Adding & subtracting decimals
🎯 In this topic you will
- Add and subtract decimals
🧠 Key Words
- decimal part
- mentally / using a mental method
- written method
Show Definitions
- decimal part: The portion of a decimal number that comes after the decimal point.
- mentally / using a mental method: Solving a calculation in your head without writing down the steps.
- written method: A step-by-step procedure written down to solve a calculation.
When you add and subtract decimal numbers mentally, there are different methods you can use.
- When you are adding, you can separate the numbers into their whole-number part and their decimal part. You can then separately add the whole-number parts and add the decimal parts, and finally add the whole-number answer to the decimal answer.
- When you are subtracting, you can separate the number you are subtracting into its whole-number part and its decimal part. Subtract the whole-number part first and then subtract the decimal part second.
- If one of the numbers you are adding or subtracting is close to a whole number, you can round it to the nearest whole number, do the addition or subtraction, then adjust your answer at the end.
When you use a written method to add and subtract decimal numbers, always write the calculation in columns, with the decimal points vertically in line. Then add and subtract as normal, but remember to write the decimal point in your answer.
❓ EXERCISES
1. Use a mental method to work out the answers to the following.
a) $3.5 4.2$
b) $12.7 4.5$
c) $4.9 - 1.5$
d) $14.6 - 6.6$
👀 Show answer
b) $12.7 4.5 = 17.2$
c) $4.9 - 1.5 = 3.4$
d) $14.6 - 6.6 = 8.0$
🔎 Reasoning Tip
Comparing decimals: Separate the numbers into their whole-number parts and their decimal parts.
🧠 Think like a Mathematician
Question: What is the easiest way to subtract a decimal from the number 1? For example: $1 - 0.3$, $1 - 0.25$, $1 - 0.405$ or $1 - 0.6839$?
Discussion task: Explore mental strategies and shortcuts for subtracting decimals from 1. Consider the role of complements and place value.
Follow-up prompts:
👀 show answer
- Think of the subtraction $1 - x$ as “what do I add to $x$ to make 1?”. This is called the complement to 1.
- Write both numbers with the same number of decimal places by adding zeros if needed.
- $1.000 - 0.300 = 0.700$
- $1.00 - 0.25 = 0.75$
- $1.000 - 0.405 = 0.595$
- $1.0000 - 0.6839 = 0.3161$
- This works because lining up decimal places helps visualise the subtraction and avoid place-value errors.
- Mental tip: For quick mental maths, subtract each decimal digit from 9, except the final digit which you subtract from 10 (adjusting for carrying if needed).
❓ EXERCISES
3. Match each green question card with its correct pink answer card.
👀 Show answer
$1 - 0.78 = 0.22$
$1 - 0.44 = 0.56$
$1 - 0.284 = 0.716$
$1 - 0.432 = 0.568$
4. Zara mentally works out $14.4 - 6.5$ like this:
Use Zara’s method to mentally work out the following.
a) $7.4 - 2.6$
b) $8.3 - 2.9$
c) $12.5 - 9.8$
d) $15.1 - 5.7$
👀 Show answer
b) Whole: $8 - 2 = 6$, Decimal: $0.3 - 0.9 = -0.6$, Combined: $6 - 0.6 = 5.4$
c) Whole: $12 - 9 = 3$, Decimal: $0.5 - 0.8 = -0.3$, Combined: $3 - 0.3 = 2.7$
d) Whole: $15 - 5 = 10$, Decimal: $0.1 - 0.7 = -0.6$, Combined: $10 - 0.6 = 9.4$
🧠 Think like a Mathematician
Scenario: Zara works out $7.5 4.8$ mentally by rounding and adjusting:
“If I round 4.8 up to 5, I can change $7.5 4.8$ to $7.5 5$, which equals 12.5. Then I must add the extra 0.2, which gives me 12.7.”
Question: Is Zara correct? Discuss in pairs or small groups.
Follow-up prompts:
👀 show answer
- Yes — Zara is correct. $7.5 4.8 = 7.5 (5 - 0.2) = 12.5 - 0.2 = 12.3$ is incorrect — wait, let's check carefully:
- She rounded 4.8 up to 5 (difference of 0.2).
- $7.5 5 = 12.5$.
- Since she added 0.2 too much, she should subtract 0.2, not add it.
- Common mistake: When rounding a number up, you must subtract the excess afterwards; when rounding down, you must add the difference afterwards.
- Correct method:$7.5 4.8 = 7.5 (5 - 0.2) = 12.5 - 0.2 = 12.3$.
❓ EXERCISES
6. Mentally work out the following. Use the method of rounding one of the numbers to a whole number. One of them has been started for you.
a) $4.3 7.9$
b) $8.9 9.6$
c) $22.8 3.3$
d) $5.4 - 1.9$
e) $14.9 - 4.4$
f) $21.1 - 6.7$
👀 Show answer
b) Round $9.6 \to 10$: $8.9 10=18.9$, then subtract $0.4 \Rightarrow \boxed{18.5}$.
c) Round $3.3 \to 3$: $22.8 3=25.8$, then add $0.3 \Rightarrow \boxed{26.1}$.
d) Round $1.9 \to 2$: $5.4-2=3.4$, then add $0.1 \Rightarrow \boxed{3.5}$.
e) Round $14.9 \to 15$: $15-4.4=10.6$, then subtract $0.1 \Rightarrow \boxed{10.5}$.
f) Round $6.7 \to 7$: $21.1-7=14.1$, then add $0.3 \Rightarrow \boxed{14.4}$.
7. Use a written method to work out the following.
a) $25.81 8.4$
b) $8.76 - 4.1$
c) $38.91 - 9.78$
👀 Show answer
b) $\boxed{4.66}$ ($8.76-4.10=4.66$)
c) $\boxed{29.13}$
8. Bo records his mass at the start and end of every month. Here are his records for June and July.
a) During which month, June or July, did Bo’s mass decrease the most?
b) At the start of August Bo’s mass was $88.35\ \text{kg}$. During August his mass decreased by $1.82\ \text{kg}$. What was Bo’s mass at the end of August?
👀 Show answer
b) $88.35 - 1.82 = \boxed{86.53\ \text{kg}}$ at end of August.
9. Use a written method to work out the following.
a) $4.76 - (-12.52)$
b) $32.6 - (-0.742)$
👀 Show answer
b) $32.6 - (-0.742) = 32.6 0.742 = \boxed{33.342}$
🔎 Reasoning Tip
Negative numbers: Subtracting a negative is the same as adding.
10. Marcus and Arun use different methods to work out $10-4.83$.
a) Critique both methods by explaining the advantages and disadvantages of each of their methods.
b) Can you improve on their methods?
c) Which method do you prefer to use to work out subtractions like these? Explain why you prefer this method.
👀 Show answer
Marcus (column subtraction with regrouping): Advantages — Algorithmic and reliable; aligns place values; scales to longer decimals; gives the exact answer $5.17$. Disadvantages — Regrouping across the decimal point (turning $10$ into $9$ units $ 9$ tenths $ 10$ hundredths) can feel fiddly; slower without pencil-and-paper.
Arun (stepwise place-value differences): He does $10-4=6$, then $6.0-0.8=5.2$, then $5.20-0.03=5.17$. Advantages — Mental‑math friendly; highlights place value; minimal writing. Disadvantages — Requires careful tracking so adjustments stay equivalent; can take more steps for numbers with many non‑zero decimals.
b) A tidy mental improvement is a compensation / counting‑up method: Count up from $4.83$ to $10$: $4.83 \rightarrow 5.00$ is $0.17$, then $5.00 \rightarrow 10.00$ is $5$. Total $=0.17 5=\boxed{5.17}$. Or use compensation directly: $10-4.83=(10-5) 0.17=5.17$. This is short, reduces error, and needs only two easy steps because $10$ is a round number.
c) Preference: the compensation/counting‑up method for problems like $10-4.83$, because working to the nearest whole/round number makes the arithmetic quick (just add $0.17$ and $5$) while keeping place value clear. For longer decimals or when writing is allowed, Marcus’s column method is my backup since it’s systematic.
11. Work out:
a) $10 - 6.42$
b) $20 - 12.83$
c) $30 - 4.55$
d) $40 - 16.782$
👀 Show answer
b) $7.17$
c) $25.45$
d) $23.218$
12. At the cinema, Priya spends $4.75$ on a ticket, $1.75$ on sweets and $0.85$ on a drink.
a) How much does she spend in total?
b) Priya pays with a $10 note. How much change will she receive?
👀 Show answer
b) $10.00 - 7.35 = 2.65$ dollars change.
13. Jed works as a plumber. He has four lengths of pipe that measure $1.8\ \text{m}$, $3.5\ \text{m}$, $2.45\ \text{m}$ and $0.85\ \text{m}$.
a) What is the total length of the four pipes?
b) Jed needs $10\ \text{m}$ of pipe in total to finish a job. How much more pipe must he buy?
👀 Show answer
b) Needs $10.0 - 8.6 = 1.4\ \text{m}$ more pipe.
14. This is how Zara works out $4.6 - 8.21$.
Use Zara’s method to work out the following.
a) $5.43 - 9.57$
b) $8.12 - 15.4$
c) $-13.8 7.92$
d) $6.582 - (4.5 5.061)$
🔎 Reasoning Tip
Rewriting expressions: For part c: \(-13.8 7.92 = 7.92 - 13.8\)
👀 Show answer
b) $15.4 - 8.12 = 7.28$, so answer $= -7.28$
c) $-13.8 7.92 = -13.8 7.92 = -(13.8 - 7.92) = -5.88$
d) $4.5 5.061 = 9.561$, then $9.561 - 6.582 = 2.979$, so answer $= -2.979$
15. Use a written method to work out the following.
a) $-5.43 - 9.57$
b) $-8.12 - 15.4$
c) $-5.43 - (-9.57)$
d) $-8.12 - (-15.4)$
👀 Show answer
b) $-8.12 - 15.4 = -(8.12 15.4) = -23.52$
c) $-5.43 - (-9.57) = -5.43 9.57 = 4.14$
d) $-8.12 - (-15.4) = -8.12 15.4 = 7.28$
⚠️ Be careful!
When adding or subtracting decimals in columns, always line up the decimal points — not the edges of the numbers. For example, write $43.6$ as $43.60$ if needed, so it lines up correctly with $5.45$. Misalignment changes place values and gives the wrong answer.