Estimate and Divide Mixed Numbers | Grade 6 Math | LevelKite

How many groups of $1\frac{1}{2}$ fit in $7\frac{1}{2}$ ? That question is a division of two mixed numbers. A mixed number has a whole-number part and a fraction part, so $7\frac{1}{2}$ means $7 + \frac{1}{2}$ . Every mixed number in this lesson is greater than 1.

You write the question as $7\frac{1}{2} \div 1\frac{1}{2}$ . The dividend $7\frac{1}{2}$ is the number you start with. The divisor $1\frac{1}{2}$ is the number you divide by. Here it is the size of one group. A divisor is never zero. The quotient is the result. Here it counts the groups.

Count up by $1\frac{1}{2}$ to find it: $1\frac{1}{2}$ , $3$ , $4\frac{1}{2}$ , $6$ , $7\frac{1}{2}$ . That takes 5 groups, so the quotient is $5$ . Multiplying confirms it: $5 \times 1\frac{1}{2} = 7\frac{1}{2}$ .

Counting like this gets slow with harder numbers. In this lesson you will first estimate a quotient. Then you will find it exactly.

Before you divide exactly, find out about how big the quotient should be. An estimate is a number close to the exact answer. You find it with easier numbers.

To estimate a quotient of mixed numbers, round each mixed number to the nearest whole number. Then divide the whole numbers. To round, compare the fraction part with $\frac{1}{2}$ :

If the fraction part is less than $\frac{1}{2}$ , keep the whole-number part.
If the fraction part is $\frac{1}{2}$ or more, go up to the next whole number.

A mixed number greater than 1 rounds to a whole number that is at least 1. So the rounded divisor is never zero, and you can divide by it.

If the whole numbers do not divide evenly, write the estimate as a mixed number. For example, $7 \div 2 = 3\frac{1}{2}$ .

An estimate is usually not the exact quotient. The table shows how five mixed numbers round.

Rounding mixed numbers to the nearest whole number

Mixed number	Fraction part compared with one half	Rounds to
$5\frac{5}{6}$	$\frac{5}{6}$ is more than $\frac{1}{2}$ , because $\frac{1}{2} = \frac{3}{6}$	$6$
$1\frac{3}{4}$	$\frac{3}{4}$ is more than $\frac{1}{2}$ , because $\frac{1}{2} = \frac{2}{4}$	$2$
$4\frac{1}{6}$	$\frac{1}{6}$ is less than $\frac{1}{2}$ , because $\frac{1}{2} = \frac{3}{6}$	$4$
$1\frac{2}{3}$	$\frac{2}{3} = \frac{4}{6}$ is more than $\frac{1}{2} = \frac{3}{6}$	$2$
$7\frac{1}{2}$	The fraction part is exactly $\frac{1}{2}$ , so round up	$8$

Worked example

Estimate a quotient

Estimate $5\frac{5}{6} \div 1\frac{3}{4}$ by rounding each mixed number to the nearest whole number.

Round the dividend. Its fraction part, five sixths, is more than one half, so the dividend rounds up to 6. The wavy equals sign means 'is about equal to.'
$5\frac{5}{6} \approx 6$
Round the divisor. Its fraction part, three fourths, is more than one half, so the divisor rounds up to 2. The rounded divisor is not zero, so you can divide by it.
$1\frac{3}{4} \approx 2$
Divide the rounded numbers. This asks how many groups of 2 fit in 6. Three groups of 2 make 6, so the estimate is 3.
$6 \div 2 = 3$
Check with multiplication. You rounded the numbers, so this check is only approximate. Three groups of about 2 make about 6, and 6 is close to the original dividend.
$3 \times 2 = 6 \approx 5\frac{5}{6}$

Answer

The quotient $5\frac{5}{6} \div 1\frac{3}{4}$ is about $3$ . About 3 groups of $1\frac{3}{4}$ fit in $5\frac{5}{6}$ .

To divide exactly, first write each mixed number as one fraction.

Start with $7\frac{1}{2}$ . One whole is $\frac{2}{2}$ , so 7 wholes are $\frac{14}{2}$ . Then $7\frac{1}{2} = \frac{14}{2} + \frac{1}{2} = \frac{15}{2}$ . In the same way, $1\frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}$ .

Here is a shortcut. Multiply the whole number by the denominator (the bottom number). Add the numerator (the top number). Keep the denominator. For $7\frac{1}{2}$ , that is $7 \times 2 + 1 = 15$ , so the fraction is $\frac{15}{2}$ .

Each fraction equals its mixed number. So $7\frac{1}{2} \div 1\frac{1}{2}$ and $\frac{15}{2} \div \frac{3}{2}$ are the same division, with the same quotient.

Division undoes multiplication, as long as the divisor is not zero. So the quotient of $\frac{15}{2} \div \frac{3}{2}$ is the number that gives $\frac{15}{2}$ when you multiply it by the divisor $\frac{3}{2}$ . Exactly one number does this.

A reciprocal helps you find that number. The reciprocal of a nonzero number is the number you multiply it by to get 1. To find the reciprocal of a nonzero fraction, switch its numerator and denominator. The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$ , because $\frac{3}{2} \times \frac{2}{3} = \frac{6}{6} = 1$ .

Now test $\frac{15}{2} \times \frac{2}{3}$ as the quotient. Multiply it by the divisor $\frac{3}{2}$ and see what comes back.

\frac{15}{2} \times \frac{2}{3} \times \frac{3}{2} = \frac{15}{2} \times 1 = \frac{15}{2}

You can group the numbers in a multiplication in any way without changing the result. Group the last two fractions. They are reciprocals, so together they make 1. The dividend $\frac{15}{2}$ comes back, so $\frac{15}{2} \times \frac{2}{3}$ is the quotient. Dividing by $\frac{3}{2}$ gives the same result as multiplying by its reciprocal, $\frac{2}{3}$ .

To multiply fractions, multiply the numerators and multiply the denominators:

$\frac{15}{2} \times \frac{2}{3} = \frac{15 \times 2}{2 \times 3} = \frac{30}{6} = 5$

This matches the 5 groups you counted at the start.

Key concept

Divide by multiplying by the reciprocal

To divide mixed numbers, write each one as a fraction. Then multiply the dividend by the reciprocal of the divisor.

This works for every divisor that is not zero. The reciprocal times the divisor is 1. So when you multiply your answer by the divisor, you get the dividend back, and that is what a quotient must do. Zero has no reciprocal, and you can never divide by zero.

Estimate first by rounding each mixed number to the nearest whole number. Check at the end: the exact quotient times the divisor equals the dividend. A check with an estimate is only approximate. Its result should be close to the dividend, but it may not match exactly.

\text{dividend} \div \text{divisor} = \text{dividend} \times \text{reciprocal of the divisor}, \quad \text{divisor} \neq 0

Worked example

Divide two mixed numbers

Find $4\frac{1}{6} \div 1\frac{2}{3}$ . Estimate first, then divide exactly and check.

Estimate. One sixth is less than one half, so the dividend rounds to 4. Two thirds is more than one half, so the divisor rounds up to 2. About 2 groups of the divisor should fit in the dividend.
$4 \div 2 = 2$
Write each mixed number as a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. The values do not change, so the quotient does not change.
$4\frac{1}{6} = \frac{4 \times 6 + 1}{6} = \frac{25}{6} \quad \text{and} \quad 1\frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3}$
Multiply the dividend by the reciprocal of the divisor. The divisor, five thirds, is not zero, so it has a reciprocal: three fifths. Dividing by five thirds gives the same result as multiplying by three fifths, because three fifths times five thirds is 1.
$\frac{25}{6} \div \frac{5}{3} = \frac{25}{6} \times \frac{3}{5}$
Multiply the numerators and multiply the denominators. Then simplify, which means writing an equal fraction with smaller numbers. You can divide both 75 and 30 by 15. Dividing the numerator and the denominator by the same nonzero number keeps the value of the fraction.
$\frac{25 \times 3}{6 \times 5} = \frac{75}{30} = \frac{75 \div 15}{30 \div 15} = \frac{5}{2}$
Write the result as a mixed number. Five halves is four halves plus one half, and four halves make 2 wholes. The exact quotient is near the estimate of 2, so it is reasonable.
$\frac{5}{2} = \frac{4}{2} + \frac{1}{2} = 2\frac{1}{2}$
Check with multiplication. This quotient is exact, so the quotient times the divisor must equal the dividend exactly.
$\frac{5}{2} \times \frac{5}{3} = \frac{5 \times 5}{2 \times 3} = \frac{25}{6} = 4\frac{1}{6}$

Answer

The quotient $4\frac{1}{6} \div 1\frac{2}{3}$ is $2\frac{1}{2}$ . Two full groups of $1\frac{2}{3}$ and half of another group fit in $4\frac{1}{6}$ . There is no remainder: the $\frac{1}{2}$ means half of a group.

Take the reciprocal of the whole divisor. In $7\frac{1}{2} \div 1\frac{1}{2}$ , it is tempting to switch the 1 and the 2 in the divisor's fraction part only. That turns $1\frac{1}{2}$ into $1\frac{2}{1} = 1 + 2 = 3$ and gives $7\frac{1}{2} \times 3 = 22\frac{1}{2}$ .

This fails because $3$ is not the reciprocal of $1\frac{1}{2}$ . A nonzero number times its reciprocal is 1, but $1\frac{1}{2} \times 3 = 4\frac{1}{2}$ . So multiplying by 3 does not undo multiplying by $1\frac{1}{2}$ .

An estimate catches the mistake. Rounding gives $8 \div 2 = 4$ , and $22\frac{1}{2}$ is far from 4. Write the divisor as one fraction first: $1\frac{1}{2} = \frac{3}{2}$ , with reciprocal $\frac{2}{3}$ . Then $\frac{15}{2} \times \frac{2}{3} = \frac{30}{6} = 5$ .

Worked example

Divide and compare with your estimate

Find $5\frac{5}{6} \div 1\frac{3}{4}$ exactly. In the first example, you estimated this quotient as about $3$ .

Write both mixed numbers as fractions. Each fraction equals its mixed number, so the quotient stays the same.
$5\frac{5}{6} = \frac{5 \times 6 + 5}{6} = \frac{35}{6} \quad \text{and} \quad 1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}$
Multiply the dividend by the reciprocal of the divisor. The divisor, seven fourths, is not zero, and its reciprocal is four sevenths.
$\frac{35}{6} \div \frac{7}{4} = \frac{35}{6} \times \frac{4}{7} = \frac{35 \times 4}{6 \times 7} = \frac{140}{42}$
Simplify, then write a mixed number. You can divide both 140 and 42 by 14, because 140 is 14 times 10 and 42 is 14 times 3. Ten thirds is nine thirds plus one third, and nine thirds make 3 wholes.
$\frac{140}{42} = \frac{140 \div 14}{42 \div 14} = \frac{10}{3} = 3\frac{1}{3}$
Compare with your estimate. The exact quotient is a little more than 3, so it agrees with the estimate of about 3.
$3\frac{1}{3} \approx 3$
Check with multiplication. The quotient is exact and the divisor is not zero, so the quotient times the divisor must equal the dividend.
$\frac{10}{3} \times \frac{7}{4} = \frac{70}{12} = \frac{70 \div 2}{12 \div 2} = \frac{35}{6} = 5\frac{5}{6}$

Answer

The quotient $5\frac{5}{6} \div 1\frac{3}{4}$ is $3\frac{1}{3}$ . Three full groups of $1\frac{3}{4}$ and one third of another group fit in $5\frac{5}{6}$ .

Takeaway. To divide mixed numbers, write each one as a fraction and multiply the dividend by the reciprocal of the divisor, which is never zero. Estimate first by rounding each mixed number to the nearest whole number. Check at the end: the exact quotient times the divisor equals the dividend. If the exact quotient is far from your estimate, look for a mistake.

In this lesson, each quotient told you how many groups the size of the divisor fit in the dividend. In a sharing situation, a quotient is the size of one equal share instead. Next, you will divide mixed numbers in word problems and choose whether to multiply or divide.