the quotient times the divisor gives back the dividend. That is also your check: when the quotient is exact, as every quotient in this lesson is, quotient $\times$ divisor $=$ dividend.

In the formula, $\frac{a}{b}$ is the dividend and $\frac{c}{d}$ is the divisor. The letters stand for whole numbers, and $b$ , $c$ and $d$ are not zero.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}

Worked example

A fraction divided by a fraction

Find $\frac{5}{6} \div \frac{2}{3}$ . Read it as: how many groups of size $\frac{2}{3}$ fit in $\frac{5}{6}$ ?

Find the reciprocal of the divisor. The divisor is not zero, so swap its numerator and its denominator.
$\text{reciprocal of } \frac{2}{3} = \frac{3}{2}$
Keep the dividend and multiply it by that reciprocal. Multiply the numerators, then multiply the denominators.
$\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{5 \times 3}{6 \times 2} = \frac{15}{12}$
Simplify the result. Divide the numerator and the denominator by 3, a number that divides both evenly. Dividing both by the same nonzero number keeps the value the same.
$\frac{15}{12} = \frac{15 \div 3}{12 \div 3} = \frac{5}{4}$
Check: the exact quotient times the nonzero divisor must give back the dividend.
$\frac{5}{4} \times \frac{2}{3} = \frac{5 \times 2}{4 \times 3} = \frac{10}{12} = \frac{5}{6}$

Answer

The quotient is $\frac{5}{4}$ , which equals $1\frac{1}{4}$ . So $1\frac{1}{4}$ groups of size $\frac{2}{3}$ fit in $\frac{5}{6}$ : one full group and $\frac{1}{4}$ of another group.

Take the reciprocal of the divisor, not the dividend. A common mistake in $\frac{5}{6} \div \frac{2}{3}$ is to swap the first fraction instead and write $\frac{6}{5} \times \frac{2}{3} = \frac{12}{15} = \frac{4}{5}$ . The check exposes it: $\frac{4}{5} \times \frac{2}{3} = \frac{8}{15}$ , but the dividend is $\frac{5}{6}$ . In thirtieths, $\frac{8}{15} = \frac{16}{30}$ and $\frac{5}{6} = \frac{25}{30}$ , so they are not equal.

It fails because only the divisor's own reciprocal multiplies with the divisor to make $1$ . The reciprocal of the dividend does not undo the divisor. In fact, $\frac{6}{5} \times \frac{2}{3}$ answers a different division, $\frac{2}{3} \div \frac{5}{6}$ .

The big idea: two numbers are reciprocals when their product is $1$ . To divide by any number that is not zero, keep the dividend and multiply it by the reciprocal of the divisor. Then check: the exact quotient times the divisor must equal the dividend. Read the quotient the way the question asks, as a count of groups the size of the divisor or as the size of one equal share.

Next, you will use this method to solve word problems, interpret quotients of fractions, and find mistakes in fraction division.