One place at a time

Before this page: know the multiplication tables to 12 × 12 and understand the value of each digit from place value.

Short division records a large division as a sequence of smaller divisions by the same one-digit divisor. It starts at the greatest occupied place and moves right, so every digit written in the answer keeps the same place value as the digit beneath it.

This page uses whole-number divisors from 2 to 9. A divisor with two or more digits needs long division; a decimal divisor is first changed using the method on dividing by decimals.

Why the remainder moves

In 984 ÷ 4, the 9 represents 9 hundreds. Dividing them equally gives 2 hundreds in each group, with 1 hundred left. That remaining hundred is exchanged for 10 tens; together with the next 8 tens, it makes 18 tens to divide.

• 1 hundred → 10 tens

  10 tens and the next 8 tens make 18 tens.

• 2 tens → 20 ones

  20 ones and the next 4 ones make 24 ones.

The small carried numerals are a compact record of those exchanges. They do not alter the dividend: they show how the remainder from one place is renamed in the next smaller place.

The short-division routine

Zeroes can belong in the quotient

After the first answer digit has been written, each remaining dividend digit must produce a digit in the quotient. In 6,048 ÷ 6, the thousands give 1. The hundreds give 0, the tens also give 0 remainder 4, and the remaining 4 tens become 40 ones. With the final 8 ones, that makes 48 ones. The quotient is 1,008.

The decimal point stays on its boundary

The decimal point is not divided or carried. Put the point in the quotient directly above the point in the dividend, then continue through the decimal places with the same routine. If the quotient is below 1, write a zero before its decimal point.

For 18.9 ÷ 7, the first two digits make 18 ones. This gives 2 ones remainder 4. Exchange the remaining 4 ones for 40 tenths; with the next 9 tenths, there are 49 tenths. Then 49 ÷ 7 = 7, so the quotient is 2.7.

Two ways to finish

First use every digit already written in the dividend. If the final remainder is zero, the division is complete. If it is not zero, the instruction in the question determines which route to take.

Following the decimal route

If the original digits run out and the question requires an exact decimal, extend the dividend with a zero after its decimal point. Add the point first if the dividend is a whole number. This does not change its value: 13 = 13.0 = 13.00 = 13.000.

Here the first digit, 1, is smaller than 8, so begin with 13. Then 13 ÷ 8 = 1 remainder 5; the remaining 5 ones become 50 tenths, and the division continues through the decimal places.

A terminating decimal finishes when its remainder reaches zero. Some divisions continue with a repeating pattern instead; those are written using recurring-decimal notation.

Divide two chosen numbers

Check by reversing the division

A failed inverse check usually points to an omitted quotient zero, an ignored carried remainder or a digit written in the wrong column.

Common mistakes

• Dividing the next digit on its ownAfter 9 ÷ 4 leaves remainder 1, the next amount is 18 tens, not 8 tens. Include the carried remainder before dividing.
• Dropping an internal zeroOnce the quotient has begun, write zero when a place makes no whole groups. Both zeroes in 1,008 hold necessary places.
• Moving the decimal pointThe point is not part of a division step. Keep it on the same place-value boundary by writing the quotient's point directly above it.
• Using the wrong finishing ruleA final remainder may be the required answer. Append zeroes only when a decimal is needed, and recognise that some decimal answers recur.

Summary

1. Place the one-digit divisor outside, the dividend inside and the quotient above the line.
2. Work from left to right: divide, write the quotient digit and carry any remainder into the following place.
3. Avoid unnecessary leading zeroes, but write the zero before a decimal below 1 and keep every internal zero.
4. After the original digits, either stop with a remainder or append zeroes and follow the decimal route.
5. Check by multiplying the quotient by the divisor, then adding any final integer remainder.