Divide Fractions by Counting Groups | Grade 6 Math | LevelKite

Start with whole numbers, which you already know. In $12 \div 3 = 4$ , the number you divide, $12$ , is the dividend. The number you divide by, $3$ , is the divisor. The result, $4$ , is the quotient.

In this lesson, every division asks one question: how many groups, each the size of the divisor, fit in the dividend? Here one group is $3$ , and $4$ groups of $3$ fit in $12$ . So the quotient counts groups.

No divisor in this lesson is zero. Multiplication and division are inverse operations: one undoes the other. Dividing $12$ by $3$ gives $4$ , and multiplying $4$ by $3$ gives back $12$ . So an exact quotient, one that is not rounded, times the divisor equals the dividend. You will use this check in every example.

With fractions, one whole is the reference unit. Every number in a problem is measured in the same whole. A fraction's denominator tells how many equal parts the whole is cut into, and its numerator counts how many of those parts you have.

To divide, think of each whole as cut into parts of one common size, so that the dividend and the divisor are each a whole number of parts. A common denominator, one denominator you can use for both numbers, sets that size. To use it, you may rename a fraction as an equivalent fraction, which names the same amount with a different denominator. Multiply the numerator and the denominator by the same nonzero whole number. That is the same as multiplying by a fraction equal to $1$ , so the amount stays the same. Since the dividend and the divisor keep their amounts, the quotient stays the same too.

Then the division becomes a count of parts: how many groups of the divisor's parts fit in the dividend's parts?

Worked example

Example 1: Groups of one part

Use a model to find $\frac{3}{4} \div \frac{1}{8}$ .

Name the roles so you know what to count. The dividend, 3/4, is the amount you start with. The divisor, 1/8, is the size of one group. It is a unit fraction, which means its numerator is 1. The quotient will count how many groups of 1/8 fit in 3/4.
$\frac{3}{4} \div \frac{1}{8}$
Use eighths as the common part size, because 1/8 is already in eighths and 8 is a multiple of 4. Rename 3/4 by multiplying its numerator and denominator by 2. That multiplies it by 2/2, which equals 1, so the amount stays the same.
$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$
Show the parts in a model so you can count groups. In the figure below, one whole has 8 equal parts, and the first 6 are shaded: that is the dividend, 6 eighths. Each solid bracket marks one complete group of 1 part, which is 1/8.
Count the groups. The 6 shaded parts make 6 complete groups. No shaded part is left outside them, so the quotient is 6 complete groups plus 0 of another group.
$\frac{6}{8} \div \frac{1}{8} = 6 \div 1 = 6$
Check by multiplying the quotient by the divisor. Multiplication undoes division, so the product should be the dividend.
$6 \times \frac{1}{8} = \frac{6}{8} = \frac{3}{4}$

Answer

$\frac{3}{4} \div \frac{1}{8} = 6$ , so $6$ groups of $\frac{1}{8}$ fit in $\frac{3}{4}$ .

One whole in eighths: the

6

shaded parts are

\frac{3}{4}

, and each bracket is one group of

\frac{1}{8}

.

Key concept

A quotient counts divisor-sized groups

When the dividend and the divisor are parts of one common size of the same whole, one group is the divisor's number of parts. The quotient counts the groups in the dividend: the complete groups, plus the fraction of one more group that any parts outside them fill. That fraction is the dividend's parts in the last group out of the parts in one group. Read it against one group, never against the whole. You keep counting groups past the end of a whole, so a group can cross into the next whole. A group can also be longer than the dividend. Because multiplication undoes division, an exact quotient times the nonzero divisor equals the dividend.

\text{quotient} \times \text{divisor} = \text{dividend}

Worked example

Example 2: Part of one more group

A loop trail in a park is $\frac{2}{3}$ mile long. You walk $1\frac{1}{2}$ miles on it. How many loops do you walk?

Name the roles. The dividend, 1 1/2 miles, is the distance you walk. The divisor, 2/3 mile, is the length of one loop, so one group is one loop. The quotient will count loops.
$1\frac{1}{2} \div \frac{2}{3}$
Let one mile be the whole. Use sixths as the common part size, because 6 is a multiple of both 2 and 3. Write 1 1/2 in halves, then rename both numbers as equivalent fractions in sixths. Each rename keeps the amount the same.
$1\frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}, \quad \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$
In the figure below, each whole is one mile with 6 equal parts, and the first 9 parts are shaded. Each bracket is one loop of 4 parts, and the group drawn by itself below the wholes shows one loop. The second loop crosses from the first mile into the second.
Count the groups. Two solid brackets mark 2 complete loops, which use 8 shaded parts. The last bracket is dashed: it is a full loop of 4 parts, but only 1 of them is shaded. That part is 1/4 of one more loop.
$\frac{9}{6} \div \frac{4}{6} = 2 + \frac{1}{4} = 2\frac{1}{4}$
Check by multiplication, which undoes division: the quotient times the divisor should equal the dividend.
$2\frac{1}{4} \times \frac{2}{3} = \frac{9}{4} \times \frac{2}{3} = \frac{9 \times 2}{4 \times 3} = \frac{18}{12} = \frac{3}{2} = 1\frac{1}{2}$

Answer

$1\frac{1}{2} \div \frac{2}{3} = 2\frac{1}{4}$ , so you walk $2\frac{1}{4}$ loops.

Each whole is one mile in sixths: the

9

shaded parts are

1\frac{1}{2}

miles, and each bracket is one

\frac{2}{3}

-mile loop of

4

parts.

Worked example

Example 3: Less than one group

A movie is $2\frac{1}{2}$ hours long. You have watched it for $2$ hours. What fraction of the movie have you watched?

Name the roles. The dividend, 2 hours, is the time you have watched. The divisor, 2 1/2 hours, is the length of the movie, so one group is one movie. The quotient counts how many movies fit in 2 hours.
$2 \div 2\frac{1}{2}$
Let one hour be the whole. Use halves as the common part size, because both numbers are a whole number of half hours.
$2 = \frac{2}{1} = \frac{2 \times 2}{1 \times 2} = \frac{4}{2}, \quad 2\frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$
In the figure below, each whole is one hour with 2 equal parts, and the first 4 parts are shaded. One group is 5 parts, so it runs into the third hour and is longer than the dividend. Its bracket is dashed because only some of its parts are shaded.
Count the groups. There are no complete groups. The 4 shaded parts fill 4 of the 5 parts in the only group. Read this against one movie, not against one hour.
$\frac{4}{2} \div \frac{5}{2} = 0 + \frac{4}{5} = \frac{4}{5}$
Check by multiplying, the inverse of dividing: the quotient times the divisor should give back the dividend, 2.
$\frac{4}{5} \times 2\frac{1}{2} = \frac{4}{5} \times \frac{5}{2} = \frac{4 \times 5}{5 \times 2} = \frac{20}{10} = 2$

Answer

$2 \div 2\frac{1}{2} = \frac{4}{5}$ , so you have watched $\frac{4}{5}$ of the movie.

Each whole is one hour in halves: the

4

shaded parts are

2

hours, and the dashed bracket is one

2\frac{1}{2}

-hour movie of

5

parts.

A drawing has to stop somewhere, but division does not. When the groups would need too many wholes to draw, count the same parts in writing. This counting works for any positive dividend and divisor, including whole numbers and numbers greater than $1$ .

Try $\frac{7}{2} \div 2$ . The dividend, $\frac{7}{2}$ , is the amount you start with. The divisor, $2$ , is the size of one group, and the quotient counts groups of $2$ . In halves, $\frac{7}{2}$ is $7$ halves, and $2 = \frac{2}{1} = \frac{2 \times 2}{1 \times 2} = \frac{4}{2}$ is $4$ halves. Drawing two groups of $4$ halves would take $4$ wholes, so count in writing. One complete group uses $4$ of the $7$ halves. The other $3$ halves fill $\frac{3}{4}$ of one more group. So $\frac{7}{2} \div 2 = 1 + \frac{3}{4} = 1\frac{3}{4}$ . Check: $1\frac{3}{4} \times 2 = \frac{7}{4} \times \frac{2}{1} = \frac{14}{4} = \frac{7}{2}$ .

Takeaway: To divide with a model, write the dividend and the nonzero divisor as parts of one common size of the same whole. Then count how many divisor-sized groups fit in the dividend. If the last group is only partly filled, name that part as a fraction of one group, never of the whole. Check that the exact quotient times the divisor equals the dividend.

Next, you will learn what the reciprocal of a number is and how to find it. You will also divide whole numbers and unit fractions, and divide two fractions.