Calculating with Integers and Decimals
Learn to calculate with whole numbers, negative numbers and decimals. Then use the agreed order of operations to work through mixed calculations accurately.
Foundation skills for both tiers. Begin with familiar examples and build one step at a time.
Start with operations
The previous lesson used place value and number lines to compare numbers. Now we will use those ideas to calculate.
Addition (+) combines amounts. Subtraction (−) finds a difference or takes an amount away.
Multiplication (×) combines equal groups. Division (÷) shares into equal groups or counts how many groups fit.
Start with positive numbers, then follow each new step carefully. A minus sign can name a negative number or tell you to subtract.
Example Question:
What is 3 + 4 × 2?
Should you add or multiply first? Example 11 explains why the answer is 11.
Exam Tip
Keep negative numbers in brackets when this makes the signs clearer.
Write one change per line. Do not try to do all the operations mentally at once.
Method — Calculate, then check
1. Decide which operation the question needs.
2. Keep signs and decimal place values visible.
3. For mixed operations, use BIDMAS and work left to right within equal priorities.
4. Use an inverse operation or an estimate to check the answer. Include units.
Worked Examples
Example 1 — Begin with positive integers
Addition combines amounts. Subtraction finds what remains or the difference.
37 + 28
37 + 20 = 57
57 + 8 = 65
Multiplication describes equal groups: 4 × 3 = 12 means four groups of three. Division shares or groups: 12 ÷ 3 = 4.
Example 2 — Add a negative number
On a number line, adding a positive number moves right. Adding a negative moves left.
3 + 4 = 7: start at 3 and move 4 right.
3 + (−5): start at 3 and move 5 left.
3 + (−5) = −2
The brackets keep the negative number together. They do not change its value.
−3 + 7 = 4: move 7 right from −3.
−4 + (−3) = −7: move 3 left from −4.
Example 3 — Subtract into a negative result
5 − 8 means move 8 units left from 5.
Five steps reach zero. Three more steps reach −3.
5 − 8 = −3
Example 4 — Subtract a negative
Subtraction undoes addition. Adding −3 moves left by 3. Subtracting −3 reverses that movement, so it moves right by 3.
4 − (−3) = 4 + 3
4 + 3 = 7
Check by adding the subtracted number back: 7 + (−3) = 4.
This reasoning applies to subtraction of a negative. It does not make −4 + (−3) positive.
Example 5 — Multiply and divide signed integers
Three groups of −2 give −2 + (−2) + (−2) = −6, so 3 × (−2) = −6.
Reversing the order of the factors leaves the product unchanged: (−2) × 3 = −6.
Now follow a pattern. Each time the second factor falls by 1, the result increases by 2.
| Calculation | Result |
|---|---|
| (−2) × 2 | −4 |
| (−2) × 1 | −2 |
| (−2) × 0 | 0 |
| (−2) × (−1) | 2 |
| (−2) × (−2) | 4 |
This explains why multiplying two negative numbers gives a positive result.
(−4) × (−3) = 12
Division reverses multiplication. Since (−4) × 3 = −12, we have (−12) ÷ 3 = −4.
Since 4 × (−3) = −12, we have (−12) ÷ (−3) = 4.
| Signs of the two numbers | Multiplication or division result |
|---|---|
| Positive, positive | Positive |
| Positive, negative | Negative |
| Negative, positive | Negative |
| Negative, negative | Positive |
These sign rules apply to multiplication and division of non-zero numbers. Any number multiplied by zero is zero. Zero divided by a non-zero number is zero. Division by zero is undefined.
Example 6 — Add decimals by matching places
4.7 + 2.35
Write 4.7 as 4.70. This adds an empty hundredths place without changing the value.
| Units | Tenths | Hundredths | |
|---|---|---|---|
| First amount | 4 | 7 | 0 |
| + second amount | 2 | 3 | 5 |
| Total | 7 | 0 | 5 |
0 hundredths + 5 hundredths = 5 hundredths.
7 tenths + 3 tenths = 10 tenths. Exchange them for 1 unit.
4 units + 2 units + the exchanged unit = 7 units.
4.70 + 2.35 = 7.05
When writing a column calculation, put the decimal points directly underneath one another. That aligns matching places.
Example 7 — Subtract decimals using exchange
6.2 − 3.47
Write 6.2 as 6.20. We need to subtract 7 hundredths, but the hundredths place is empty.
Exchange 1 of the 2 tenths for 10 hundredths. There is now 1 tenth and 10 hundredths.
10 hundredths − 7 hundredths = 3 hundredths.
There is only 1 tenth left. Exchange 1 unit for 10 tenths: we now have 5 units and 11 tenths.
11 tenths − 4 tenths = 7 tenths.
5 units − 3 units = 2 units.
6.20 − 3.47 = 2.73
Example 8 — Multiply decimals using size
First, 2.4 × 3 is three groups of 2.4.
24 tenths × 3 = 72 tenths = 7.2.
Now find 2.4 × 0.3. The second factor is one tenth of 3, so the answer is one tenth of 7.2.
7.2 ÷ 10 = 0.72
Another route starts with 24 × 3 = 72. To turn 24 into 2.4 we divide by 10; to turn 3 into 0.3 we divide by 10 again.
The product must therefore be divided by 100: 72 ÷ 100 = 0.72.
Sense check: multiplying a positive amount by 0.3 gives less than the original amount.
Example 9 — Divide decimals fairly
8.4 ÷ 2
84 tenths shared between two groups gives 42 tenths.
8.4 ÷ 2 = 4.2
Now find 6.3 ÷ 0.3. This asks how many groups of 0.3 fit into 6.3.
Measure both amounts in tenths: that is 63 tenths in groups of 3 tenths.
6.3 ÷ 0.3 = 63 ÷ 3
63 ÷ 3 = 21
Multiplying both numbers by the same non-zero number leaves the quotient, or division answer, unchanged. Scaling only one number would change it.
Signed decimal division
The scaling method also works with negative decimals. First calculate the size, then use the sign rule: different signs give a negative quotient; matching signs give a positive quotient.
For (−6.3) ÷ 0.3, scale both values by 10: (−63) ÷ 3 = −21. For (−6.3) ÷ (−0.3), the same size calculation gives 21. Check: −21 × 0.3 = −6.3.
Quick practice — Calculate (−4.8) ÷ 0.6.
Scale both values by 10: (−48) ÷ 6 = −8. Check: −8 × 0.6 = −4.8.
Example 10 — Check with an inverse operation
37 + 28 = 65
Undo adding 28 by subtracting 28:
65 − 28 = 37
For division, multiply back: 21 × 0.3 = 6.3, which checks 6.3 ÷ 0.3 = 21.
These checks help catch errors. Also check that you chose the right operation for the question.
Example 11 — Use the agreed operation order
In a mixed calculation, use brackets first, then indices, then multiplication/division, then addition/subtraction.
BIDMAS is a reminder. An index is a power: 3² means 3 × 3 = 9.
3 + 4 × 2
= 3 + 8
= 11
Multiplication comes before addition.
2 + 3² × 4
= 2 + 9 × 4
= 2 + 36
= 38
| Priority | What to do |
|---|---|
| 1 — B | Brackets |
| 2 — I | Indices (powers) |
| 3 — D and M | Division and multiplication, left to right |
| 4 — A and S | Addition and subtraction, left to right |
Example 12 — Brackets and equal priorities
(3 + 4) × 2
= 7 × 2
= 14
The brackets change which operation comes first.
Multiplication and division have equal priority:
20 ÷ 5 × 2
= 4 × 2
= 8
Do not turn this into 20 ÷ (5 × 2). Those brackets were not in the question.
Addition and subtraction also have equal priority:
10 − 3 + 2
= 7 + 2
= 9
Do not add 3 and 2 first. Work left to right within each priority level.
Example 13 — Choose the right operation in context
A temperature is −2°C. It falls by 5°C. What is the new temperature? °C means degrees Celsius.
A fall means subtract:
−2 − 5 = −7°C
A notebook costs £2.35. You buy three and pay with £10. Find the change.
Cost = 3 × £2.35 = £7.05
Change = £10.00 − £7.05 = £2.95
Estimate: three notebooks cost about £7, so change of about £3 makes sense.
Money, bills and account balances
A cost price is what a seller pays; the selling price is what a customer pays. Selling an item bought for £48 for £60 gives a profit of £60 − £48 = £12. Selling it for £43 instead gives a loss of £48 − £43 = £5.
On a bank statement, a credit adds money to the account and a debit takes money out. The balance is the amount in the account; it can be negative if the account is overdrawn.
Worked example: a bill contains two items at £12.50 each and delivery of £8. The bill is 2 × £12.50 + £8 = £33. A balance of £50 becomes £17 after this debit. A £6 refund is a credit, so the new balance is £23.
Keep pounds and pence consistent. Percentage charges such as VAT and income tax are applied in Percentages of an Amount.
Quick practice — A £40 account pays for three items at £8.75 each plus £4 delivery, then receives a £5 refund. Find the final balance.
The bill is 3 × £8.75 + £4 = £30.25. The balance is £40 − £30.25 + £5 = £14.75.
Written methods, units and exact answers
Place value also explains written multiplication and division. Keep each part in its correct column rather than relying on a calculator.
Worked example — multiply 247 by 36
36 = 30 + 6. Find the six lots first, then the thirty lots.
| Part | Working | Result |
|---|---|---|
| 247 × 6 | 6 × 7 = 42: write 2, carry 4. Then 6 × 4 + 4 = 28: write 8, carry 2. Finally 6 × 2 + 2 = 14. | 1,482 |
| 247 × 30 | 247 × 3 = 741. Multiplying by 30 needs ten times this amount. | 7,410 |
| Add the partial products | 1,482 + 7,410 | 8,892 |
For 2.47 × 3.6, the two factors are 100 times and 10 times smaller than 247 and 36. Divide 8,892 by 1,000: 8.892. This is sensible: about 2.5 × 3.6 = 9.
Worked example — divide 936 by 24
Share 936 into 24 equal groups. The 9 hundreds cannot give every group a whole hundred, so regroup them with the 3 tens to make 93 tens.
Share the 93 tens between 24 groups. Each group gets 3 tens, using 3 × 24 = 72 tens. Subtract: 93 − 72 = 21 tens left.
Bring down the 6 units: 21 tens and 6 units make 216. Then 216 ÷ 24 = 9.
The quotient is 3 tens and 9 units: 936 ÷ 24 = 39. Check: 39 × 24 = 936.
Quick practice — calculate 326 × 24 and 864 ÷ 16 without a calculator.
326 × 4 = 1,304. 326 × 20 = 6,520. Add the partial products: 7,824.
16 fits into 86 five times, with 86 − 80 = 6 left. Bring down 4 to make 64. 64 ÷ 16 = 4, so the quotient is 54. Check: 54 × 16 = 864.
Use the right units
A measurement needs a number and a unit. Convert to the same unit before adding or comparing measurements. A smaller unit needs more units to describe the same amount.
| Quantity | Useful relationships |
|---|---|
| Length | 1 cm = 10 mm; 1 m = 100 cm; 1 km = 1,000 m |
| Mass | 1 kg = 1,000 g |
| Capacity | 1 litre = 1,000 millilitres; 1 millilitre = 1 cm³ |
| Time | 1 minute = 60 seconds; 1 hour = 60 minutes |
| Area | 1 m² = 100 cm × 100 cm = 10,000 cm² |
| Volume | 1 m³ = 100 cm × 100 cm × 100 cm = 1,000,000 cm³ |
For example, 2.4 kg = 2.4 × 1,000 = 2,400 g. But 1.5 hours = 1.5 × 60 = 90 minutes: time does not convert in hundreds.
A compound unit combines units. A speed of 32 km/h means 32 kilometres per hour. If speed = distance ÷ time, 48 km in 1.5 hours gives 48 ÷ 1.5 = 32 km/h. More detailed speed, density and pressure problems belong with measures and rates.
Quick practice — convert 0.65 kg to grams and 2.25 hours to minutes.
0.65 × 1,000 = 650 g.
2.25 × 60 = 135 minutes. Two hours is 120 minutes and one quarter of an hour is 15 minutes.
Exact answers can include π — both tiers
π (pi) is the exact constant used for circles. It is approximately 3.142, but that decimal is rounded. Keep π in your working when an exact answer is requested.
Given circumference = π × diameter, a circle with diameter 6 cm has circumference 6π cm. Two lengths of 2π cm and 3π cm have exact total 5π cm. Do not replace π by 3.14 part-way through exact working.
Quick practice — use circumference = π × diameter to give an exact circumference for diameter 9 cm.
Substitute the given diameter: π × 9 = 9π cm. This is exact; a rounded decimal would be an approximation.
Key ideas to remember
Adding a negative moves left. Subtracting a negative reverses that movement.
Align decimal points for addition and subtraction. Explain the size of a decimal product or quotient.
Division and multiplication have equal priority. Addition and subtraction have equal priority.
Calculator entry and checking
Enter brackets and powers exactly as written. Use your calculator’s negative-number entry for a negative value; the button varies by model.
For (−3) × (−5), make both negative values clear. The result should be 15.
Enter 2 + 3² × 4 with the square applied only to 3. It should give 38.
Estimate before accepting a display. 4.7 + 2.35 is about 7, so 70.5 would be unreasonable.
An inverse check should return you to the starting amount. A correct calculation still needs to answer the actual question.
Think Like a Mathematician
Why does subtracting a negative increase the value?
Start with 4 − (−3). The answer must be a number which becomes 4 when −3 is added back.
7 + (−3) = 4, so the answer is 7.
On a number line, adding −3 moves left. Undoing it moves right. That is why subtracting −3 has the same effect as adding 3.
Common mistakes to avoid
Losing signs: copy the entire negative number, using brackets when helpful.
Turning any two minus signs into plus: −4 + (−3) is −7. Look at the operation, not just the number of signs.
Misaligning decimals: add units to units and tenths to tenths. Placeholder zeros help keep columns clear.
Guessing a decimal point: explain the scaling and check the size of the answer.
Scaling only one number in division: multiply both the dividend and divisor by the same power of 10.
Reading BIDMAS as D always before M: work left to right for multiplication/division, and likewise for addition/subtraction.
Missing calculator brackets: enter the intended grouping before calculating.
Try it yourself
Work through these in order. Write your working before opening each answer.
1. Calculate 26 + 17.
26 + 10 = 36.
36 + 7 = 43.
2. Calculate (a) 4 + (−7), (b) −5 + 8.
(a) Move 7 left from 4: −3.
(b) Move 8 right from −5: 3.
3. Calculate 6 − 10.
Move 6 left to reach zero, then another 4 left.
6 − 10 = −4.
4. Calculate −2 − (−6).
Subtracting −6 undoes adding −6, so move 6 right.
−2 − (−6) = −2 + 6 = 4.
5. Calculate (a) (−3) × 5, (b) (−3) × (−5), (c) 18 ÷ (−3), (d) (−18) ÷ (−3).
Multiply or divide the sizes, then use the signs.
(a) Different signs: −15.
(b) Same signs: 15.
(c) Different signs: −6. Check (−6) × (−3) = 18.
(d) Same signs: 6. Check 6 × (−3) = −18.
6. Calculate 3.8 + 1.47.
Write 3.8 as 3.80 and align decimal points.
8 tenths + 4 tenths = 12 tenths: exchange 10 tenths for 1 unit.
3.80 + 1.47 = 5.27.
7. Calculate 5.1 − 2.76.
Write 5.10. Exchange 1 tenth for 10 hundredths: 10 − 6 = 4 hundredths.
Exchange 1 unit for 10 tenths: 10 − 7 = 3 tenths.
4 − 2 = 2 units.
5.10 − 2.76 = 2.34.
8. Calculate 1.6 × 0.4.
16 × 4 = 64.
Each factor was made ten times larger for that calculation. Divide the product by 100.
64 ÷ 100 = 0.64. This is below 1.6, as expected.
9. Calculate (a) 9.6 ÷ 3, (b) 4.8 ÷ 0.6.
(a) 96 tenths ÷ 3 = 32 tenths = 3.2.
(b) Scale both numbers by 10: 48 ÷ 6 = 8.
Check: 8 × 0.6 = 4.8.
10. Calculate 54 + 29 and check using the inverse.
54 + 20 = 74.
74 + 9 = 83.
Check: 83 − 29 = 54.
11. Calculate (a) 6 + 3 × 4, (b) (6 + 3) × 4.
(a) Multiply first: 6 + 12 = 18.
(b) Brackets first: 9 × 4 = 36.
12. Calculate (a) 24 ÷ 6 × 3, (b) 12 − 5 + 2, (c) 1 + 2² × 5.
(a) Work left to right: 24 ÷ 6 = 4, then 4 × 3 = 12.
(b) Work left to right: 12 − 5 = 7, then 7 + 2 = 9.
(c) Power first: 2² = 4. Then 4 × 5 = 20. Finally 1 + 20 = 21.
13. The temperature is −4°C and rises by 9°C. Find the new temperature.
A rise means add. Move 9 right from −4.
−4 + 9 = 5°C.
14. Three pens cost £1.25 each. How much change is due from £5?
Total cost = 3 × £1.25 = £3.75.
Change = £5.00 − £3.75 = £1.25.
Check: £3.75 + £1.25 = £5.00.
Frequently Asked Questions
Does adding always make a number bigger?
No. Adding a negative number makes it smaller. For example, 3 + (−5) = −2.
Why can I write 4.7 as 4.70?
Both represent the same amount. The extra zero records an empty hundredths place.
Do two negative numbers always give a positive answer?
Their product or quotient is positive. Their sum is negative. The operation matters.
Why scale both numbers in decimal division?
You change both to the same smaller unit, so the number of groups is unchanged. 6.3 ÷ 0.3 equals 63 ÷ 3.
Does division always come before multiplication?
No. They have equal priority, so work from left to right. Addition and subtraction work the same way.
Can I divide by zero?
No. Division by zero is undefined. For example, no number multiplied by zero can give 8.