Solve, Interpret, and Check Fraction Division | Grade 6 Math | LevelKite

You already know how to divide one fraction by another. In this lesson you put that skill to work. You will solve word problems, explain what each answer means, and find mistakes in division work.

Three words name the parts of a division. In $\frac{3}{4} \div \frac{1}{8} = 6$ , the dividend is $\frac{3}{4}$ , the number being divided. The divisor is $\frac{1}{8}$ , the number you divide by. The quotient is $6$ , the result. The divisor is never $0$ in this lesson, because division by $0$ is not defined.

The word problems in this lesson ask one of two questions. The numbers alone do not tell you which one. The situation does.

How many groups? You know a total and the size of one group. You divide the total by the group size. The quotient is a count of groups, and each group is the size of the divisor.
How big is one share? You know a total and a number of equal shares. You divide the total by the number of shares. The quotient is the size of one equal share.

In both questions the total is the dividend. So begin each problem by asking: What is the total? Am I dividing by a group size or by a number of shares?

One ribbon, two different questions

Situation	Division	What the quotient means
You cut a $\frac{3}{4}$ -yard ribbon into $\frac{1}{8}$ -yard pieces. How many pieces do you get?	$\frac{3}{4} \div \frac{1}{8} = 6$	A count of groups: $6$ pieces, each $\frac{1}{8}$ yard long
You cut a $\frac{3}{4}$ -yard ribbon into $6$ equal pieces. How long is each piece?	$\frac{3}{4} \div 6 = \frac{1}{8}$	The size of one share: each piece is $\frac{1}{8}$ yard long

\left(\frac{3}{4} \times \frac{8}{1}\right) \times \frac{1}{8} = \frac{3}{4} \times \left(\frac{8}{1} \times \frac{1}{8}\right) = \frac{3}{4} \times 1 = \frac{3}{4}

Key concept

A quotient has a meaning you can check

In this lesson's problems, a quotient means one of two things. It is a count of groups when you divide a total by the size of one group. It is the size of one equal share when you divide a total by a number of shares. A share size is an amount, not a count of groups. Either way, when the divisor is not $0$ and the quotient is exact (not rounded), the quotient times the divisor equals the dividend. Use that fact to check your work.

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

Worked example

How many cups? A count of groups

A bottle holds $\frac{3}{5}$ liter of juice. Each cup holds $\frac{3}{20}$ liter. How many cups can you fill?

Find the total and what you divide by. The total is 3/5 liter. One group is one cupful, 3/20 liter. The question asks how many groups fit in the total, so divide the total by the group size.
$\frac{3}{5} \div \frac{3}{20}$
The divisor 3/20 is not zero, so you can multiply by its reciprocal, 20/3. This gives the quotient because 3/20 times 20/3 is 1: multiply the result by the divisor and the dividend 3/5 comes back.
$\frac{3}{5} \div \frac{3}{20} = \frac{3}{5} \times \frac{20}{3}$
Multiply the numerators, and multiply the denominators. Then simplify: 60 divided by 15 is 4.
$\frac{3 \times 20}{5 \times 3} = \frac{60}{15} = 4$
Check with multiplication. The quotient 4 is exact, so 4 times the divisor must equal the dividend. Dividing the numerator and the denominator of 12/20 by the same number, 4, gives the equal fraction 3/5, so the check passes.
$4 \times \frac{3}{20} = \frac{12}{20} = \frac{3}{5}$
Say what the 4 means. You divided by a group size, so 4 is a count of groups: 4 cups that each hold 3/20 liter.

Answer

You can fill $4$ cups. The quotient $4$ is a count of cups, not an amount of juice.

Worked example

How much for each? The size of one share

Four students share $2\frac{2}{3}$ pounds of clay equally. How much clay does each student get?

Find the total and what you divide by. The total is 2 and 2/3 pounds. The number of equal shares is 4. The question asks for the size of one share, so divide the total by the number of shares.
$2\frac{2}{3} \div 4$
Write the mixed number (a whole number plus a fraction) as one fraction. 2 wholes are 6 thirds, and 6 thirds plus 2 thirds is 8 thirds.
$2\frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3}$
The divisor 4 is not zero. Its reciprocal is 1/4, because 4 times 1/4 is 1. Multiply the dividend by 1/4. Then divide the numerator and the denominator of 8/12 by 4 to get the equal fraction 2/3.
$\frac{8}{3} \div 4 = \frac{8}{3} \times \frac{1}{4} = \frac{8}{12} = \frac{2}{3}$
Check with multiplication. The quotient 2/3 is exact, so 2/3 times the divisor 4 must equal the dividend 8/3. It does.
$\frac{2}{3} \times 4 = \frac{8}{3} = 2\frac{2}{3}$
Say what 2/3 means. You divided by a number of shares, so 2/3 is the size of one share. It is an amount of clay in pounds. It is not a count of groups.

Answer

Each student gets $\frac{2}{3}$ pound of clay. The quotient $\frac{2}{3}$ is the size of one equal share.

You have now seen both meanings. In the bottle problem, $4$ was a count of cups. In the clay problem, $\frac{2}{3}$ pound was the size of one student's share. To interpret a quotient fully, ask two more questions.

What does a fraction part mean? A quotient that counts groups does not have to be a whole number. A quotient of $3\frac{1}{3}$ groups means $3$ whole groups and $\frac{1}{3}$ of one more group. The fraction part is a fraction of one group. It is not measured in the unit of the dividend.
Is the size sensible? For a positive dividend, the quotient can be greater than, equal to, or less than the dividend. The divisor decides which.

The next example answers the first question. The mistake and the table after it answer the second.

Worked example

What does the fraction part mean?

It takes $\frac{3}{4}$ hour to paint one birdhouse. You have $2\frac{1}{2}$ hours. How many birdhouses can you paint, and what does the quotient mean?

The total is 2 and 1/2 hours. One group is the 3/4 hour that one birdhouse takes. You want the number of groups, so divide the total by the group size. First write 2 and 1/2 as 5/2: 2 wholes are 4 halves, and 1 more half makes 5 halves.
$2\frac{1}{2} \div \frac{3}{4} = \frac{5}{2} \div \frac{3}{4}$
The divisor 3/4 is not zero, so multiply by its reciprocal, 4/3. Divide the numerator and the denominator of 20/6 by 2 to get the equal fraction 10/3. Since 9 thirds make 3 wholes, 10/3 is 3 and 1/3.
$\frac{5}{2} \div \frac{3}{4} = \frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = \frac{10}{3} = 3\frac{1}{3}$
Check the exact quotient with multiplication. 10/3 times the divisor 3/4 must equal the dividend 5/2. Since 30/12 simplifies to 5/2, the check passes.
$\frac{10}{3} \times \frac{3}{4} = \frac{30}{12} = \frac{5}{2}$
Interpret the quotient. It counts groups, so 3 and 1/3 means 3 whole birdhouses and 1/3 of another one. The 1/3 is a fraction of one birdhouse's painting time, not 1/3 hour. To write it in hours, find 1/3 of 3/4 hour.
$\frac{1}{3} \times \frac{3}{4} = \frac{3}{12} = \frac{1}{4}$
The exact quotient 3 and 1/3 leaves nothing over. A leftover appears only when you count whole birdhouses: 3 birdhouses with 1/4 hour left. Check that kind of answer with the divisor times the whole count, plus the leftover. The sum must equal the dividend.
$\frac{3}{4} \times 3 + \frac{1}{4} = \frac{9}{4} + \frac{1}{4} = \frac{10}{4} = \frac{5}{2}$

Answer

The quotient is $3\frac{1}{3}$ . You can paint $3$ whole birdhouses and $\frac{1}{3}$ of a fourth one. If only finished birdhouses count, you paint $3$ and have $\frac{1}{4}$ hour left, not $\frac{1}{3}$ hour.

Mistake: thinking that division always makes a number smaller. A student who thinks this works out $\frac{3}{5} \div \frac{3}{20}$ , gets $4$ , and crosses it out because $4$ is greater than $\frac{3}{5}$ . The same student multiplies $\frac{3}{5} \times \frac{3}{20}$ with no reciprocal and gets $\frac{9}{100}$ . They accept that wrong answer because it is small. This belief fails because it is true only in some cases. For a positive dividend and a positive divisor, the quotient is less than the dividend only when the divisor is greater than $1$ . Here the divisor $\frac{3}{20}$ is between $0$ and $1$ . Dividing by it is the same as multiplying by its reciprocal, $\frac{20}{3}$ . Multiplying a positive number by a number greater than $1$ makes it greater.

How the size of the quotient compares with the dividend

Dividend	Divisor	Same as multiplying by	The quotient is	Example
Positive	Between $0$ and $1$	a number greater than $1$	greater than the dividend	$\frac{3}{5} \div \frac{3}{20} = 4$ , and $4$ is greater than $\frac{3}{5}$
Positive	Equal to $1$	$1$	equal to the dividend	$\frac{3}{5} \div 1 = \frac{3}{5}$
Positive	Greater than $1$	a number between $0$ and $1$	less than the dividend	$2\frac{2}{3} \div 4 = \frac{2}{3}$ , and $\frac{2}{3}$ is less than $2\frac{2}{3}$
Zero	Any nonzero number	the divisor's reciprocal	$0$ , because zero stays zero	$0 \div \frac{3}{20} = 0$

The last skill is finding mistakes in division work. You can test a finished answer without redoing all the work. When the divisor is not $0$ , use three checks.

Size check. Compare the quotient with the dividend, as in the table. If the size is wrong, there is a mistake. If the size is right, the answer may still be wrong, so keep checking.
Multiplication check. An exact quotient times the divisor equals the dividend. A rounded quotient gives a product that is only close to the dividend. For a whole count with a leftover, the divisor times the whole count, plus the leftover, equals the dividend.
Meaning check. Ask if the answer should be a count of groups or the size of one share, and if its unit fits.

When a check fails, read the work one line at a time. Is the total the dividend? Is each mixed number written correctly as a fraction? Is the divisor, and only the divisor, replaced by its reciprocal? Is the multiplication correct?

Worked example

Find and fix the mistake

A rope is $1\frac{1}{2}$ meters long. Kai cuts it into pieces that are each $\frac{3}{10}$ meter long. To find the number of pieces, Kai wrote three lines. Line 1: $1\frac{1}{2} \div \frac{3}{10} = \frac{3}{2} \div \frac{3}{10}$ . Line 2: $\frac{3}{2} \div \frac{3}{10} = \frac{2}{3} \times \frac{3}{10}$ . Line 3: $\frac{2}{3} \times \frac{3}{10} = \frac{6}{30} = \frac{1}{5}$ . Find Kai's mistake and correct it.

Start with a size check. The dividend 3/2 is positive and the divisor 3/10 is between 0 and 1, so the quotient must be greater than 3/2. Kai's answer, 1/5, is less than 3/2. It also does not fit the story: one piece is shorter than the rope, so at least 1 whole piece must fit. The work has a mistake.
$\frac{1}{5} < \frac{3}{2}$
Confirm with a multiplication check. The divisor is not zero, so an exact quotient times the divisor equals the dividend. Kai's answer times 3/10 is 3/50, which is not 3/2. So 1/5 is not the quotient.
$\frac{1}{5} \times \frac{3}{10} = \frac{3}{50} \neq \frac{3}{2}$
Read the lines in order. Line 1 is true: 1 and 1/2 equals 3/2. Line 2 is the first line that is not true. Kai took the reciprocal of the dividend, 3/2, and left the divisor, 3/10, alone. You replace only the divisor with its reciprocal, because the divisor times its own reciprocal makes 1 and brings the dividend back. Line 3 multiplies correctly, but it starts from the wrong line.
Correct Line 2. Keep the dividend 3/2 and multiply by 10/3, the reciprocal of the divisor. Then simplify: 30 divided by 6 is 5.
$\frac{3}{2} \div \frac{3}{10} = \frac{3}{2} \times \frac{10}{3} = \frac{30}{6} = 5$
Check the corrected quotient both ways. The quotient 5 is greater than 3/2, as the size check requires. And 5 times the divisor is 15/10, which simplifies to the dividend 3/2.
$5 \times \frac{3}{10} = \frac{15}{10} = \frac{3}{2}$

Answer

In Line 2, Kai took the reciprocal of the dividend instead of the divisor. The correct quotient is $5$ , so the rope makes $5$ pieces that are each $\frac{3}{10}$ meter long.

Takeaway. A fraction quotient is more than a number. First decide what it means in the situation: a count of groups that are each the size of the divisor, or the size of one equal share. Then test it. When the divisor is not $0$ , an exact quotient times the divisor gives back the dividend. When the dividend is positive, a divisor between $0$ and $1$ gives a quotient greater than the dividend, and a divisor greater than $1$ gives a quotient less than the dividend. If a check fails, read the work line by line to find the first line that is not true.