Multiplying decimals booklet
Multiplying decimals booklet
- Unit 1: Integers
- Unit 2: Place value & Rounding
- Unit 3: Decimals
- Unit 4: Fractions
- Unit 5: Percentages
- Unit 6: Ratio & Proportion
🎯 In this topic you will
- Multiply decimals by whole numbers and decimals
🧠 Key Words
- estimation
- fill in
- mentally
- place value
- written method
Show Definitions
- estimation: Finding an approximate value close to the exact answer.
- fill in: To complete missing parts of a number, statement, or calculation.
- mentally: Solving a calculation in your head without writing it down.
- place value: The value of a digit depending on its position in a number.
- written method: A step-by-step calculation recorded on paper or digitally.
Follow these steps when you multiply a decimal by a whole number:
- First, work out the multiplication without the decimal point.
- Finally, put the decimal point in the answer. There must be the same number of digits after the decimal point in the answer as there were in the question.
❓ EXERCISES
1. Use a mental method to work out the following.
a) $0.1 \times 8$
b) $0.5 \times 5$
c) $0.9 \times 2$
👀 Show answer
b) $0.5\times 5=\boxed{2.5}$
c) $0.9\times 2=\boxed{1.8}$
2. Use a mental method to work out the following. All the answers are shown in the cloud.
a) $6\times 0.02$
b) $4\times 0.3$
c) $3\times 0.004$
d) $120\times 0.1$
👀 Show answer
3. Copy this diagram and fill in the missing numbers. All the calculations give the answer in the centre oval. All the missing numbers are given in the rectangle.
👀 Show answer
4. Kai works out that $521\times 53=27613$. Use this information to write down the answer to the following.
a) $521\times 5.3$
b) $521\times 0.53$
👀 Show answer
b) $521\times 0.53=\dfrac{27613}{100}=\boxed{276.13}$
🧠 Think like a Mathematician
Question:
- a) What is the answer to $52.1 \times 53$? What do you notice about this answer and the answer to part a of Question 4?
- b) Why is the answer to $521 \times 0.53$ the same as the answer to $5.21 \times 53$?
Follow-up prompts:
👀 show answer
- a:$52.1 \times 53 = 2761.3$. This answer is exactly 10 times the answer to $5.21 \times 53$ because 52.1 is 10 times larger than 5.21.
- b:$521 \times 0.53 = (521 \times 53) \div 100$ and $5.21 \times 53 = (521 \times 53) \div 100$. Both involve multiplying 521 by 53 and then dividing by 100, so the results are identical.
- Key idea: Moving the decimal point in one factor changes the size of the number by a power of ten, which you can compensate for in the other factor to keep the product the same.
❓ EXERCISES
6. a) Work out $162\times 34$.
b) Use your answer to part a to write down the answers to the following.
i) $162\times 3.4$ ii) $162\times 0.34$ iii) $162\times 0.034$
iv) $16.2\times 34$ v) $1.62\times 34$ vi) $0.162\times 34$
👀 Show answer
b) Scale by powers of $10$ from part a: i) $162\times 3.4=\dfrac{5508}{10}=\boxed{550.8}$ ii) $\dfrac{5508}{100}=\boxed{55.08}$ iii) $\dfrac{5508}{1000}=\boxed{5.508}$
iv) $16.2\times 34=\dfrac{5508}{10}=\boxed{550.8}$ v) $\dfrac{5508}{100}=\boxed{55.08}$ vi) $\dfrac{5508}{1000}=\boxed{5.508}$
7. Raj uses these methods to work out and check his answer.
a) Write down the advantages and disadvantages of Raj’s method.
b) Can you improve his method?
👀 Show answer
Disadvantages: The grid uses many partial products (time‑consuming); the rounding check is crude—$200$ is quite far from $167.4$, so it doesn’t tightly validate the answer.
b) Possible improvements (any one of the following, with working):
• Use compensation: $0.45=0.5-0.05$, so $0.45\times372=(0.5\times372)-(0.05\times372)=186-18.6=\boxed{167.4}$. • Scale first to reduce steps: $372\times45=\boxed{16740}$ then divide by $100\Rightarrow \boxed{167.4}$ (do $372\times(40 5)=14880 1860=16740$). • Use $4.5\times37.2$ (both ×$10$ then ÷$100$): $4.5\times37.2=(4.5\times30) (4.5\times7.2)=135 32.4=\boxed{167.4}$. • Tighter check: round to $0.45\times370=166.5$ or $0.45\times380=171$; $167.4$ lies reasonably between these.
8. Work out these multiplications. Show how to check your answers.
🔎 Reasoning Tip
Checking your work: For the check for part c, work out \( 0.8 \times 40 \).
a) $3.2 \times 52$
b) $8.1 \times 384$
c) $0.78 \times 41$
👀 Show answer
b) $8.1 \times 384 = 8.1 \times (400 - 16) = 3240 - 129.6 = \boxed{3110.4}$ Check: $8 \times 384 = 3072$ (close to $3110.4$).
c) $0.78 \times 41 = 0.78 \times (40 1) = 31.2 0.78 = \boxed{31.98}$ Check: $0.8 \times 41 = 32.8$ (close to $31.98$).
9. This is part of Anna’s homework.
a) Without checking the method and working out the answer, how can you tell that Anna is incorrect?
b) Work out the correct answer, showing all your working.
👀 Show answer
b) Step 1: Ignore the decimal: $4735\times 18$ $4735 \times 10 = 47350$ $4735 \times 8 = 37880$ Sum: $47350 37880 = 85230$
Step 2: Adjust for two decimal places in $47.35$: $85230 \div 100 = \boxed{852.30}$ So the correct answer is $\mathbf{852.3}$.
10. In 1 gram of green gold there is 0.23 g of copper.
How many grams of copper are there in 36 g of green gold?
👀 Show answer
11. Darren exchanges some British pounds (£) to US dollars ($).
For every £1 he receives $1.29.
Darren says, ‘If I exchange £275, I should receive about $350.’
Is Darren correct? Explain your answer.
👀 Show answer
12. Samir manages a hotel. The table shows the cost of items that he buys for the hotel bathrooms.
Samir buys:
• 350 bottles of shampoo
• 425 bottles of shower gel
• 275 bottles of hand lotion
• 600 bars of soap.
What is the total cost of these items?
👀 Show answer
Total cost: $91 97.75 88 108 = \boxed{\$384.75}$.
🍬 Learning Bridge
Now that you can confidently multiply decimals by whole numbers and decimals using place value, you’re ready to take it a step further. Next, you’ll apply the same skills to situations with negative numbers and more varied decimal combinations — where understanding signs, place value, and estimation becomes even more important for getting accurate results.
Follow these steps when you multiply a decimal by a whole number or a decimal:
- First, work out the multiplication without the decimal points.
- Finally, put the decimal point in the answer. There must be the same number of digits after the decimal point in the answer as there were in the question.
❓ EXERCISES
13. Use a mental method to work out:
a) $0.1 \times (-8)$
b) $0.2 \times 3$
c) $0.3 \times (-7)$
d) $0.7 \times 8$
e) $0.9 \times (-4)$
👀 Show answer
b) $0.2 \times 3 = 0.6$
c) $0.3 \times (-7) = -2.1$
d) $0.7 \times 8 = 5.6$
e) $0.9 \times (-4) = -3.6$
14. Use a mental method to work out:
a) $-6 \times 0.03$
b) $-9 \times 0.2$
c) $-18 \times 0.001$
d) $-20 \times 0.9$
👀 Show answer
b) $-9\times 0.2 = \boxed{-1.8}$
c) $-18\times 0.001 = \boxed{-0.018}$
d) $-20\times 0.9 = \boxed{-18}$
15. Here are five calculation cards:
a) Work out the answers to the calculations on the cards.
b) Write the answers in order of size, starting with the smallest.
👀 Show answer
B) $0.039\times (-180) = -7.02$
C) $0.85\times (-9) = -7.65$
D) $0.44\times (-16) = -7.04$
E) $0.04\times (-182) = -7.28$
Order from smallest to largest: $-7.65$ (C), $-7.28$ (E), $-7.2$ (A), $-7.04$ (D), $-7.02$ (B)
🧠 Think like a Mathematician
Scenario: Arun says: “I don’t understand why $0.2 \times 0.3$ is $0.06$ and not $0.6$.”
Zara shows Arun this pattern:
- $2 \times 3 = 6$
- $0.2 \times 3 = 0.6$
- $0.2 \times 0.3 = 0.06$
- $0.2 \times 0.03 = 0.006$
- $0.2 \times 0.003 = 0.0006$
Question: How can you use the place value of the digits in 0.2 and 0.3 to explain to Arun why the answer is 0.06 and not 0.6?
Follow-up prompts:
👀 show answer
- Explanation: - $0.2 = \dfrac{2}{10}$ and $0.3 = \dfrac{3}{10}$. - Multiplying: $\dfrac{2}{10} \times \dfrac{3}{10} = \dfrac{6}{100}$. - $\dfrac{6}{100} = 0.06$.
- Each factor has one decimal place, so in the product you must have two decimal places in total. This is why 0.2 × 0.3 gives 0.06, not 0.6.
- Key idea: Multiplying by a decimal less than 1 reduces the value, not increases it.
❓ EXERCISES
17. a) Copy and complete these patterns.
i $2\times 4 = 8$
$0.2\times 4 = \_\_\_$
$0.2\times 0.4 = \_\_\_$
$0.2\times 0.04 = \_\_\_$
$0.2\times 0.004 = \_\_\_$
ii $3\times 5 = 15$
$0.3\times 5 = \_\_\_$
$0.3\times 0.5 = \_\_\_$
$0.3\times 0.05 = \_\_\_$
$0.3\times 0.005 = \_\_\_$
👀 Show answer
ii: $0.3\times 5 = 1.5$ $0.3\times 0.5 = 0.15$ $0.3\times 0.05 = 0.015$ $0.3\times 0.005 = 0.0015$
17. b) Work out:
i) $0.1\times 0.09$
ii) $0.6\times 0.8$
iii) $0.07\times 0.4$
iv) $0.03\times 0.05$
v) $0.12\times 0.3$
vi) $0.06\times 0.11$
👀 Show answer
ii) $0.6\times 0.8 = 0.48$
iii) $0.07\times 0.4 = 0.028$
iv) $0.03\times 0.05 = 0.0015$
v) $0.12\times 0.3 = 0.036$
vi) $0.06\times 0.11 = 0.0066$
18. Fill in the missing spaces in this spider diagram. All the calculations give the answer in the middle ($0.36$). All the answers are in the yellow rectangle on the right.
👀 Show answer
$0.9\times 0.4 = 0.36$ → missing: $0.4$
$36\times 0.01 = 0.36$ → missing: $36$
$3\times 0.12 = 0.36$ → missing: $3$
$0.06\times 6 = 0.36$ → missing: $0.06$
$0.04\times 9 = 0.36$ → missing: $0.04$
$0.3\times 1.2 = 0.36$ → missing: $0.3$
🧠 Think like a Mathematician
Given:$42 \times 87 = 3654$
a) Use this information to find the results of:
- $42 \times 8.7$
- $42 \times 0.87$
- $4.2 \times 87$
- $4.2 \times 8.7$
b) Explain why your answers to (i) and (iii) are the same, and why your answers to (ii) and (iv) are the same.
c) Generalise a method that can be used to adjust the calculation $42 \times 87 = 3654$ to solve other similar problems with decimals, such as $0.42 \times 0.87$ or $0.042 \times 8.7$.
👀 show answer
- a)
- (i) $42 \times 8.7 = 365.4$
- (ii) $42 \times 0.87 = 36.54$
- (iii) $4.2 \times 87 = 365.4$
- (iv) $4.2 \times 8.7 = 36.54$
- b) In (i) and (iii), one number is 10 times smaller while the other is 10 times larger, so the product stays the same. The same reasoning applies for (ii) and (iv).
- c) Start from the known product $42 \times 87 = 3654$. - If one factor is divided by 10, divide the product by 10. - If one factor is multiplied by 10, multiply the product by 10. Apply these adjustments according to the decimal shift in each factor.
❓ EXERCISES
20. a) Work out $158 \times 46$.
b) Use your answer to part a to write the answers to these multiplications:
i) $15.8 \times 46$
ii) $158 \times 4.6$
iii) $15.8 \times 4.6$
iv) $1.58 \times 4.6$
v) $15.8 \times 0.46$
vi) $1.58 \times 0.046$
👀 Show answer
b) i) $15.8 \times 46 = \dfrac{7268}{10} = \boxed{726.8}$
ii) $158 \times 4.6 = \dfrac{7268}{10} = \boxed{726.8}$
iii) $15.8 \times 4.6 = \dfrac{7268}{100} = \boxed{72.68}$
iv) $1.58 \times 4.6 = \dfrac{7268}{1000} = \boxed{7.268}$
v) $15.8 \times 0.46 = \dfrac{7268}{1000} = \boxed{7.268}$
vi) $1.58 \times 0.046 = \dfrac{7268}{100000} = \boxed{0.07268}$
21. Sam uses this method to work out and check her answer.
🔎 Reasoning Tip
Estimation check: Use estimation to check your answers by rounding all the numbers in the question to one significant figure.
a) Write the advantages and disadvantages of Sam’s method.
b) Can you improve her method?
c) Which method do you prefer to use to multiply decimals? Write why you prefer this method.
👀 Show answers
b) Possible improvements: • Compute $67\times 428=28\,676$ then place 4 decimal places → $\boxed{2.8676}$. • Or use compensation: $(0.7-0.03)\times 4.28=2.996-0.1284=\boxed{2.8676}$. • Or split: $0.67\times 4 0.67\times 0.28 = 2.68 0.1876 = \boxed{2.8676}$. • Tighter check: $0.67\times 4.3\approx 2.881$ and $0.67\times 4.2\approx 2.814$ → answer should be between $2.814$ and $2.881$.
c) Sample preference: “Multiply as integers then fix the decimal places.” It’s fast (one main multiplication) and reduces chance of misplacing partial products; I use grid/area only for teaching place value.
22. Work out these multiplications. Show how to check your answers.
a) $6.7 \times 9.4$
b) $0.56 \times 8.3$
c) $0.23 \times 8.15$
d) $0.69 \times 0.254$
👀 Show answers
b) $0.56\times 8.3=\dfrac{56\times 83}{1000}=\dfrac{4648}{1000}=\boxed{4.648}$. Check: $0.6\times 8.3\approx 4.98$ (close).
c) $0.23\times 8.15=\dfrac{23\times 815}{10000}=\dfrac{18\,745}{10000}=\boxed{1.8745}$. Check: $0.2\times 8\approx 1.6$; $0.25\times 8\approx 2$ → answer between, as expected.
d) $0.69\times 0.254=\dfrac{69\times 254}{100000}=\dfrac{17\,526}{100000}=\boxed{0.17526}$. Check: $0.7\times 0.25=0.175$ (very close).
23. This is part of Syra’s homework. Use estimation to check if Syra’s answers could be correct. If not, explain why.
👀 Show answers
b) Exact $= \dfrac{78\times 93}{100000}=0.07254$. Reasonable estimate $\approx 8\times0.01=0.08$. Syra is correct.
c) Exact $= \dfrac{65\times 43}{10^6}=0.002795$. Estimate $\approx 0.07\times0.04=0.0028$. Syra’s $0.02795$ is ×10 too big → incorrect.
24. A vet needs to work out how much medicine to give to a cat.
The cat has a mass of 5.8 kg.
a) Estimate the number of milligrams (mg) needed.
b) Calculate the accurate number of milligrams (mg) needed.
👀 Show answers
b) Accurate: $7.3\times 5.8=\dfrac{73\times 58}{100}=\boxed{42.34\ \text{mg}}$.
25. A coin is made from silver and copper. The mass of the coin is 4.2 g.
a) Estimate the mass of the silver in this coin.
b) Calculate the accurate mass of the silver in this coin.
👀 Show answers
b) Accurate: $0.775\times 4.2=\boxed{3.255\ \text{g}}$.
⚠️ Be careful!
When multiplying decimals, never place the decimal point by eye — always count the total number of decimal places in the question and make sure the answer has the same total. For example, $3.2 \times 0.04$ has three decimal places in total, so the product must be $0.128$, not $1.28$.