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Dividing decimals

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visibility 117update 8 months agobookmarkshare

🎯 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.
 
📘 Worked example

Work out:

a. $4.258 \div 2$

b. $41.481 \div 18$

Answer:

a.

2┌────── 2.┌────── 2.12┌────── 2.129┌──────
2│ 4.258‌ ‌ ‌2│ 4.258‌ ‌‌ ‌ ‌ ‌2│ 4.258‌‌ ‌ ‌ ‌ ‌ ‌2│ 4.258

b.

‌ 2┌──────── 2.┌──────── 2.30┌──────── 2.304┌────────
18│ 41.481‌ ‌ ‌‌18│ 41.481‌ ‌ ‌ ‌ ‌‌18│ 41.481‌ ‌ ‌ ‌ ‌‌ ‌‌18│ 41.4810

a. First, work out $4 \div 2 = 2$. Write $2$ above the $4$.
Write the decimal point in the answer.
Continue the division: $2 \div 2 = 1$, write $1$ above the $2$.
$5 \div 2 = 2 \ r1$. Write $2$ above the $5$ and carry the $1$ onto the $8$.
$18 \div 2 = 9$ exactly, write $9$ above the $18$.

b. First, work out $41 \div 18 = 2 \ r5$. Write $2$ above the $1$ and carry the $5$ onto the $4$.
Write the decimal point in the answer.
Continue the division: $54 \div 18 = 3$ exactly, write $3$ above the $54$.
$8 \div 18 = 0 \ r8$. Write $0$ above the $8$ and carry the $8$ onto the $1$.
$81 \div 18 = 4 \ r9$. Write $4$ above the $81$ and carry the $9$ onto a new zero.
$90 \div 18 = 5$ exactly.

 

🧠 PROBLEM-SOLVING Strategy

Dividing Decimals by Whole Numbers or Decimals

Convert decimal divisors to whole numbers when possible, then use normal division rules while keeping decimal places accurate.

  1. Set up the division — dividend inside bracket, divisor outside.
  2. Align decimal points in working and in the answer.
  3. If divisor is a whole number — use standard short or long division.
  4. If divisor has decimals — multiply both dividend and divisor by the same power of 10 to make the divisor a whole number, then divide.
  5. Write zeros if needed to reach required accuracy.
  6. Estimate first to check if your answer will be reasonable.
  7. Check the answer by multiplying quotient × divisor — it should match the dividend within rounding limits.
  8. Apply sign rules — same signs → positive; different signs → negative.

Quick examples

  • $4.258 \div 2 = 2.129$
  • $41.481 \div 18 \approx 2.305$
  • $9.28 \div 8 = 1.16$
 

❓ 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
a. $2.138$
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
a. $4.327$
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
$\$9.28 \div 8 = \$1.16$ per metre

4. In a supermarket, five chickens cost $\$18.25$. What is the cost of one chicken?

👀 Show answer
$\$18.25 \div 5 = \$3.65$ per chicken

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
$\$145.50 \div 6 = \$24.25$ each

6. Copy and complete these divisions.

a. $27.3852 \div 12$

b. $46.1875 \div 15$

c. $78.82825 \div 25$

👀 Show answer
a. $2.2821$
b. $3.0783$
c. $3.15313$

7. Lara works out $112.4 \div 16$. She writes:

Lara's working for 112.4 ÷ 16

a. Explain the mistake that Lara has made.

b. Write down the correct answer.

👀 Show answer
a. Lara made a place value error when bringing down the digits after the decimal point.
b. The correct answer is $7.025$.

8. Kyle works out $251.55 \div 26$. He writes:

Kyle's working for 251.55 ÷ 26

a. Instead of stopping the division and writing ‘remainder 13’, what should Kyle have done?

b. Work out the correct answer.

👀 Show answer
a. He should have continued the division by adding a zero after the decimal point and carrying on until the remainder was eliminated or a repeating decimal was established.
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:

$56.322 \div 9 = 6.258$

  1. a) is approximately correct?
  2. 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
Completed table: $14,\ 28,\ 42,\ 56,\ 70,\ 84,\ 98,\ 112,\ 126$
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 i. $235$
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.

Zara's division example

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
a. $1.5$
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
a. $19.1$
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 $0.7$.

 
📘 Worked example

Work out:

a. $-32 \div 0.2$

b. $3.468 \div 0.8$

Answer:

a.
$-32 \div 0.2=\dfrac{-32}{0.2}$
$\dfrac{-32\times 10}{0.2\times 10}=\dfrac{-320}{2}$
$-320 \div 2=-160$

b.
$3.468 \div 0.8=\dfrac{3.468}{0.8}$
$\dfrac{3.468\times 10}{0.8\times 10}=\dfrac{34.68}{8}$

3 4 . 6 8
          3 2
          ───
            2 6
            2 4
            ───
              2 8
              2 4
              ───
                4 8
                4 8
                ───
                  0

$3.468 \div 0.8 = 4.335$

a. First write the division as a fraction. Multiply numerator and denominator by $10$ to remove the decimal in the denominator: $\dfrac{-32}{0.2}=\dfrac{-320}{2}$. Finally, divide to get $-160$.

b. Write the division as a fraction and multiply top and bottom by $10$ to make the denominator an integer: $\dfrac{34.68}{8}$. Now perform short division of $34.68$ by $8$ to get $4.335$.

 

🧠 PROBLEM-SOLVING Strategy

Dividing by Decimals

Remove the decimal from the divisor by scaling both numbers, then divide as normal.

  1. Write the division as a fraction — \(\dfrac{\text{dividend}}{\text{divisor}}\).
  2. Eliminate the decimal from the divisor by multiplying both numerator and denominator by the same power of 10.
  3. Perform the equivalent whole-number division using short or long division.
  4. Apply sign rules — same signs → positive; different signs → negative.
  5. Check your answer — multiply the quotient by the original divisor to recover the dividend (within rounding limits).
  6. Estimate first to ensure the result is reasonable.

Quick example

  • $5.67 \div 0.7 = \dfrac{5.67}{0.7} = \dfrac{56.7}{7} = 8.1$
  • $-32 \div 0.2 = \dfrac{-32}{0.2} = \dfrac{-320}{2} = -160$
 

❓ 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
a. $6$
b. $8$
c. $-70$
d. $-90$

15. Which of these calculation cards is the odd one out? Explain why.

Calculation cards

👀 Show answer
The odd one out is card B ($1.4 \div 0.2 = 7$) because its answer is not a whole number multiple of $9$, while the others give whole number multiples of $7$ or $9$.
 

🧠 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:

  1. Write the division as a fraction.
  2. Multiply the numerator and denominator by 10.
  3. 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
a. $2.3$
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.

Formula for cost per metre

What is the cost of the string per metre?

👀 Show answer
$\frac{\$1.08}{0.8} = \$1.35$ per metre
 

🧠 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.”

Jamal's homework example for estimating and calculating division

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
a. Estimate: $27.6 \approx 28$, $0.3 \approx 0.3$ → $28 \div 0.3 \approx 93$; Accurate: $92$
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.

Isla's working for 50.46 ÷ 1.2

a. Explain the mistake that Isla has made.

b. Write the correct answer.

👀 Show answer
a. After multiplying both numbers by $10$ to get $504.6 \div 12$, she misplaced the decimal in the long division and treated $0.6 \div 12$ as $0.5$ instead of $0.05$ (place‑value/bringing‑down error).
b. $50.46 \div 1.2 = 42.05$.

22. Raffa works out $461.7 \div 1.8$. This is what he writes.

Raffa's working for 461.7 ÷ 1.8

a. What should Raffa have done, instead of stopping the division and writing ‘remainder $9$’?

b. Work out the correct answer.

👀 Show answer
a. Continue the division past the decimal point (add a decimal and a zero) after $4617 \div 18$ so the remainder is expressed as a decimal.
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
a. Completed row: $19,\ 38,\ 57,\ 76,\ 95,\ 114,\ 133,\ 152,\ 171$.
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
a. Completed row: $25,\ 50,\ 75,\ 100,\ 125,\ 150,\ 175,\ 200,\ 225$.
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.

Rectangle diagram with area and width

👀 Show answer
$\text{Length} = \frac{50.15}{3.4} \approx 14.75\,\text{m}$.

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 i. $425$
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.

Marcus's division example

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
Example: $1.798 \div 0.7 = 2.56\ldots \approx 2.6$ (1 d.p.)

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.

 
📘 Worked example

Work out:

a. $12 \times 0.6$  b. $0.3 \times 0.15$  c. $-16 \div 0.4$
d. $9.6 \div -0.12$  e. $\dfrac{36 \times 0.5}{0.2 \times 4.5}$

Answer:

a.
$12 \times 6 = 72$
$12 \times 0.6 = 7.2$

b.
$3 \times 15 = 45$
$0.3 \times 0.15 = 0.045$

c.
$\dfrac{-16 \times 10}{0.4 \times 10} = \dfrac{-160}{4} = -40$

d.
$\dfrac{9.6 \times 100}{-0.12 \times 100} = \dfrac{960}{-12} = -80$

e.
$36 \times 0.5 = 18$
$2 \times 4.5 = 9$
$0.2 \times 4.5 = 0.9$
$\dfrac{36 \times 0.5}{0.2 \times 4.5} = \dfrac{18}{0.9} = \dfrac{180}{9} = 20$

a. Ignore the decimal point and calculate $12 \times 6 = 72$. Since $6$ is $10$ times bigger than $0.6$, divide by $10$ to get $7.2$.

b. Ignore the decimals and calculate $3 \times 15 = 45$. Now place the decimal — $0.3$ has $1$ decimal place and $0.15$ has $2$, so the answer has $3$: $0.045$.

c. Think of the division as a fraction. Multiply numerator and denominator by $10$ to eliminate the decimal: $\dfrac{-16}{0.4} = \dfrac{-160}{4} = -40$.

d. Again think of the division as a fraction. Multiply numerator and denominator by $100$ to remove the decimal: $\dfrac{9.6}{-0.12} = \dfrac{960}{-12} = -80$.

e. First find the numerator: $36 \times 0.5 = 18$. Then the denominator: $2 \times 4.5 = 9$. Since $0.2$ is $10$ times smaller than $2$, divide $9$ by $10$ to get $0.9$. Rewrite the fraction as $\dfrac{18}{0.9}$ and multiply top and bottom by $10$ to get $\dfrac{180}{9} = 20$.

 

🧠 SOLVING Strategy

Multiplying and Dividing by Decimals

Use place value to turn decimal operations into simpler whole-number calculations.

  1. Understand place value — multiplying or dividing by a decimal changes the scale of the number.
  2. For mental work — remove the decimal by scaling both numbers by powers of 10, calculate, then replace the decimal point.
  3. For written work — if dividing, write as a fraction, multiply top and bottom by a power of 10 to make the divisor a whole number, then divide as normal.
  4. For multiplication — ignore decimals, multiply as integers, then insert the decimal point according to the total decimal places in the question.
  5. Check with estimation — round to 1 s.f. and confirm the result is reasonable.
  6. Mind the signs — same signs → positive; different signs → negative.

Quick examples

  • $12 \times 0.6 = 7.2$   (12 × 6 = 72, then ÷10)
  • $-16 \div 0.4 = -40$   ($\frac{-16}{0.4} = \frac{-160}{4}$)
  • $\frac{36 \times 0.5}{0.2 \times 4.5} = 20$
 

❓ 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
a. $1.6$   b. $-5.6$   c. $-5.4$   d. $6$
e. $0.3$   f. $-0.66$   g. $3.6$   h. $-0.44$

29. Copy and complete

a.

Row a: copy-and-complete products with 0.08×0.2, 8×2, 0.008×0.2

b.

Row b: copy-and-complete products with 0.4×0.007, 4×7, 0.4×0.007 etc.

👀 Show answer
a. $0.08 \times 0.2 = 0.016$,   $8 \times 2 = 16$,   $0.008 \times 0.2 = 0.0016$
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.

Cards A–L with decimal products to group by equal value

👀 Show answer
Example groupings (any equivalent grouping earns credit):
• $3\times0.05$, $0.3\times0.5$, $0.03\times5$ 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
a. $20$   b. $-50$   c. $-30$   d. $600$   e. $40$   f. $-400$

32. Copy and complete.

a. $0.81 \div 0.09 = \, \_\_$

Scaffold showing ×100 method for 0.81 ÷ 0.09

b. $6.4 \div 0.004 = \, \_\_$

Scaffold showing ×1000 method for 6.4 ÷ 0.004

👀 Show answer
a. $9$   (since $\frac{0.81}{0.09}=\frac{81}{9}$)
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
a. D ($40$)   b. B ($9$)   c. C ($30$)   d. D ($600$)
 

🧠 Think like a Mathematician

c) Work out mentally:

  1. $12 \div 0.2$
  2. $12 \div 0.4$
  3. $12 \div 0.6$
  4. $12 \div 0.8$
  5. $12 \div 1.0$
  6. $12 \div 1.2$

d) Use your answers to part c to answer these questions:

  1. 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 $0.6$?
  2. 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
a. False (multiplying by $0.62$ gives a smaller number)
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.

Hassan's division example

Is Hassan correct? Explain your answer. Show your working.

👀 Show answer
Yes, Hassan is correct.
$\frac{24 \times 0.25}{0.2 \times 0.6} = \frac{6}{0.12} = 50$.
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
a. $\frac{24}{0.12} = 200$
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.

Six rectangular question cards

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
Matching:
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
a. $123 \times 57 = 7011$
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.

Hugo's estimation and calculation example

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
a. Hugo’s estimate ($0.2 \times 40 = 8$) is reasonable, but his written calculation is inefficient. He calculates $378 \times 20$ directly instead of adjusting decimal places earlier to simplify.
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 $0.171\,\text{m}^2$.

Rectangle with area 0.171 m² and width 0.38 m

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
a. Estimate: $\frac{0.17}{0.4} \approx 0.425\,\text{m}$
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.

 

📘 What we've learned — Dividing & Multiplying Decimals

  • Dividing by whole numbers: set up short/long division with the dividend inside the bracket, divisor outside; align decimal points in working and answer.
  • Dividing by decimals: multiply both dividend and divisor by the same power of 10 to make the divisor a whole number, then divide normally.
  • Multiplying by decimals: ignore decimals, multiply as integers, then place the decimal point according to the total decimal places in the question.
  • Sign rules:
    • Positive ÷/× Positive = Positive
    • Positive ÷/× Negative = Negative
    • Negative ÷/× Negative = Positive
  • Estimation check: round to 1 significant figure before calculating to see if the answer is reasonable.
  • Check answers: multiply the quotient by the divisor to confirm the dividend (within rounding limits).
  • Place value sense-check:
    • Dividing by a decimal < 1 gives a larger answer than the starting number.
    • Multiplying by a decimal < 1 gives a smaller answer than the starting number.
    • Both factors < 1 → product smaller than each factor.
  • Common mistakes to avoid:
    • Forgetting to shift both dividend and divisor when removing decimals.
    • Miscounting total decimal places when placing the decimal in the product.
    • Ignoring sign rules when working with negatives.
  • Quick examples:
    • $4.258 \div 2 = 2.129$
    • $5.67 \div 0.7 = \frac{56.7}{7} = 8.1$
    • $0.002 \times 4 = 0.008$
    • $476 \times 3.7 = 1761.2$
    • $-16 \div 0.4 = -40$
    • $\frac{36 \times 0.5}{0.2 \times 4.5} = 20$
 
 

 

 

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