Long division, one digit at a time

How do you share ₹3408 equally among 4 peo­ple? Or 7700 sheets of paper among 4 classes? The times tables can­not answer that in one step, because the num­bers are too big. Long divi­sion can. It breaks a big divi­sion into small ones, one digit at a time, and every small step uses a times table you already know.

How long divi­sion is writ­ten#

We write the divi­sor, a bracket, the div­i­dend, another bracket, and then the quo­tient as it grows:

$$4 \; \big) \; 3408 \; \big( \; 852$$

Read it as "3408 divided by 4 gives 852". The work­ing goes under­neath the div­i­dend, one step at a time.

Before you start, write the times table of the divi­sor at the side of your page. Then each step is a mat­ter of look­ing at the table.

The four steps, again and again#

1. Look. Take the dig­its on the left until the num­ber is big enough for the divi­sor to go into it.
2. Find the table fact. Find the biggest fact in the table that is not more than that num­ber. Write its mul­ti­plier in the quo­tient.
3. Take away. Write the prod­uct under the num­ber and take it away.
4. Bring down. Bring down the next digit of the div­i­dend beside what is left. Then go back to step 1.

When there are no more dig­its to bring down, what is left is the remain­der.

Your first long divi­sions#

Exam­ple 1#

Work out $100 \div 2$.

The first digit is 1, and 2 does not go into 1. So look at the first two dig­its together: 10. From the table, $2 \times 5 = 10$. Write 5 in the quo­tient and take away: $10 - 10 = 0$.

Bring down the last 0. Now the num­ber is 0, and $2 \times 0 = 0$, so write 0 in the quo­tient. Noth­ing is left.

$$\begin{array}{rl} 2\,\big) & 100\,\big(\,50 \\ - & \underline{10} \\ & \phantom{10}0 \end{array}$$

So $100 \div 2 = 50$.

Exam­ple 2#

Work out $123 \div 3$.

3 does not go into 1, so look at 12. $3 \times 4 = 12$, so write 4 and take away to leave 0. Bring down the 3. $3 \times 1 = 3$, so write 1. Noth­ing is left.

$$\begin{array}{rl} 3\,\big) & 123\,\big(\,41 \\ - & \underline{12} \\ & \phantom{1}03 \\ - & \phantom{12}\underline{3} \\ & \phantom{12}0 \end{array}$$

So $123 \div 3 = 41$. Check: $3 \times 41 = 123$.

Exam­ple 3#

Work out $3408 \div 4$.

4 does not go into 3, so look at 34. In the table, $4 \times 8 = 32$ and $4 \times 9 = 36$. 36 is too big, so use $4 \times 8 = 32$. Write 8, and $34 - 32 = 2$ is left. Bring down the 0 to make 20. $4 \times 5 = 20$, so write 5, leav­ing 0. Bring down the 8. $4 \times 2 = 8$, so write 2, leav­ing 0.

$$\begin{array}{rl} 4\,\big) & 3408\,\big(\,852 \\ - & \underline{32} \\ & \phantom{3}20 \\ - & \phantom{3}\underline{20} \\ & \phantom{34}08 \\ - & \phantom{340}\underline{8} \\ & \phantom{340}0 \end{array}$$

So $3408 \div 4 = 852$.

Exam­ple 4#

Work out $7700 \div 4$.

This time 4 goes into the first digit. $4 \times 1 = 4$, so write 1, and $7 - 4 = 3$ is left. Bring down 7 to make 37. $4 \times 9 = 36$, so write 9, leav­ing 1. Bring down 0 to make 10. $4 \times 2 = 8$, so write 2, leav­ing 2. Bring down the last 0 to make 20. $4 \times 5 = 20$, so write 5, leav­ing 0.

$$\begin{array}{rl} 4\,\big) & 7700\,\big(\,1925 \\ - & \underline{4} \\ & 37 \\ - & \underline{36} \\ & \phantom{7}10 \\ - & \phantom{77}\underline{8} \\ & \phantom{77}20 \\ - & \phantom{77}\underline{20} \\ & \phantom{770}0 \end{array}$$

So $7700 \div 4 = 1925$.

When the divi­sor does not go in: write a zero#

Some­times, after you bring down a digit, the num­ber is still smaller than the divi­sor. Then you must write 0 in the quo­tient before you bring down the next digit. For­get­ting this zero is the most com­mon mis­take in long divi­sion.

Exam­ple 5#

Work out $1224 \div 6$.

6 does not go into 1, so look at 12. $6 \times 2 = 12$, so write 2, leav­ing 0. Bring down 2. The num­ber is 2, and 6 does not go into 2. Write 0 in the quo­tient. Bring down the 4 as well, to make 24. $6 \times 4 = 24$, so write 4, leav­ing 0.

$$\begin{array}{rl} 6\,\big) & 1224\,\big(\,204 \\ - & \underline{12} \\ & \phantom{1}024 \\ - & \phantom{12}\underline{24} \\ & \phantom{122}0 \end{array}$$

So $1224 \div 6 = 204$. With­out the zero you would write 24, which is far too small. Check: $6 \times 204 = 1224$.

Exam­ple 6#

Work out $210012 \div 6$.

Look at 21. $6 \times 3 = 18$, so write 3, leav­ing 3. Bring down 0 to make 30. $6 \times 5 = 30$, so write 5, leav­ing 0. Bring down 0: 6 does not go into 0, so write 0. Bring down 1: 6 does not go into 1, so write another 0. Bring down 2 to make 12. $6 \times 2 = 12$, so write 2, leav­ing 0.

$$\begin{array}{rl} 6\,\big) & 210012\,\big(\,35002 \\ - & \underline{18} \\ & \phantom{2}30 \\ - & \phantom{2}\underline{30} \\ & \phantom{21}0012 \\ - & \phantom{2100}\underline{12} \\ & \phantom{21001}0 \end{array}$$

So $210012 \div 6 = 35002$. Count the dig­its as a check: every digit you bring down puts exactly one digit in the quo­tient. Answers such as 3502 or 352 have lost a zero.

When some­thing is left over#

Exam­ple 7#

Work out $8114 \div 11$.

Write the table of 11 first: 11, 22, 33, 44, 55, 66, 77, 88, 99. 11 does not go into 8, so look at 81. $11 \times 7 = 77$, so write 7, leav­ing 4. Bring down 1 to make 41. $11 \times 3 = 33$, so write 3, leav­ing 8. Bring down 4 to make 84. $11 \times 7 = 77$, so write 7, leav­ing 7. There are no more dig­its.

$$\begin{array}{rl} 11\,\big) & 8114\,\big(\,737 \\ - & \underline{77} \\ & \phantom{8}41 \\ - & \phantom{8}\underline{33} \\ & \phantom{81}84 \\ - & \phantom{81}\underline{77} \\ & \phantom{811}7 \end{array}$$

So the quo­tient is 737 and the remain­der is 7. Check: $(11 \times 737) + 7 = 8107 + 7 = 8114$.

Exam­ple 8#

Work out $180 \div 12$.

12 does not go into 1, so look at 18. $12 \times 1 = 12$, so write 1, leav­ing 6. Bring down 0 to make 60. $12 \times 5 = 60$, so write 5, leav­ing 0.

$$\begin{array}{rl} 12\,\big) & 180\,\big(\,15 \\ - & \underline{12} \\ & \phantom{1}60 \\ - & \phantom{1}\underline{60} \\ & \phantom{18}0 \end{array}$$

So $180 \div 12 = 15$.

Big­ger num­bers, the same steps#

How­ever long the div­i­dend is, the steps do not change. Here are some longer ones to try. Cover the answers, work them out, and then check.

The last one has three zeros in a row in the mid­dle of its quo­tient. Each time you bring down a digit and 2 does not go into the num­ber, a 0 goes into the quo­tient.

Divi­sion writ­ten as a frac­tion#

A divi­sion can also be writ­ten with a line, one num­ber over the other. $\displaystyle \frac{8}{2}$ means $8 \div 2$. To work it out, ask which table fact gives the top num­ber.

You will use this way of writ­ing divi­sion a great deal when you meet frac­tions.

Com­mon mis­takes#

• For­get­ting a zero in the quo­tient. In $1224 \div 6$ the answer is 204, not 24.
• Using a table fact that is too big. For 34 divided by 4, $4 \times 9 = 36$ is more than 34, so use $4 \times 8 = 32$.
• Leav­ing too much behind. If what is left after tak­ing away is as big as the divi­sor, you chose a fact that was too small.
• Not check­ing. Mul­ti­ply the quo­tient by the divi­sor and add the remain­der. You should get the div­i­dend back.

Key terms#

• Long divi­sion: divid­ing a big num­ber one digit at a time.
• Bring down: write the next digit of the div­i­dend beside what is left.
• Quo­tient: the answer to a divi­sion.
• Remain­der: what is left at the very end.

Prac­tice#

1. $86 \div 2$
2. $502 \div 2$
3. $303 \div 3$
4. $936 \div 4$
5. $5205 \div 5$
6. $13248 \div 2$
7. $2412 \div 6$
8. $7215 \div 5$
9. $\displaystyle \frac{18}{2}$
10. Find the quo­tient and remain­der when 125 is divided by 4.
11. $252 \div 12$
12. ₹1236 is shared equally among 4 friends. How much does each friend get?

Answers#