Dividing decimals
🎯 In this topic you will
- Divide decimals by whole numbers
- Divide decimals by numbers with one decimal place
- Estimate, multiply, and divide decimals by integers and decimals
🧠 Key Words
- equivalent calculation
- estimation
- inverse calculation
- reverse calculation
- short division
Show Definitions
- equivalent calculation: A different calculation that produces the same result as another.
- estimation: Finding an approximate value close to the exact answer.
- inverse calculation: A calculation that undoes the effect of another, such as subtraction undoing addition.
- reverse calculation: Working backwards from a result to find the original numbers or values.
- short division: A quick method of division that works directly on the digits without writing out the full long division steps.
❓ EXERCISES
1. Copy and complete these divisions.
a. $6.414 \div 3$
b. $9.43385 \div 5$
c. $6.65128 \div 8$
d. $5.1232 \div 4$
e. $7.6544 \div 7$
f. $0.846 \div 9$
👀 Show answer
b. $1.88677$
c. $0.83141$
d. $1.2808$
e. $1.09349$
f. $0.094$
2. Work out:
a. $8.654 \div 2$
b. $8.922 \div 6$
c. $32.925 \div 5$
d. $58.912 \div 8$
👀 Show answer
b. $1.487$
c. $6.585$
d. $7.364$
3. Maggie pays $\$9.28$ for $8\ \text{m}$ of ribbon.
What is the cost of the ribbon, per metre?
👀 Show answer
4. In a supermarket, five chickens cost $\$18.25$. What is the cost of one chicken?
👀 Show answer
5. Six friends have a meal in a restaurant. The total bill is $\$145.50$.
They share the bill equally between them. How much do they each pay?
👀 Show answer
6. Copy and complete these divisions.
a. $27.3852 \div 12$
b. $46.1875 \div 15$
c. $78.82825 \div 25$
👀 Show answer
b. $3.0783$
c. $3.15313$
7. Lara works out $112.4 \div 16$. She writes:
a. Explain the mistake that Lara has made.
b. Write down the correct answer.
👀 Show answer
b. The correct answer is $7.025$.
8. Kyle works out $251.55 \div 26$. He writes:
a. Instead of stopping the division and writing ‘remainder 13’, what should Kyle have done?
b. Work out the correct answer.
👀 Show answer
b. The correct answer is approximately $9.675$.
🧠 Think like a Mathematician
Question: What calculations could you do to check that the answer to a division is correct?
For example, how can you check that:
$56.322 \div 9 = 6.258$
- a) is approximately correct?
- b) is exactly correct?
Discuss in pairs or small groups.
👀 show answer
- a) Approximately correct: Round $56.322$ to $56.3$ and $9$ to $9$. $56.3 \div 9 \approx 6.26$, which is close to $6.258$, so the answer is reasonable.
- b) Exactly correct: Multiply $6.258 \times 9$ and check if it equals $56.322$ exactly. Since $6.258 \times 9 = 56.322$, the answer is correct.
❓ EXERCISES
10. Copy and complete the table below, which shows the $14$ times table.
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ |
|---|---|---|---|---|---|---|---|---|
| $14$ | $28$ | $42$ |
b. Use the table to help you to work out $126.392 \div 14$.
c. Show how to check that your answer to part b is correct. Use estimation and an inverse calculation.
🔎 Reasoning Tip
Answer check: Your rounded answer to part b multiplied by 14 should be approximately 126.
👀 Show answer
b. $126.392 \div 14 = 9.028$
c. Check: $9.028 \times 14 = 126.392$ ✔
11. Work with a partner to answer this question.
a. Mair works out that $235 \times 47 = 11045$.
Use this information to work out:
i. $11045 \div 47$
ii. $1104.5 \div 47$
iii. $110.45 \div 47$
iv. $11.045 \div 47$
b. Explain the method you used to work out the answers to parts a i, ii, iii and iv.
c. Use this method to work out the answers to the following.
i. $1104.5 \div 235$
ii. $110.45 \div 235$
iii. $11045 \div 235$
d. Check your answers with those of other learners in your class to see if you agree.
If you disagree on any of the answers, discuss where any mistakes have been made.
👀 Show answer
a ii. $23.5$
a iii. $2.35$
a iv. $0.235$
b. Method: Use proportional reasoning — each time you divide the dividend by $10$, the quotient also divides by $10$.
c i. $4.7$
c ii. $0.47$
c iii. $47$
12. This is part of Zara’s homework.
Use Zara’s method to work out the following. Round each of your answers to the required degree of accuracy.
a. $7.62 \div 5$ (1 d.p.)
b. $9.428 \div 7$ (2 d.p.)
c. $8.6 \div 13$ (3 d.p.)
👀 Show answer
b. $1.35$
c. $0.662$
13. Copy and complete these divisions.
a. $38.2 \div 2$
b. $50.7 \div 6$
c. $1.9 \div 8$
🔎 Reasoning Tip
Division accuracy: Work out the division to only one decimal place more than the degree of accuracy you need.
👀 Show answer
b. $8.45$
c. $0.2375$
🍬 Learning Bridge
Now that you’ve practised dividing decimals by whole numbers and learned to keep decimal places accurate, you’re ready to take on a new challenge — dividing by decimals. This builds directly on your division skills but adds the clever trick of creating an equivalent calculation with a whole-number divisor, making the process simpler and more efficient.
When you divide a number by a decimal, you can use the place value of the decimal to work out an easier equivalent calculation. An easier equivalent calculation is to divide by a whole number instead of a decimal.
For example, you can write $5.67 \div 0.7$ as $\frac{5.67}{0.7}$.
Multiplying the numerator and denominator of the fraction by $10$ gives:
$\frac{5.67 \times 10}{0.7 \times 10} = \frac{56.7}{7}$
This makes an equivalent calculation that is much easier to do because dividing by $7$ is much easier than dividing by $0.7$.
❓ EXERCISES
14. Copy and complete these divisions.
a. $2.4 \div 0.4 = \frac{2.4}{0.4} = \frac{2.4 \times 10}{0.4 \times 10} = \frac{24}{4} = \ \_\_$
b. $7.2 \div 0.9 = \frac{7.2}{0.9} = \frac{7.2 \times 10}{0.9 \times 10} = \frac{72}{9} = \ \_\_$
c. $-42 \div 0.6 = \frac{-42}{0.6} = \frac{-42 \times 10}{0.6 \times 10} = \frac{-420}{6} = \ \_\_$
d. $-45 \div 0.5 = \frac{-45}{0.5} = \frac{-45 \times 10}{0.5 \times 10} = \frac{-450}{5} = \ \_\_$
👀 Show answer
b. $8$
c. $-70$
d. $-90$
15. Which of these calculation cards is the odd one out? Explain why.
👀 Show answer
🧠 Think like a Mathematician
Question: Arun is working out $3 \div 0.6$. He says:
I understand that I must do $ \frac{3}{0.6} = \frac{3 \times 10}{0.6 \times 10} = \frac{30}{6} = 5$ because that makes the calculation a lot easier.
But I don’t understand why we don’t divide the answer at the end by 10 as we multiplied the numbers at the start by 10.
I think the answer should be $5 \div 10 = 0.5$.
Task: What explanation could you give to Arun to show that he is wrong?
👀 show answer
When you multiply both numbers in a division by the same amount, the value of the division does not change.
In $3 \div 0.6$, multiplying both $3$ and $0.6$ by $10$ gives $30 \div 6$. This is the same calculation, just without decimals.
Since you already adjusted both numbers equally, there is no need to divide the result by $10$ again — doing so would make the answer too small.
Therefore, the correct answer is $5$, not $0.5$.
❓ EXERCISES
17. Work out
🔎 Reasoning Tip
Follow these steps:
- Write the division as a fraction.
- Multiply the numerator and denominator by 10.
- Use short division to work out the answer.
a. $0.92 \div 0.4$
b. $5.74 \div 0.7$
c. $-774 \div 0.9$
d. $-288 \div 0.3$
👀 Show answer
b. $8.2$
c. $-860$
d. $-960$
18. Artur pays $\$1.08$ for a piece of string $0.8\,\text{m}$ long. Artur uses this formula to work out the cost of the string per metre.
What is the cost of the string per metre?
👀 Show answer
🧠 Think like a Mathematician
Question: What calculations can you do to check that the answer to a division is probably correct?
Example: How can you check that $20.504 \div 0.8 = 25.63$ is probably correct?
👀 show answer
To check if $20.504 \div 0.8 \approx 25.63$ is reasonable:
1. Estimate: Round $20.504$ to $20.5$ and $0.8$ to $0.8$. $20.5 \div 0.8$ is the same as $205 \div 8 \approx 25.625$. This is very close to $25.63$, so the answer seems reasonable.
2. Reverse operation: Multiply $25.63 \times 0.8$. $25.63 \times 0.8 = 20.504$, which matches the original number exactly.
✔ This confirms that the answer is correct.
❓ EXERCISES
20. This is part of Jamal’s homework.
🔎 Reasoning Tip
Approximation symbol: Remember, the symbol \( \approx \) means “is approximately equal to.”
Use Jamal’s method to work out each of these divisions.
i First, estimate the answer.
ii Then calculate the accurate answer.
a. $27.6 \div 0.3$
b. $-232 \div 0.4$
c. $306 \div 0.9$
d. $-483 \div 0.7$
e. $43.76 \div 0.8$
f. $-33972 \div 0.6$
👀 Show answer
b. Estimate: $-232 \approx -230$, $0.4 \approx 0.4$ → $-230 \div 0.4 \approx -575$; Accurate: $-580$
c. Estimate: $306 \approx 300$, $0.9 \approx 1$ → $300 \div 1 \approx 300$; Accurate: $340$
d. Estimate: $-483 \approx -480$, $0.7 \approx 0.7$ → $-480 \div 0.7 \approx -686$; Accurate: $-690$
e. Estimate: $43.76 \approx 44$, $0.8 \approx 0.8$ → $44 \div 0.8 \approx 55$; Accurate: $54.7$
f. Estimate: $-33972 \approx -34000$, $0.6 \approx 0.6$ → $-34000 \div 0.6 \approx -56667$; Accurate: $-56620$
21. Isla works out $50.46 \div 1.2$. This is what she writes.
a. Explain the mistake that Isla has made.
b. Write the correct answer.
👀 Show answer
b. $50.46 \div 1.2 = 42.05$.
22. Raffa works out $461.7 \div 1.8$. This is what he writes.
a. What should Raffa have done, instead of stopping the division and writing ‘remainder $9$’?
b. Work out the correct answer.
👀 Show answer
b. $461.7 \div 1.8 = 256.5$.
23. a. Copy and complete the table below showing the $19$ times table.
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ |
|---|---|---|---|---|---|---|---|---|
| $19$ | $38$ | $57$ |
b. Use the table to help you work out $59.375 \div 1.9$.
c. Show how to check that your answer to part b is correct. Use estimation with a reverse calculation.
🔎 Reasoning Tip
Answer check: In part c, your rounded answer to part b × 2 should be about 60.
👀 Show answer
b. $59.375 \div 1.9 = 31.25$ (since $\frac{59.375}{1.9}=\frac{593.75}{19}$ and $19 \times 31.25 = 593.75$).
c. Estimate: $\frac{60}{2}\approx 30$. Reverse check: $31.25 \times 1.9 = 59.375$ ✔
24. a. Complete the table below showing the $25$ times table.
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ |
|---|---|---|---|---|---|---|---|---|
| $25$ | $50$ | $75$ |
b. Helen buys a piece of wood for $\$58.90$. The piece of wood is $2.5\,\text{m}$ long.
Work out the cost per metre of the wood.
c. Show how to check your answer to part b is correct. Use estimation with a reverse calculation.
👀 Show answer
b. $\frac{58.90}{2.5} = 23.56$ dollars per metre.
c. Reverse check: $23.56 \times 2.5 = 58.90$ ✔
25. The diagram shows a rectangle with an area of $50.15\,\text{m}^2$. The width of the rectangle is $3.4\,\text{m}$.
Work out the length of the rectangle.
👀 Show answer
26. Work with a partner to answer this question.
a. Balsem works out that $425 \times 27 = 11475$.
Use this information to work out:
i. $11475 \div 27$
ii. $11475 \div 425$
iii. $11475 \div 2.7$
iv. $11475 \div 42.5$
b. Explain the method you used to work out the answers to part a.
c. Work out:
i. $1147.5 \div 2.7$
ii. $114.75 \div 2.7$
iii. $11.475 \div 2.7$
iv. $1.1475 \div 2.7$
👀 Show answer
a ii. $27$
a iii. $4250$
a iv. $270$
b. Used factor relationships from the given product: dividing the product by one factor gives the other factor; scaling factors adjusts the quotient proportionally.
c i. $425$
c ii. $42.5$
c iii. $4.25$
c iv. $0.425$
27. This is part B of Marcus’s homework.
Use Marcus’s method to work out these calculations. Round each of your answers to the given degree of accuracy.
🔎 Reasoning Tip
Division accuracy: You only need to work out the division to one decimal place more than the degree of accuracy you need.
a. $3.79 \div 0.6$ (1 d.p.)
b. $82.35 \div 1.1$ (2 d.p.)
c. $-5689 \div 2.3$ (1 d.p.)
👀 Show answer
a. $3.79 \div 0.6 \approx 6.316\ldots \Rightarrow 6.3$ (1 d.p.)
b. $82.35 \div 1.1 \approx 74.8636\ldots \Rightarrow 74.86$ (2 d.p.)
c. $-5689 \div 2.3 \approx -2473.478\ldots \Rightarrow -2473.5$ (1 d.p.)
🍬 Learning Bridge
Now that you can divide by decimals by converting them into easier whole-number calculations, it’s time to bring all your decimal skills together. In the next part, you’ll see how multiplying and dividing by decimals share the same place-value principles, letting you tackle both operations with a single, efficient strategy.
When you multiply or divide a number by a decimal, use the place value of the decimal to work out an equivalent calculation. For simple questions, you can do this ‘in your head’, or mentally. For more difficult questions, you will need to write down the steps in your working.
❓ EXERCISES
28. Work out mentally
a. $8 \times 0.2$
b. $8 \times (-0.7)$
c. $-0.6 \times 9$
d. $-0.4 \times (-15)$
e. $6 \times 0.05$
f. $-22 \times 0.03$
g. $0.12 \times 30$
h. $0.11 \times (-4)$
👀 Show answer
e. $0.3$ f. $-0.66$ g. $3.6$ h. $-0.44$
29. Copy and complete
a.
b.
👀 Show answer
b. $0.4 \times 0.007 = 0.0028$, $4 \times 7 = 28$, $0.04 \times 0.007 = 0.00028$
30. Sort these cards into groups that have the same answer.
👀 Show answer
• $3\times0.05$, $0.3\times0.5$, $0.03\times5$ all $=0.15$
• $30\times0.05$, $0.5\times3$ $=1.5$
• $500\times0.03$, $5\times0.3$ $=15$
• $0.003\times5$, $0.005\times3$, $0.03\times0.5$ $=0.015$
31. Work out mentally
a. $4 \div 0.2$
b. $-25 \div 0.5$
c. $12 \div (-0.4)$
d. $-60 \div (-0.1)$
e. $2 \div 0.05$
f. $28 \div (-0.07)$
👀 Show answer
32. Copy and complete.
a. $0.81 \div 0.09 = \, \_\_$
b. $6.4 \div 0.004 = \, \_\_$
👀 Show answer
b. $1600$ (since $\frac{6.4}{0.004}=\frac{6400}{4}$)
33. Which answer is correct, A, B, C or D?
a. $0.8 \div 0.02 =$ A $0.04$ B $0.4$ C $4$ D $40$
b. $4.5 \div 0.5 =$ A $0.9$ B $9$ C $90$ D $900$
c. $0.09 \div 0.003 =$ A $0.3$ B $3$ C $30$ D $300$
d. $3.6 \div 0.006 =$ A $0.6$ B $6$ C $60$ D $600$
👀 Show answer
🧠 Think like a Mathematician
c) Work out mentally:
- $12 \div 0.2$
- $12 \div 0.4$
- $12 \div 0.6$
- $12 \div 0.8$
- $12 \div 1.0$
- $12 \div 1.2$
d) Use your answers to part c to answer these questions:
- When you divide a number by $0.7$, do you expect your answer to be larger or smaller than when you divide the same number by $0.6$?
- When you divide a number by a decimal between $0$ and $1$, do you expect the answer to be larger or smaller than the number you started with?
👀 show answer
c) i) $12 \div 0.2 = 60$ ii) $12 \div 0.4 = 30$ iii) $12 \div 0.6 = 20$ iv) $12 \div 0.8 = 15$ v) $12 \div 1.0 = 12$ vi) $12 \div 1.2 = 10$
d) i) Dividing by $0.7$ gives a smaller answer than dividing by $0.6$, because the larger the divisor (when between 0 and 1), the smaller the quotient. ii) Dividing by a decimal between 0 and 1 makes the result larger than the original number.
❓ EXERCISES
35. Write True or False for each of these statements.
a. $5.729 \times 0.62 > 5.729$
b. $4.332 \div 0.95 > 4.332$
c. $12.664 \times 1.002
d. $45.19 \div 1.45
👀 Show answer
b. True (dividing by less than $1$ increases the value)
c. False (multiplying by more than $1$ increases the value)
d. True (dividing by more than $1$ decreases the value)
36. This is part of Hassan’s homework.
Is Hassan correct? Explain your answer. Show your working.
👀 Show answer
$\frac{24 \times 0.25}{0.2 \times 0.6} = \frac{6}{0.12} = 50$.
His working simplifies correctly to $5$ after dividing $6$ by $1.2$.
37. Work out
a. $\frac{48 \times 0.5}{0.04 \times 3}$
b. $\frac{120 \times 0.3}{0.2 \times 1.5}$
c. $\frac{84 \times 0.25}{35 \times 0.002}$
d. $\frac{120 \times 0.4 \times 0.1}{0.8 \times 0.15}$
👀 Show answer
b. $\frac{36}{0.3} = 120$
c. $\frac{21}{0.07} = 300$
d. $\frac{4.8}{0.12} = 40$
38. Here are six rectangular question cards and seven oval answer cards.
a. Match each question card with the correct answer card.
b. There is one answer card left over. Write a question card to go with that answer card.
c. Ask a partner to check that your question gives the correct answer.
👀 Show answer
A → v ($-24$)
B → vi ($-0.24$)
C → ii ($0.024$)
D → vii ($-2.4$)
E → i ($2.4$)
F → iii ($0.24$)
Leftover card: iv ($24$)
Example new question: $12 \times 2$ or $\frac{48}{2}$
🧠 Think like a Mathematician
Here is a calculation:
$28 \times 0.57 = 15.96$
What other calculations can you deduce from this calculation?
👀 show answer
From $28 \times 0.57 = 15.96$, we can deduce:
• $0.57 \times 28 = 15.96$ (commutative property)
• $15.96 \div 28 = 0.57$ (inverse of multiplication)
• $15.96 \div 0.57 = 28$ (inverse of multiplication)
• $280 \times 0.57 = 159.6$ (scaling both sides by 10)
• $2.8 \times 0.57 = 1.596$ (scaling both sides by 0.1)
• $28 \times 5.7 = 159.6$ (scaling one factor by 10)
• $2800 \times 0.057 = 159.6$ (shifting decimal places in both factors)
• Many other equivalent equations can be formed by proportional scaling of both factors and the product.
❓ EXERCISES
40.
a. Work out $123 \times 57$.
b. Use your answer to part a to write the answers to these calculations.
i. $12.3 \times 57$
ii. $123 \times 5.7$
iii. $12.3 \times 5.7$
iv. $1.23 \times 5.7$
v. $12.3 \times 0.57$
vi. $0.123 \times 0.57$
👀 Show answer
b. i. $701.1$ ii. $7011 \times 0.1 = 701.1$
iii. $70.11$ iv. $7.011$
v. $7.011 \times 0.1 = 0.7011$
vi. $0.07011$
41. Hugo uses these methods to estimate and work out the answer to this question.
a. Critique Hugo’s methods.
b. Can you improve his methods? If you can, write down your method(s).
c. Estimate and work out the answers to these calculations. Use your favourite methods.
i. $4.35 \times 27.5$
ii. $11.78 \div 0.19$
iii. $\frac{64 \times 3.6}{0.012}$
👀 Show answer
b. Improved method: Convert to $\frac{23 \times 378}{100 \times 10}$, or $2.3 \times 37.8$ and adjust decimal places at the end. Using $0.23 \times 37.8$, multiply $23 \times 378 = 8694$ and divide by $1000$ to get $8.694$, which rounds to $8.69$.
c.
i. $4.35 \times 27.5 = 119.625$
ii. $11.78 \div 0.19 \approx 62$
iii. $\frac{64 \times 3.6}{0.012} = \frac{230.4}{0.012} = 19200$
42. The diagram shows a rectangle. The area of the rectangle is $0.171\,\text{m}^2$.
a. Estimate the length of the rectangle.
b. Work out the length of the rectangle.
c. Compare your answers to parts a and b. Do you think your answer to part b is correct? Explain why.
👀 Show answer
b. Exact: $\frac{0.171}{0.38} \approx 0.45\,\text{m}$
c. The exact answer is slightly greater than the estimate because $0.38$ is slightly less than $0.4$, so dividing by it produces a slightly larger value.
⚠️ Common Mistake
When dividing by a decimal, students often forget to scale both numbers by the same power of 10 to make the divisor a whole number. For example, in $5.67 \div 0.7$, multiplying only the divisor by 10 gives $\frac{5.67}{7}$, which changes the problem and produces the wrong answer. The correct step is to multiply both numerator and denominator by 10: $\frac{5.67 \times 10}{0.7 \times 10} = \frac{56.7}{7} = 8.1$.
⚡ Tip: Always check your answer by multiplying it back by the original divisor — it should match the dividend.