Math Core

A charity game raises $8,556, and the money is shared equally among 23 schools. How much does each school get? Problems like this come up all the time, and a calculator isn't always in your pocket. In this lesson you'll make long division fast and reliable, even with big dividends and three-digit divisors.

The words you need

In $8{,}556 \div 23 = 372$ :

The dividend is the number being divided: $8{,}556$ .
The divisor is the number you divide by: $23$ .
The quotient is the answer: $372$ .
If something is left over, it is the remainder.

The standard algorithm

Long division repeats the same four steps, one place value at a time.

Divide, multiply, subtract, bring down

Divide: Find how many times the divisor fits into the current number. Write that digit in the quotient.
Multiply that digit by the divisor.
Subtract. The result must be smaller than the divisor.
Bring down the next digit of the dividend, then repeat.

When no digits are left, whatever remains is the remainder. Check: $\text{quotient} \times \text{divisor} + \text{remainder} = \text{dividend}$ .

To guess each digit, round the divisor. For 23, think "about 20." For 207, think "about 200." Then test your guess with a real multiplication.

Worked example: A two-digit divisor

Find $8{,}556 \div 23$ .

\begin{array}{r} 372 \\ 23\,\overline{)\,8556} \\ \underline{-\,69\phantom{00}} \\ 165\phantom{0} \\ \underline{-\,161\phantom{0}} \\ 46 \\ \underline{-\,46} \\ 0 \end{array}

23 doesn't fit into 8, so start with 85. $23 \times 3 = 69$ and $85 - 69 = 16$ .
Bring down the 5 to make 165. $23 \times 7 = 161$ and $165 - 161 = 4$ .
Bring down the 6 to make 46. $23 \times 2 = 46$ and $46 - 46 = 0$ .

So $8{,}556 \div 23 = 372$ . Check: $23 \times 372 = 8{,}556$ . ✓ Each school gets $372.

Estimate before you divide

An estimate tells you how many digits the quotient should have, so you'll notice a missing digit right away. Round to compatible numbers that divide easily. For $25{,}930 \div 64$ , use $25{,}600 \div 64 = 400$ . The quotient should be a little more than 400.

Worked example: A zero in the quotient

Find $25{,}930 \div 64$ .

\begin{array}{r} 405 \\ 64\,\overline{)\,25930} \\ \underline{-\,256\phantom{00}} \\ 33\phantom{0} \\ \underline{-\,0\phantom{0}} \\ 330 \\ \underline{-\,320} \\ 10 \end{array}

64 doesn't fit into 2 or 25, so start with 259. $64 \times 4 = 256$ and $259 - 256 = 3$ .
Bring down the 3 to make 33. 64 doesn't fit into 33, so write 0 in the quotient.
Bring down the 0 to make 330. $64 \times 5 = 320$ and $330 - 320 = 10$ .

So $25{,}930 \div 64 = 405 \text{ R } 10$ . Check: $64 \times 405 + 10 = 25{,}920 + 10 = 25{,}930$ . ✓

Common mistake

Every time you bring down a digit, you must write a digit in the quotient, even if it is 0. Skipping the 0 above would give 45, which is nowhere near the estimate of 400.

Three-digit divisors

Nothing new happens with a bigger divisor. The first "divide" step just uses more digits of the dividend.

Worked example: A three-digit divisor

Find $47{,}196 \div 207$ .

\begin{array}{r} 228 \\ 207\,\overline{)\,47196} \\ \underline{-\,414\phantom{00}} \\ 579\phantom{0} \\ \underline{-\,414\phantom{0}} \\ 1656 \\ \underline{-\,1656} \\ 0 \end{array}

207 doesn't fit into 47, so start with 471. Think $471 \div 200 \approx 2$ . $207 \times 2 = 414$ and $471 - 414 = 57$ .
Bring down the 9 to make 579. $579 \div 200 \approx 2$ . $207 \times 2 = 414$ and $579 - 414 = 165$ .
Bring down the 6 to make 1,656. $1{,}656 \div 200 \approx 8$ . $207 \times 8 = 1{,}656$ , so nothing is left.

So $47{,}196 \div 207 = 228$ .

What to do with a remainder

The question decides what the remainder means. Sometimes you round up, sometimes you drop it, and sometimes you write it as a fraction of the divisor.

Worked example: Reading the remainder

A library has 3,125 books to pack in boxes that hold 48 books each.

Long division gives $3{,}125 \div 48 = 65 \text{ R } 5$ , since $48 \times 65 = 3{,}120$ .

How many boxes can be filled completely? 65. The extra 5 books don't fill a box.
How many boxes are needed for all the books? 66. The last 5 books still need a box.
As a mixed number: the remainder is 5 out of a group of 48, so $3{,}125 \div 48 = 65\tfrac{5}{48}$ .

Tip

If a remainder is ever as big as the divisor, your last quotient digit was too small. Go back and try one more.

Practice

Practice 1

Find $7{,}992 \div 36$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Find $5{,}418 \div 42$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Find $9{,}648 \div 24$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Which is the best estimate for $61{,}850 \div 29$ ?

Practice 5

Find $6{,}000 \div 53$ . Write your answer like 12 R 3.

Type your answer

Practice 6

Find $38{,}556 \div 153$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

A bakery makes 1,850 cookies and packs them in tins of 24. After filling as many tins as possible, how many cookies are left over?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Find $7{,}245 \div 60$ . Write the answer as a mixed number in simplest form.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.