Math Core

A school has 672 chairs to set up in rows of 32. How many rows will there be? You know how to divide by a one-digit number. Dividing by a two-digit number uses the same steps, but you need one new skill: making a good estimate for each digit of the quotient.

Estimate first

Before you divide, round the numbers to make an easy problem. Choose numbers that divide evenly, called compatible numbers.

$672 \div 32$ is about $600 \div 30 = 20$ .
$3{,}215 \div 79$ is about $3{,}200 \div 80 = 40$ .

The estimate tells you about how big the answer should be. It also tells you how many digits the quotient will have.

Partial quotients

One friendly way to divide is to take away easy groups of the divisor, one chunk at a time. Each chunk is a partial quotient. At the end, add up the chunks.

Worked example: Taking away groups

Find $672 \div 32$ .

Take away big groups of 32 first.

Take away	Groups of 32	Left over
$32 \times 20 = 640$	20	$672 - 640 = 32$
$32 \times 1 = 32$	1	$32 - 32 = 0$

Add the groups: $20 + 1 = 21$ . So $672 \div 32 = 21$ . There will be 21 rows.

Check: $32 \times 21 = 672$ . ✓

Long division

Long division uses the same divide, multiply, subtract, bring down steps you already know. To guess each digit, round the divisor to the nearest ten.

Dividing by a two-digit number

Divide: Look at the first part of the dividend that is at least as big as the divisor. Estimate how many times the divisor fits. Rounding the divisor helps.
Multiply that digit by the divisor.
Subtract. The difference must be less than the divisor.
Bring down the next digit and repeat.

Check by multiplying: quotient $\times$ divisor $+$ remainder $=$ dividend.

Worked example: No remainder

Find $1{,}548 \div 43$ .

\begin{array}{r} 36 \\ 43\,\overline{)\,1548} \\ \underline{-\,129\phantom{0}} \\ 258 \\ \underline{-\,258} \\ 0 \end{array}

43 does not fit into 1 or 15, so start with 154 tens.
Estimate: 43 is about 40, and $154 \div 40$ is about 3. $43 \times 3 = 129$ . $154 - 129 = 25$ , which is less than 43. ✓
Bring down the 8 to make 258.
Estimate: $258 \div 40$ is about 6. $43 \times 6 = 258$ . $258 - 258 = 0$ .

So $1{,}548 \div 43 = 36$ . Check: $43 \times 36 = 1{,}548$ . ✓

When the estimate is off

Rounding the divisor gives a good guess, but not always the right one. Test your guess:

If the product is bigger than the number you are dividing, the digit is too big. Try one less.
If what is left after subtracting is as big as the divisor or bigger, the digit is too small. Try one more.

Worked example: Fixing a guess, with a remainder

Find $2{,}183 \div 27$ .

\begin{array}{r} 80 \\ 27\,\overline{)\,2183} \\ \underline{-\,216\phantom{0}} \\ 23 \\ \underline{-\,0} \\ 23 \end{array}

Start with 218 tens. 27 is about 30, and $218 \div 30$ is about 7.
Test 7: $27 \times 7 = 189$ and $218 - 189 = 29$ . But 29 is bigger than 27, so 7 is too small.
Try 8: $27 \times 8 = 216$ and $218 - 216 = 2$ . ✓
Bring down the 3 to make 23. 27 does not fit into 23, so write 0 in the quotient. $23 - 0 = 23$ .

So $2{,}183 \div 27 = 80 \text{ R } 23$ . Check: $27 \times 80 + 23 = 2{,}160 + 23 = 2{,}183$ . ✓

Common mistake

Don't skip the 0! In the last example, 27 did not fit into 23, but you still need a digit in the ones place. Without it you would write 8 instead of 80, which is ten times too small. Your estimate helps here: the answer should be close to $2{,}100 \div 30 = 70$ , not 8.

Worked example: Using the remainder

There are 500 students going on a field trip. Each bus holds 28 students. How many buses are needed?

Find $500 \div 28$ . 28 is about 30, and $50 \div 30$ is about 1. $28 \times 1 = 28$ , and $50 - 28 = 22$ . Bring down the 0 to make 220. $220 \div 30$ is about 7. $28 \times 7 = 196$ , and $220 - 196 = 24$ .

So $500 \div 28 = 17 \text{ R } 24$ . Seventeen buses carry 476 students, but 24 students are still waiting. They need one more bus, so the school needs 18 buses.

Tip

A remainder must always be smaller than the divisor. If your remainder is 30 when dividing by 28, one more group of 28 fits, so your quotient is too small.

Practice

Practice 1

Find $480 \div 16$ .

Type a number.

Practice 2

Find $391 \div 17$ .

Type a number.

Practice 3

Find $936 \div 24$ .

Type a number.

Practice 4

Which is the best estimate for $3{,}215 \div 79$ ?

Practice 5

Find $1{,}275 \div 25$ .

Type a number.

Practice 6

Find $845 \div 36$ . Write your answer like 12 R 3.

Type your answer

Practice 7

Find $4{,}209 \div 61$ .

Type a number.

Practice 8

A farm packs 1,000 eggs into cartons that hold 12 eggs each. How many cartons can the farm fill completely?

Type a number.