Check Fraction Quotients and Fix Mistakes | Grade 5 Math | LevelKite

You already know how to divide with whole numbers (0, 1, 2, 3, and so on) and unit fractions. A unit fraction is a fraction with $1$ as its top number, such as $\frac{1}{2}$ or $\frac{1}{8}$ . In this lesson, you will learn how to check a division answer and how to fix one that is wrong.

Start with a division you know well: $12 \div 3 = 4$ . Each number has a name.

The dividend is the number being divided. Here it is $12$ .
The divisor is the number you divide by. Here it is $3$ .
The quotient is the answer to the division. Here it is $4$ .

This division splits $12$ into $3$ equal groups, with $4$ in each group. Now put the groups back together: $3$ groups of $4$ make $12$ . As a number sentence, that is $4 \times 3 = 12$ . Here, multiplying the quotient by the divisor rebuilds the dividend.

The same idea works when a unit fraction is in the division. In this lesson, that happens in exactly two ways.

A unit fraction divided by a whole number that is not zero, such as $\frac{1}{2} \div 4$ . This splits a piece of size $\frac{1}{2}$ into $4$ equal parts. The quotient is the size of one part.
A whole number divided by a unit fraction, such as $5 \div \frac{1}{3}$ . This asks how many pieces of size $\frac{1}{3}$ fit in $5$ . The quotient is the number of pieces.

In both kinds of division, putting the parts back together should give you the dividend again. That is the idea behind checking a quotient.

Key concept

Multiply to check a quotient

This check works for both kinds of division in this lesson: a unit fraction divided by a whole number that is not zero, and a whole number divided by a unit fraction.

Multiply the quotient by the divisor. The answer to a multiplication is called the product.

If the product equals the dividend, the quotient is right.
If the product does not equal the dividend, the quotient is wrong.

This multiplication is the related multiplication for the division. It uses the same three numbers as the division.

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

Worked example

Check a unit fraction divided by a whole number

Is $\frac{1}{2} \div 4 = \frac{1}{8}$ right? Use multiplication to check the quotient.

Name the three numbers so you know what to multiply. The dividend is one half, the divisor is 4, and the quotient is one eighth.
$\frac{1}{2} \div 4 = \frac{1}{8}$
Multiply the quotient by the divisor to see if the parts rebuild the dividend. Think of it as 4 copies of one eighth. That makes 4 eighths.
$\frac{1}{8} \times 4 = \frac{4}{8}$
Write 4 eighths in a simpler way so you can compare it with the dividend. Divide the top number and the bottom number by 4. The amount stays the same, because 4 out of 8 equal parts is half of the whole.
$\frac{4}{8} = \frac{1}{2}$
Compare the product with the dividend. The product is one half, and the dividend is one half. They match, so the quotient is right.
$\frac{1}{8} \times 4 = \frac{1}{2}$

Answer

Yes, $\frac{1}{2} \div 4 = \frac{1}{8}$ is right, because $\frac{1}{8} \times 4 = \frac{1}{2}$ .

Worked example

Check a whole number divided by a unit fraction

Is $5 \div \frac{1}{3} = 15$ right? Use multiplication to check the quotient.

The dividend is 5, the divisor is one third, and the quotient is 15. Multiply the quotient by the divisor to see if the pieces rebuild the dividend. Think of it as 15 copies of one third. That makes 15 thirds.
$15 \times \frac{1}{3} = \frac{15}{3}$
Turn 15 thirds into wholes so you can compare with the dividend. It takes 3 thirds to make 1 whole, and 15 divided by 3 is 5. So 15 thirds make 5 wholes.
$\frac{15}{3} = 5$
Compare the product with the dividend. Both are 5, so the quotient is right. Notice that the quotient 15 is bigger than the dividend 5. That is fine here, because 15 pieces of size one third fit in 5 wholes.
$15 \times \frac{1}{3} = 5$

Answer

Yes, $5 \div \frac{1}{3} = 15$ is right, because $15 \times \frac{1}{3} = 5$ .

The check does more than say yes or no. It also gives you a reason. Suppose someone asks, How do you know your quotient is right? Say the division, then the word because, then the related multiplication.

This kind of reason works for both kinds of division in this lesson: a unit fraction divided by a whole number that is not zero, and a whole number divided by a unit fraction. It works because the related multiplication shows that the parts rebuild the dividend. Here is the reason for the first example.

\frac{1}{2} \div 4 = \frac{1}{8} \quad \text{because} \quad \frac{1}{8} \times 4 = \frac{1}{2}

Each division and the multiplication that explains it

Division	Related multiplication	The reason in words
$\frac{1}{2} \div 4 = \frac{1}{8}$	$\frac{1}{8} \times 4 = \frac{1}{2}$	$4$ parts of size $\frac{1}{8}$ make $\frac{1}{2}$ .
$5 \div \frac{1}{3} = 15$	$15 \times \frac{1}{3} = 5$	$15$ pieces of size $\frac{1}{3}$ make $5$ .
$\frac{1}{5} \div 2 = \frac{1}{10}$	$\frac{1}{10} \times 2 = \frac{1}{5}$	$2$ parts of size $\frac{1}{10}$ make $\frac{1}{5}$ .
$4 \div \frac{1}{6} = 24$	$24 \times \frac{1}{6} = 4$	$24$ pieces of size $\frac{1}{6}$ make $4$ .

Now use the check to find mistakes. In this lesson, a division with a fraction is always a unit fraction divided by a whole number that is not zero, or a whole number divided by a unit fraction. For both kinds, a wrong quotient fails the check. You multiply it by the divisor, and you do not get the dividend.

You can also ask, Does the size of my quotient make sense? Be careful with this question. Many people expect every quotient to be smaller than the dividend. That is not always true.

What size to expect for each kind of division in this lesson

Kind of division	What it asks	Example	Size of the quotient
A unit fraction divided by a whole number greater than $1$	How big is one part when you split the piece into that many equal parts?	$\frac{1}{3} \div 2 = \frac{1}{6}$	Smaller than the dividend: $\frac{1}{6}$ is less than $\frac{1}{3}$ .
A whole number that is not zero, divided by a unit fraction less than $1$	How many pieces of that size fit in the whole number?	$2 \div \frac{1}{3} = 6$	Bigger than the dividend: $6$ is greater than $2$ .

Worked example

Find and fix a mistake

A classmate wrote $\frac{1}{4} \div 3 = 12$ . Check the work. If it is wrong, find the mistake and fix it.

This is a unit fraction divided by the whole number 3, so check it with multiplication. Multiply the quotient 12 by the divisor 3. The product is 36, but the dividend is one fourth. They do not match, so 12 is wrong. The size is a clue too: one part of one fourth cannot be bigger than one fourth.
$12 \times 3 = 36$
Find the mistake. The number 12 answers a different question: how many fourths fit in 3? Each whole has 4 fourths, so 3 wholes have 12 fourths. Your classmate swapped the dividend and the divisor. In this problem, one fourth comes first, so one fourth is the dividend. It is the amount being split.
$3 \div \frac{1}{4} = 12$
Solve the problem that was asked. Split one fourth into 3 equal parts. If you cut each of the 4 fourths in a whole into 3 equal parts, the whole has 12 equal parts. So one part is one twelfth.
$\frac{1}{4} \div 3 = \frac{1}{12}$
Check the new quotient. Multiply it by the divisor: 3 copies of one twelfth make 3 twelfths. Divide the top number and the bottom number by 3 to get one fourth. This matches the dividend, so one twelfth is right. The size makes sense too, because one twelfth is smaller than one fourth.
$\frac{1}{12} \times 3 = \frac{3}{12} = \frac{1}{4}$

Answer

The classmate's answer is wrong, because $12$ is the quotient for $3 \div \frac{1}{4}$ . The correct quotient is $\frac{1}{4} \div 3 = \frac{1}{12}$ , because $\frac{1}{12} \times 3 = \frac{1}{4}$ .

Here is the big idea to keep. In this lesson, division with a unit fraction has two kinds: a unit fraction divided by a whole number that is not zero, and a whole number divided by a unit fraction. For both kinds, multiply the quotient by the divisor. If you get the dividend, the quotient is right, and that related multiplication is your reason. If you do not get the dividend, look for a mistake. Make sure you did not swap the dividend and the divisor. Also, do not expect every quotient to be smaller than the dividend. When a whole number that is not zero is divided by a unit fraction less than $1$ , the quotient is bigger than the dividend.