Checkpoint 6 Maths · Number · Decimals

Adding, Subtracting, Multiplying and Dividing Decimals

Estimate and calculate with decimals: add and subtract with different numbers of decimal places, and multiply and divide decimals by whole numbers — with written methods, worked examples and practice.

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

In the curriculum

Estimate, then add and subtract numbers with the same or a different number of decimal places, e.g. 14.6 + 7.85 and 20 − 6.37 (6Nf.09)
Estimate, then multiply numbers with one or two decimal places by 1-digit and 2-digit whole numbers, e.g. 6.28 × 4 and 3.6 × 27 (6Nf.10)
Estimate, then divide numbers with one or two decimal places by whole numbers, e.g. 9.72 ÷ 4 and 45.6 ÷ 12 (6Nf.11)

Decimals are everywhere — money, lengths, masses, times in a race. The good news: you already know how to add, subtract, multiply and divide whole numbers. With decimals you use exactly the same methods; you just have to keep the decimal point in the right place. Estimating first is your safety net: it tells you roughly where the point should go.

Key words

Word / phrase	Meaning / use
decimal point	العلامة العشرية — separates the whole number part from the tenths and hundredths
decimal places	الخانات العشرية — digits after the point; 3.47 has two decimal places
tenths / hundredths	أجزاء من عشرة / من مئة — 0.4 is 4 tenths, 0.07 is 7 hundredths
placeholder zero	صفر لحفظ الخانة — a 0 that fills an empty place: 5.6 = 5.60
estimate	تقدير — round the numbers to get an easy rough answer
exchange (regroup)	الاستلاف / التبديل — 1 one = 10 tenths, 1 tenth = 10 hundredths
inverse	العملية العكسية — + and − undo each other; × and ÷ undo each other
short division	القسمة المختصرة — the 'bus stop' method

1. Estimate first

Round each number to something easy (usually the nearest whole number, or a number that fits a times table), then calculate in your head. Do this before the exact calculation and compare at the end.

Calculation	Round to	Estimate	Exact
14.6 + 7.85	15 + 8	23	22.45
20 − 6.37	20 − 6	14	13.63
6.28 × 4	6 × 4	24	25.12
3.6 × 27	4 × 25	100	97.2
9.72 ÷ 4	8 ÷ 4 or 10 ÷ 4	about 2 to 2.5	2.43
45.6 ÷ 12	48 ÷ 12	4	3.8

2. Adding and subtracting decimals

Estimate.
Write the numbers in columns with the decimal points in a straight line (so tenths are under tenths, hundredths under hundredths).
Fill empty places with placeholder zeros: 14.6 → 14.60. A whole number has its point at the end: 20 → 20.00.
Add or subtract from the right, carrying or exchanging just like whole numbers.
Put the decimal point in the answer straight under the other points. Compare with your estimate.

14.6 + 7.85 = 22.45. Tenths: 6 + 8 = 14, write 4 and carry 1. Ones: 4 + 7 + 1 = 12, write 2 and carry 1. Tens: 1 + 1 = 2.

For subtraction, placeholder zeros are even more important, because you need something to exchange from. 20 − 6.37 is written 20.00 − 6.37. Exchange 1 ten → 10 ones, 1 one → 10 tenths, 1 tenth → 10 hundredths, so the top number becomes 1 ten, 9 ones, 9 tenths and 10 hundredths. Then 10 − 7 = 3, 9 − 3 = 6, 9 − 6 = 3, 1 − 0 = 1: 13.63.

A quicker mental method for subtracting from a whole number is counting up on an empty number line: jump to the next tenth, then to the next whole number, then to the target, and add the jumps.

20 − 6.37 by counting up: 6.37 → 6.4 (+0.03) → 7 (+0.6) → 20 (+13). Total: 13.63.

Check with the inverse

After subtracting, add your answer to the number you took away: 13.63 + 6.37 should give 20.00. After adding, subtract one number from the total.

3. Multiplying decimals by whole numbers

Method 1 — remove the point, multiply, put it back. 6.28 is 628 hundredths. So 6.28 × 4 is 628 hundredths × 4 = 2512 hundredths = 25.12. In practice:

Estimate: 6.28 × 4 ≈ 6 × 4 = 24.
Ignore the point and multiply whole numbers: 628 × 4 = 2512.
The decimal had 2 decimal places, so the answer has 2 decimal places: 25.12.
Check against the estimate: 25.12 is close to 24 ✓

Method 2 — the grid method splits both numbers by place value. It is a good way to see what is happening and works well for 2-digit multipliers.

3.6 × 27: split 27 into 20 + 7 and 3.6 into 3 + 0.6, multiply each pair, then add the four parts.

With long multiplication, multiply as whole numbers and then place the point: 36 × 27 = 36 × 7 + 36 × 20 = 252 + 720 = 972, and 3.6 has one decimal place, so 3.6 × 27 = 97.2.

Calculation	Whole-number calculation	Decimal places	Answer
0.8 × 7	8 × 7 = 56	1	5.6
3.46 × 7	346 × 7 = 2422	2	24.22
4.7 × 23	47 × 23 = 1081	1	108.1
2.35 × 14	235 × 14 = 3290	2	32.90 = 32.9
0.25 × 12	25 × 12 = 300	2	3.00 = 3

Zeros at the end

2.35 × 14 = 32.90. The zero at the end of a decimal can be dropped: 32.90 = 32.9. But for money, write two decimal places: $32.90.

4. Dividing decimals by whole numbers

Use short division (bus stop) exactly as for whole numbers. Put the decimal point in the answer directly above the point in the number you are dividing. If there is a remainder at the end, write a zero after the last digit and keep going.

8.7 ÷ 6 = 1.45. A placeholder zero (8.70) lets the division finish.

For a 2-digit divisor such as 12, write out a few multiples first (12, 24, 36, 48, 60 …). 45.6 ÷ 12: 12 into 45 goes 3 times (36), remainder 9. Bring down the 6 tenths to make 96 tenths. 96 ÷ 12 = 8. So 45.6 ÷ 12 = 3.8. Estimate: 48 ÷ 12 = 4 ✓. Check: 3.8 × 12 = 45.6 ✓

Calculation	Steps	Answer	Check
9.6 ÷ 4	9 ÷ 4 = 2 r 1; 16 ÷ 4 = 4	2.4	2.4 × 4 = 9.6
8.46 ÷ 3	8 ÷ 3 = 2 r 2; 24 ÷ 3 = 8; 6 ÷ 3 = 2	2.82	2.82 × 3 = 8.46
7.5 ÷ 6	7 ÷ 6 = 1 r 1; 15 ÷ 6 = 2 r 3; 30 ÷ 6 = 5	1.25	1.25 × 6 = 7.5
0.84 ÷ 7	0 ÷ 7 = 0; 8 ÷ 7 = 1 r 1; 14 ÷ 7 = 2	0.12	0.12 × 7 = 0.84

سر العلامة العشرية

في الجمع والطرح: رتّب الأرقام بحيث العلامات العشرية تيجي تحت بعض بالظبط، وكمّل الخانات الفاضية بأصفار (5.6 = 5.60 و 20 = 20.00). في الضرب: اضرب من غير العلامة، وبعدين رجّعها بنفس عدد الخانات العشرية. في القسمة: العلامة العشرية في الناتج بتطلع فوق العلامة اللي في المقسوم على طول. ودايماً قدّر الإجابة الأول علشان تعرف مكان العلامة صح.

5. Worked examples

Worked example 1 — Work out 14.6 + 7.85 [1]

Estimate 15 + 8 = 23. Write 14.60 + 7.85 with the points lined up (see the diagram in section 2). Answer: 22.45. Common mistake: lining up the 6 under the 5 to get 9.31.

Worked example 2 — Work out 20 − 6.37 [1]

Estimate 20 − 6 = 14. Either column subtraction with 20.00, or counting up: 0.03 + 0.6 + 13 = 13.63. Check: 13.63 + 6.37 = 20 ✓

Worked example 3 — Work out 6.28 × 4 [1]

628 × 4: 600 × 4 = 2400, 28 × 4 = 112, total 2512. Two decimal places → 25.12. Estimate 6 × 4 = 24 ✓

Worked example 4 — Work out 3.6 × 27. Show your working. [2]

Grid: 20 × 3 = 60, 7 × 3 = 21, 20 × 0.6 = 12, 7 × 0.6 = 4.2. Total 60 + 21 + 12 + 4.2 = 97.2. Estimate 4 × 25 = 100 ✓. Mark scheme style: M1 for a complete correct method (grid or 36 × 27 = 972 seen), A1 for 97.2. Common mistake: writing 0.6 × 7 = 42 instead of 4.2 — check each part: 0.6 × 7 is 6 tenths × 7 = 42 tenths = 4.2.

Worked example 5 — Work out 9.72 ÷ 4 [1]

9 ÷ 4 = 2 r 1; 17 ÷ 4 = 4 r 1; 12 ÷ 4 = 3. Point above the point: 2.43. Check: 2.43 × 4 = 9.72 ✓

Worked example 6 — Work out 8.7 ÷ 6 [2]

Estimate 9 ÷ 6 = 1.5. 8 ÷ 6 = 1 r 2; 27 ÷ 6 = 4 r 3; write 8.70 and 30 ÷ 6 = 5. Answer: 1.45. Common mistake: stopping at 1.4 remainder 3 — at Stage 6 you continue with a placeholder zero.

Worked example 7 — Pens cost $4.75 each. Karim buys 12 pens. He pays with a $100 note. How much change should he get? [2]

Cost: 4.75 × 12. 475 × 12 = 4750 + 950 = 5700. Two decimal places → $57.00. (Estimate 5 × 12 = 60 ✓)
Change: 100.00 − 57.00 = $43.00.
M1 for 4.75 × 12 or 57 seen; A1 for $43 (or $43.00).

Common mistakes

Lining up the right-hand digits instead of the decimal points (3.5 + 1.25 written as 0.35 + 1.25).
Forgetting that a whole number has a hidden point: 7 − 2.85 → 7.00 − 2.85, not 2.85 − 7 or 0.7 − 2.85.
Losing or moving the point when multiplying: 5.4 × 16 = 864 ✗ — the estimate (about 80) shows the answer must be 86.4.
In division, not keeping the point directly above the point, or stopping with a remainder instead of adding a zero.
Dropping digits: 0.84 ÷ 7 = 0.12, not 1.2 or 0.012 — write the 0 in the ones place.
In money problems, writing $43.0 or 43 pence/cents confusion — use two decimal places for money ($43.00).

How Checkpoint asks this

Straight calculations with an answer box on Paper 1 (no calculator) — show your working in the space, because a 2-mark question can give 1 mark for a correct method.
Complete a written method: fill the missing digits in a column addition, a grid or a short division.
Ring the best estimate or 'Tick the calculation with the largest answer' — round first.
Use a given fact: 'Use 23 × 17 = 391 to work out 2.3 × 17 or 39.1 ÷ 17.'
Spot and explain a mistake in someone's working (lined-up digits, misplaced point).
Multi-step word problems with money, litres, metres and kilograms — on Paper 2 a calculator is allowed, but you still need to choose the right operations and interpret the display (e.g. 43 means $43.00).

تقدير الإجابة

قبل ما تحسب، قرّب الأرقام: 3.6 × 27 تقريباً 4 × 25 = 100، يبقى الإجابة الصح لازم تكون قريبة من 100 (وهي 97.2) مش 972 ولا 9.72. التقدير ده هو اللي بيحميك من غلطة مكان العلامة العشرية.

Practice

Remember

Always estimate first by rounding — it tells you where the decimal point goes.
Add and subtract: line up the decimal points; fill gaps with zeros (5.6 = 5.60, 20 = 20.00).
Multiply by a whole number: multiply without the point, then give the answer the same number of decimal places.
Divide by a whole number: short division with the answer's point straight above; add zeros to finish.
Check with the inverse operation.

All pages in this course

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Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

Number: place value, decimals, fractions, percentages and ratio

Geometry and measure

Statistics and probability

Practice tests