Dividing Fractions and Mixed Numbers | Grade 7 Math | LevelKite

You can already multiply fractions. Here you use that skill to divide them. Start with what a division asks.

A division such as $6 \div 2 = 3$ asks how many copies of $2$ make $6$ . The number being divided, $6$ , is the dividend. The number you divide by, $2$ , is the divisor. The answer, $3$ , is the quotient. It is correct because $3 \times 2 = 6$ .

Fractions work the same way. $\frac{3}{4} \div \frac{3}{8}$ asks how many copies of $\frac{3}{8}$ make $\frac{3}{4}$ . Write $\frac{3}{4}$ in eighths: $\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$ . Six eighths is two groups of three eighths, so the quotient is $2$ . Multiplying confirms it: $2 \times \frac{3}{8} = \frac{6}{8} = \frac{3}{4}$ .

A divisor is never zero. For any other divisor, the quotient is the number that you multiply by the divisor to get the dividend.

Counting groups gets hard when the numbers do not line up so neatly. You need a method that works every time, and it uses the reciprocal. As you know, the reciprocal of a nonzero number (any number except zero) is the number you multiply it by to get $1$ . For a nonzero fraction, the numerator (top number) and the denominator (bottom number) trade places. So the reciprocal of $\frac{3}{8}$ is $\frac{8}{3}$ , because $\frac{3}{8} \times \frac{8}{3} = \frac{24}{24} = 1$ .

Let $q$ stand for the unknown quotient of $\frac{3}{4} \div \frac{3}{8}$ . By the meaning of division, $q \times \frac{3}{8} = \frac{3}{4}$ . Now multiply both sides by $\frac{8}{3}$ . The two sides stay equal because you do the same thing to each. On the left, $\frac{3}{8} \times \frac{8}{3} = 1$ and $q \times 1 = q$ , so only $q$ is left. On the right, you get $\frac{3}{4} \times \frac{8}{3}$ . The line below finishes the work, and the result is the same $2$ you found by counting groups.

q = \frac{3}{4} \times \frac{8}{3} = \frac{3 \times 8}{4 \times 3} = \frac{24}{12} = 2

Key concept

Divide by multiplying by the reciprocal

To divide by a nonzero number, multiply by its reciprocal. Keep the dividend as it is and replace only the divisor. This works because a nonzero number times its reciprocal is $1$ , so multiplying by the reciprocal undoes multiplying by the divisor.

The divisor does not have to be a fraction with numerator $1$ . It can be a fraction such as $\frac{2}{3}$ , a mixed number (a whole number together with a fraction) such as $1\frac{3}{4}$ , a decimal such as $0.4$ , or a negative number such as $-\frac{4}{9}$ . It can never be $0$ , because $0$ has no reciprocal: no number times $0$ equals $1$ .

In the formula, $a$ , $b$ , $c$ and $d$ stand for numbers. The denominators $b$ and $d$ are not $0$ . The number $c$ is not $0$ either, so the divisor $\frac{c}{d}$ is not $0$ .

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \qquad (b \neq 0,\ c \neq 0,\ d \neq 0)

Worked example

A fraction divided by a fraction

Find the quotient $\frac{5}{6} \div \frac{2}{3}$ .

Find the divisor, the number after the division sign. It is two thirds. It is not zero, so it has a reciprocal, three halves. Multiplying the two gives 1.
$\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1$
Rewrite the division as multiplication by that reciprocal, and leave the dividend, five sixths, unchanged. This is valid because multiplying by the reciprocal undoes multiplying by the nonzero divisor.
$\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2}$
Multiply the numerators together and the denominators together, as you do whenever you multiply two fractions.
$\frac{5}{6} \times \frac{3}{2} = \frac{5 \times 3}{6 \times 2} = \frac{15}{12}$
Simplify. Both 15 and 12 are divisible by 3, and dividing the numerator and the denominator by the same nonzero number does not change the value. Four fourths make 1, so five fourths is 1 and one fourth.
$\frac{15}{12} = \frac{15 \div 3}{12 \div 3} = \frac{5}{4} = 1\frac{1}{4}$
Check with the meaning of division: the quotient times the nonzero divisor should give back the dividend. Dividing 10 and 12 by 2 gives five sixths, so it does.
$\frac{5}{4} \times \frac{2}{3} = \frac{5 \times 2}{4 \times 3} = \frac{10}{12} = \frac{5}{6}$

Answer

The quotient $\frac{5}{6} \div \frac{2}{3}$ is $\frac{5}{4}$ , which equals $1\frac{1}{4}$ .

Watch out for the wrong reciprocal. For $\frac{3}{4} \div \frac{3}{8}$ , a common mistake is to take the reciprocal of the dividend instead of the divisor: $\frac{4}{3} \times \frac{3}{8} = \frac{12}{24} = \frac{1}{2}$ . The check fails, because $\frac{1}{2} \times \frac{3}{8} = \frac{3}{16}$ , which is not the dividend $\frac{3}{4}$ .

Why does it fail? The reciprocal is there to undo multiplying by the divisor, so it must be the reciprocal of the divisor. Using the reciprocal of the dividend answers the reverse question, $\frac{3}{8} \div \frac{3}{4}$ , which does equal $\frac{1}{2}$ . Keep the dividend, and replace only the nonzero divisor with its reciprocal: $\frac{3}{4} \times \frac{8}{3} = \frac{24}{12} = 2$ .

The numbers in a division are not always plain fractions. The rule needs each number written as a single fraction, so get the dividend and the divisor ready first.

Write a mixed number as one fraction, as you did when you multiplied mixed numbers.
Write a whole number over $1$ , so $5$ becomes $\frac{5}{1}$ .
Write a decimal that ends, such as $0.6$ , as a fraction by using its place value: $0.6$ is six tenths, or $\frac{6}{10}$ . Then simplify.
Keep a negative sign with its number. A negative number times a positive number is negative, so the result could never be $1$ . That is why the reciprocal of a negative number is also negative.

None of this changes the rule. As long as the divisor is not zero, you still multiply by its reciprocal.

Worked example

A mixed number divided by a mixed number

Find the quotient $4\frac{2}{3} \div 1\frac{3}{4}$ .

Write each mixed number as one fraction, because the rule needs a single numerator and a single denominator for each number. Four wholes are 12 thirds, and 2 more thirds make 14 thirds. One whole is 4 fourths, and 3 more fourths make 7 fourths.
$4\frac{2}{3} = \frac{4 \times 3 + 2}{3} = \frac{14}{3}, \qquad 1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}$
The divisor, seven fourths, is not zero, so it has a reciprocal, four sevenths. Keep the dividend and multiply by that reciprocal.
$\frac{14}{3} \div \frac{7}{4} = \frac{14}{3} \times \frac{4}{7}$
Multiply the numerators together and the denominators together.
$\frac{14}{3} \times \frac{4}{7} = \frac{14 \times 4}{3 \times 7} = \frac{56}{21}$
Simplify. Both 56 and 21 are divisible by 7, so divide the numerator and the denominator by 7. Six thirds make 2, so eight thirds is 2 and two thirds.
$\frac{56}{21} = \frac{56 \div 7}{21 \div 7} = \frac{8}{3} = 2\frac{2}{3}$
Check: the quotient times the nonzero divisor should give back the dividend. Dividing 56 and 12 by 4 gives fourteen thirds, so it does.
$\frac{8}{3} \times \frac{7}{4} = \frac{8 \times 7}{3 \times 4} = \frac{56}{12} = \frac{14}{3} = 4\frac{2}{3}$

Answer

The quotient $4\frac{2}{3} \div 1\frac{3}{4}$ is $\frac{8}{3}$ , which equals $2\frac{2}{3}$ .

Worked example

A negative decimal divisor

Find the quotient $1\frac{1}{5} \div (-0.4)$ .

Write both numbers as fractions. One whole is 5 fifths, and 1 more fifth makes 6 fifths. The decimal is negative four tenths, and dividing 4 and 10 by 2 simplifies it to negative two fifths.
$1\frac{1}{5} = \frac{1 \times 5 + 1}{5} = \frac{6}{5}, \qquad -0.4 = -\frac{4}{10} = -\frac{2}{5}$
The divisor is not zero, so it has a reciprocal. Swap the numerator and the denominator and keep the negative sign. A negative times a negative is positive, so the two multiply to 1.
$\left(-\frac{2}{5}\right) \times \left(-\frac{5}{2}\right) = \frac{2 \times 5}{5 \times 2} = \frac{10}{10} = 1$
Keep the dividend and multiply by the reciprocal of the divisor.
$\frac{6}{5} \div \left(-\frac{2}{5}\right) = \frac{6}{5} \times \left(-\frac{5}{2}\right)$
Multiply. A positive number times a negative number is negative, so the result is negative. Then simplify: 30 divided by 10 is 3.
$\frac{6}{5} \times \left(-\frac{5}{2}\right) = -\frac{6 \times 5}{5 \times 2} = -\frac{30}{10} = -3$
Check: the quotient times the nonzero divisor should give back the dividend. A negative times a negative is positive, and six fifths is 1 and one fifth, so it does.
$-3 \times \left(-\frac{2}{5}\right) = \frac{3 \times 2}{5} = \frac{6}{5} = 1\frac{1}{5}$

Answer

The quotient $1\frac{1}{5} \div (-0.4)$ is $-3$ .

One rule for every kind of nonzero divisor

Kind of divisor	Division	Divisor as a fraction	Multiply by its reciprocal	Quotient, simplified
Fraction	$6 \div \frac{3}{4}$	$\frac{3}{4}$	$\frac{6}{1} \times \frac{4}{3} = \frac{24}{3}$	$8$
Mixed number	$\frac{5}{6} \div 1\frac{2}{3}$	$\frac{1 \times 3 + 2}{3} = \frac{5}{3}$	$\frac{5}{6} \times \frac{3}{5} = \frac{15}{30}$	$\frac{1}{2}$
Whole number	$\frac{4}{5} \div 2$	$\frac{2}{1}$	$\frac{4}{5} \times \frac{1}{2} = \frac{4}{10}$	$\frac{2}{5}$
Decimal	$\frac{9}{10} \div 0.6$	$\frac{6}{10} = \frac{3}{5}$	$\frac{9}{10} \times \frac{5}{3} = \frac{45}{30}$	$\frac{3}{2} = 1\frac{1}{2}$
Negative fraction	$\frac{2}{3} \div \left(-\frac{4}{9}\right)$	$-\frac{4}{9}$	$\frac{2}{3} \times \left(-\frac{9}{4}\right) = -\frac{18}{12}$	$-\frac{3}{2} = -1\frac{1}{2}$

Here is the idea to keep. Dividing by a nonzero number gives the same result as multiplying by its reciprocal, because a nonzero number times its reciprocal is $1$ . So write both numbers as fractions, keep the dividend, multiply by the reciprocal of the divisor, simplify, and check that the quotient times the divisor gives the dividend. This works for every nonzero divisor you can write as a fraction, whether it is a fraction, a mixed number, a decimal or a negative number. It never works for a divisor of $0$ , because $0$ has no reciprocal.

Next, you will estimate the results of adding, subtracting, multiplying and dividing fractions, and you will solve word problems with fractions and mixed numbers.