Making decimal calculations easier
🎯 In this topic you will
- Simplify calculations containing decimals
🧠 Key Words
- equivalent fraction
- factor
- partitioning
- place value
Show Definitions
- equivalent fraction: A fraction that represents the same value as another fraction, even if it looks different.
- factor: A number that divides another number exactly, with no remainder.
- partitioning: Splitting a number or shape into parts to make calculations easier.
- place value: The value of a digit depending on its position in a number.
When you are calculating using decimals, you can often make a calculation easier using a variety of methods, such as:
- using the place value of the decimal
- using the correct order of operations
❓ EXERCISES
1. Complete the workings to make the following calculations easier. Use the place value method.
🔎 Reasoning Tip
Using partitioning: Remember to use partitioning to make whole number multiplication easier. For example: \( 7 \times 18 = 7 \times 10 7 \times 8 \).
a. $0.7 \times 180$
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b. $0.04 \times 7600$
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2. Work out the answers to the following. Use the same method as in Question 1.
a. $0.6 \times 410$
b. $0.9 \times 320$
c. $0.02 \times 3200$
d. $0.08 \times 5300$
👀 Show answer
b. $288$
c. $64$
d. $424$
🧠 Think like a Mathematician
Look at the answers to questions 1 and 2. Compare the questions to the final multiplication you had to do.
Examples:
1a. $0.7 \times 180 = 7 \times 18$
1b. $0.04 \times 7600 = 4 \times 76$
2a. $0.6 \times 410 = 6 \times 41$
2c. $0.02 \times 3200 = 2 \times 32$
Can you see a pattern? Can you write a general rule that you could follow? Explain how your rule works. Will it always work for any numbers?
👀 show answer
Pattern: In each case, multiplying a decimal by a whole number is made easier by removing the decimal point and adjusting the other factor proportionally.
General Rule: If you multiply a number by a decimal, you can multiply both numbers by the same power of 10 to make the decimal into a whole number. This gives an equivalent multiplication that is often easier to calculate.
Example: $0.7 \times 180$ → multiply both numbers by 10 → $7 \times 1800/10$ = $7 \times 18$.
$0.04 \times 7600$ → multiply both numbers by 100 → $4 \times 760000/100$ = $4 \times 76$.
Why it works: Multiplying one factor by a power of 10 and dividing the other factor by the same power of 10 keeps the product the same.
Always works? Yes, for any real numbers — because it’s based on the fundamental property of multiplication.
❓ EXERCISES
4. Alek has $\$3450$ to invest. He decides to buy some gold, silver and precious stones. The table shows the amount he will spend on each item.
| Item | Amount |
|---|---|
| Gold | $0.6 \times 3450 = \_\_$ |
| Silver | $0.3 \times 3450 = \_\_$ |
| Precious stones | $0.1 \times 3450 = \_\_$ |
👀 Show answer
Silver: $\$1035$
Precious stones: $\$345$
Check: $2070 1035 345 = 3450$
5. Complete the workings to make these calculations easier.
a. $4.6 \times 9$
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b. $7.3 \times 9$
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6. Work out the answers to the following. Use the same method as in Question 5.
a. $6.8 \times 9$
b. $4.7 \times 9$
c. $12.6 \times 9$
👀 Show answer
b. $4.7 \times 9 = 47 - 4.7 = 42.3$
c. $12.6 \times 9 = 126 - 12.6 = 113.4$
🧠 Think like a Mathematician
Question: Work out $7.6 \times 8$
Solution:
Use: $7.6 = 7 0.6$
$7 \times 8 = 56$
$0.6 \times 8 = 4.8$
$56 4.8 = 60.8$
- Do you understand the method that Pedro has used?
- Do you think this is an easy or difficult method to use? Explain your answer.
Pedro has used this method to multiply a two-digit decimal ($7.6$) by a one-digit whole number ($8$).
Do you think it would be easy to use this method to answer questions such as $5.67 \times 7$ or $5.6 \times 45$? Explain your answer.
👀 show answer
a) Yes. Pedro split the decimal into a whole number and a fractional part, multiplied each by $8$, then added the results.
b) For simple decimals like $7.6$, this is easy because mental multiplication is straightforward. However, for numbers like $5.67$ or large multipliers like $45$, the calculations become more cumbersome as you’d have to split and multiply more parts.
Why it works: This is an application of the distributive property: $a \times (b c) = a \times b a \times c$.
Example: $5.6 \times 45 = (5 \times 45) (0.6 \times 45) = 225 27 = 252$.
❓ EXERCISES
8. Use Pedro’s method to work out the following.
a. $4.2 \times 6$
b. $7.8 \times 5$
c. $6.3 \times 8$
👀 Show answer
b. $39$
c. $50.4$
9. A square has a side length of $8.6\,\text{m}$. The formula to work out the perimeter of a square is $P = 4L$, where $P$ is the perimeter and $L$ is the side length. Use the formula to work out the perimeter of the square.
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10. Complete the workings to make these divisions easier. Then work out the answer.
a. $14.55 \div 30$
b. $67.35 \div 50$
c. $45.85 \div 700$
d. $893.6 \div 200$
👀 Show answer
b. $1.347$
c. $0.0655$
d. $4.468$
11. Write an explanation to convince that the answer to $\frac{45.6}{30}$ is the same as the answer to $\frac{4.56}{3}$.
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12. Twenty members of a football club go out for dinner at a restaurant. The total cost of the meal is $\$564.25$. The total cost is shared equally between them.
a. How much does each member pay? Round your answer to the nearest:
i. cent ii. dollar
b. Which of your answers in part a i or ii is the most suitable amount for each member to pay? Explain your answer.
👀 Show answer
a ii. $\$28$
b. $\$28.21$ is more accurate, but if cash payment in whole dollars is preferred, $\$28$ may be more practical.
🍬 Learning Bridge
After simplifying decimal calculations by using place value, partitioning, and reordering operations, you’re ready to turn those ideas into quick mental moves. Next, you’ll factor decimals into easier parts, pair “friendly” numbers, and use small adjustments (like thinking “one minus a little”) to choose the fastest route—then confirm your results with smart estimation.
When you are calculating using decimals, there are several methods you can use to make a calculation easier, such as
- using the place value of the decimal
- breaking a decimal into parts using factors
- using the correct order of operations
❓ EXERCISES
13. Copy and complete the workings for these questions.
a. $(0.3 0.4) \times 0.2$
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b. $(1.1 - 0.8) \times 0.4$
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14. Use the same method as in Question 1 to work out:
a. $(0.7 0.1) \times 0.6$
b. $(0.3 0.9) \times 0.5$
c. $(0.6 - 0.4) \times 1.2$
d. $(1.8 - 1.5) \times 1.1$
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b. $0.6$
c. $0.24$
d. $0.33$
15. Complete the workings to make these calculations easier.
a. $52 \times 0.9 = 52 \times (1 - 0.1)$
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b. $8.3 \times 0.9 = 8.3 \times (1 - 0.1)$
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16. Use the same method as in Question 3 to work out:
a. $28 \times 0.9$
b. $17 \times 0.9$
c. $4.9 \times 0.9$
👀 Show answer
b. $17 - 1.7 = 15.3$
c. $4.9 - 0.49 = 4.41$
🧠 Think like a Mathematician
Question:
Work in pairs or small groups to discuss this question. Look again at the working and method for Question 3a.
- Describe a similar method you could use to work out $52 \times 9.9$
- Describe a similar method you could use to work out $52 \times 0.99$
- Use your methods to work out:
- $26 \times 9.9$
- $26 \times 0.99$
👀 show answer
a) To calculate $52 \times 9.9$, think of $9.9$ as $10 - 0.1$: $52 \times 9.9 = 52 \times (10 - 0.1) = 520 - 5.2 = 514.8$.
b) To calculate $52 \times 0.99$, think of $0.99$ as $1 - 0.01$: $52 \times 0.99 = 52 \times (1 - 0.01) = 52 - 0.52 = 51.48$.
c-i) $26 \times 9.9 = 26 \times (10 - 0.1) = 260 - 2.6 = 257.4$.
c-ii) $26 \times 0.99 = 26 \times (1 - 0.01) = 26 - 0.26 = 25.74$.
Reasoning: This method uses the distributive property $a \times (b - c) = a \times b - a \times c$, which simplifies mental arithmetic by using nearby whole numbers.
❓ EXERCISES
18. The diagram shows a rectangle. The width is $6\,\text{m}$ and the length is $9.9\,\text{m}$. Work out the area of the rectangle.
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19. A new medicine was given to $3200$ patients. It made $99\%$ of them feel better. For the other $1\%$ the medicine made no difference. Work out how many of the patients the new medicine made better.
🔎 Reasoning Tip
Percentage as a decimal: \( 99\% = 0.99 \), so \( 3200 \times 0.99 = \square \).
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20. Work out the answers to these calculations. Look for different ways to make the calculations easier.
a. $2.5 \times 32.7 \times 4$
b. $\frac{0.2 \times 2.3}{0.1}$
c. $(420 360) \times 0.7$
d. $8 \times 1.32 \times 2.5$
e. $(720 - 120) \times 0.07$
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b. $\frac{0.2 \times 2.3}{0.1} = \frac{0.46}{0.1} = 4.6$
c. $(420 360) \times 0.7 = 780 \times 0.7 = 546$
d. $(8 \times 2.5) \times 1.32 = 20 \times 1.32 = 26.4$
e. $(720 - 120) \times 0.07 = 600 \times 0.07 = 42$
21. Complete the workings. Use factors to make these calculations easier.
a. $24 \times 0.35 = 24 \times 0.5 \times 0.7$
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$12 \times 0.7 = 8.4$
b. $32 \times 0.45 = 32 \times 0.5 \times 0.9$
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$16 \times 0.9 = 14.4$
22. Use the same method as in Question 9 to work out:
a. $80 \times 0.15$
b. $116 \times 0.25$
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b. $116 \times 0.25 = 116 \div 4 = 29$
23. This is part of Pedro’s classwork.
Use Pedro’s method to work out:
🔎 Reasoning Tip
Alternative calculations: For part b, you could use \( 0.2 \times 0.8 \) or \( 0.4 \times 0.4 \).
a. $180 \times 0.14$ (use $0.14 = 0.2 \times 0.7$)
b. $120 \times 0.16$
c. $250 \times 0.18$
d. $450 \times 0.24$
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b. $120 \times 0.2 = 24$, $24 \times 0.8 = 19.2$
c. $250 \times 0.2 = 50$, $50 \times 0.9 = 45$
d. $450 \times 0.2 = 90$, $90 \times 1.2 = 108$
🧠 Think like a Mathematician
Question:
In questions 9 and 10, all the decimals had a factor of $0.5$. Did you always multiply by $0.5$ first? If you did, explain why you did this.
Look at Question 11. Pedro has used a factor of $0.2$ and multiplied by this first. Why do you think he has not used $0.5$ in these questions?
👀 show answer
Multiplying by $0.5$ first can make calculations easier because multiplying by $0.5$ is the same as halving the number, which is a simple mental operation. In Questions 9 and 10, all the decimals were multiples of $0.5$, so this approach simplified the process.
In Question 11, Pedro used $0.2$ because the decimal in that question was related to fifths rather than halves. Multiplying by $0.2$ first was the most direct way to work with the numbers, and using $0.5$ would not have simplified the calculation in this context.
❓ EXERCISES
25. A grandmother gives $\$12600$ to be shared between her grandchildren.
Abdul gets $0.35$ of the money. Zhi gets $0.25$ of the money. Paula gets $0.22$ of the money. Yola gets $0.18$ of the money.
a. Work out how much they each get.
b. Show how you can check that your answers are correct.
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Zhi: $0.25 \times 12600=\$3150$
Paula: $0.22 \times 12600=\$2772$
Yola: $0.18 \times 12600=\$2268$
Check: $4410 3150 2772 2268=\$12600$ (fractions also sum to $0.35 0.25 0.22 0.18=1$) ✔
26. Zara and Sofia are trying to solve this puzzle.
$\dfrac{32 \times (0.8 - \square)}{0.02}=800$
Zara: I think the missing number is 0.4.
Sofia: I think the missing number is 0.3.
Who is correct? Explain how you worked out your answer.
👀 Show answer
Divide by $32$: $0.8-x=0.5 \Rightarrow x=0.8-0.5=0.3$.
Sofia is correct: the missing number is $\boxed{0.3}$.
⚠️ Common Mistake
When simplifying decimal calculations, learners often change the value of the question by scaling or reordering incorrectly. For example, in $12.56 \div 40$, dividing only the numerator by 10 but not the denominator gives $\frac{1.256}{40}$ instead of the easier (and equivalent) $\frac{1.256}{4}$. This mistake happens when the same factor isn’t applied to both numbers in a division.
⚡ Tip: In multiplication or division, you can change one number to make the calculation easier, but you must adjust the other number by the same factor to keep the value the same.