Cambridge Primary Checkpoint · Mathematics 0096 · Stage 6 · Number
Multiplying Whole Numbers (up to 10 000 × 2-digit)
Estimate, then multiply whole numbers up to 10 000 by 1-digit and 2-digit numbers using the grid method, short and long multiplication, and check your answers.
British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course
- Estimate the answer to a multiplication before you work it out, and use the estimate to check your answer (6Ni.04).
- Multiply any whole number up to 10 000 by a 1-digit number (for example 7 386 × 8) or by a 2-digit number (for example 4 286 × 37), using a written method (6Ni.04).
In Stage 6 you multiply big numbers: up to 10 000 times a 1-digit or 2-digit number. You will learn three things on this page: how to estimate quickly, how to use the grid method and the column method (short and long multiplication), and how to check that your answer makes sense. Paper 1 of Checkpoint is without a calculator, so you must be fast and accurate with written methods.
Key words
| Word / phrase | Meaning / use |
|---|---|
| product | حاصل الضرب — the answer to a multiplication. The product of 6 and 7 is 42. |
| multiply / times | يضرب — find the total of equal groups; sign × |
| estimate | تقدير — a quick, close answer made by rounding the numbers first; sign ≈ means "is approximately equal to" |
| round | يقرّب — change a number to a nearby "easy" number, e.g. 3 186 → 3 000 |
| digit | رقم — one of 0, 1, 2 … 9. 37 is a 2-digit number. |
| partition | يجزّئ — split a number into parts by place value: 346 = 300 + 40 + 6 |
| carry (regroup) | الحمل / الباقي الصاعد — when a column makes 10 or more, the tens move to the next column on the left |
| place holder (zero) | صفر حافظ للخانة — the 0 you write first when you multiply by the tens digit |
| inverse | العملية العكسية — the opposite operation; division is the inverse of multiplication |
1. Estimate first
An estimate tells you roughly how big the answer should be. Round each number so that it has one non-zero digit (for example 3 186 → 3 000, 47 → 50), then multiply the easy numbers. To multiply numbers ending in zeros, multiply the non-zero digits and then write all the zeros back.
| Calculation | Round | Easy calculation | Estimate |
|---|---|---|---|
| 4 873 × 6 | 5 000 × 6 | 5 × 6 = 30, add 3 zeros | ≈ 30 000 |
| 718 × 9 | 700 × 9 | 7 × 9 = 63, add 2 zeros | ≈ 6 300 |
| 3 186 × 47 | 3 000 × 50 | 3 × 5 = 15, add 4 zeros | ≈ 150 000 |
| 8 245 × 62 | 8 000 × 60 | 8 × 6 = 48, add 4 zeros | ≈ 480 000 |
| 96 × 31 | 100 × 30 | 1 × 3 = 3, add 3 zeros | ≈ 3 000 |
Is my estimate too big or too small? If you rounded both numbers up, the estimate is bigger than the exact answer. If you rounded both down, the estimate is smaller. Example: 6 812 × 39 ≈ 7 000 × 40 = 280 000, and both numbers went up, so the exact answer (265 668) is less than 280 000.
You can also make an estimate that is closer by rounding only one number, or by rounding to the nearest hundred: 2 645 × 48 ≈ 2 600 × 50 = 130 000 (exact answer 126 960). Any sensible estimate gets the mark in a test, as long as your working is shown.
2. Useful mental strategies
These short cuts use the laws of arithmetic. They help with estimates and with checking.
| Strategy | Example |
|---|---|
| Multiply by a multiple of 10, 100 or 1000 | 4 000 × 7 = 28 000 (4 × 7 = 28, then add 3 zeros); 300 × 60 = 18 000 |
| Partition and multiply each part | 2 300 × 6 = 2 000 × 6 + 300 × 6 = 12 000 + 1 800 = 13 800 |
| × 4 = double, then double again | 1 350 × 4: 1 350 → 2 700 → 5 400 |
| × 5 = × 10, then halve | 846 × 5 = 8 460 ÷ 2 = 4 230 |
| × 9 = × 10, then subtract one lot | 762 × 9 = 7 620 − 762 = 6 858 |
| × 25 = × 100, then divide by 4 | 48 × 25 = 4 800 ÷ 4 = 1 200 |
| × 99 = × 100, then subtract one lot | 57 × 99 = 5 700 − 57 = 5 643 |
| Use a fact you know | If 37 × 24 = 888, then 37 × 48 = 888 × 2 = 1 776 and 37 × 25 = 888 + 37 = 925 |
3. The grid method
The grid method shows why multiplication works. You partition the numbers by place value, multiply every part by every part, then add.
- Partition both numbers: 346 = 300 + 40 + 6 and 27 = 20 + 7.
- Draw a grid with the parts of one number along the top and the parts of the other down the side.
- Multiply to fill each cell: 300 × 20 = 6 000, 40 × 20 = 800, 6 × 20 = 120, 300 × 7 = 2 100, 40 × 7 = 280, 6 × 7 = 42.
- Add each row, then add the row totals.
The grid method is safe but slow for 4-digit × 2-digit numbers (8 cells). Learn the column method for speed, and keep the grid method as a way to check.
4. Short multiplication (× a 1-digit number)
- Write the big number on top and the 1-digit number underneath, lined up on the right (ones under ones).
- Start with the ones. Multiply, write the ones digit of the result, and carry the tens digit to the next column.
- Move left one column at a time. Multiply, then add the carry. Write the digit and carry again.
- In the last column write the whole result.
- Compare with your estimate.
5. Long multiplication (× a 2-digit number)
Long multiplication is two short multiplications and one addition. For 4 286 × 37 you work out 4 286 × 7 and 4 286 × 30, then add.
- Estimate: 4 000 × 40 = 160 000.
- Row 1 — multiply by the ones digit (7): 4 286 × 7 = 30 002.
- Row 2 — multiply by the tens digit (3 tens = 30): first write 0 in the ones column (the place holder), because you are multiplying by 30, not 3. Then work out 4 286 × 3 = 12 858 and write it to the left of the zero: 128 580.
- Add the two rows: 30 002 + 128 580 = 158 582.
- Check: 158 582 is close to 160 000 ✓
قبل ما تضرب، قرّب الأرقام واعمل تقدير سريع (مثلاً 4 286 × 37 ≈ 4 000 × 40 = 160 000). في الضرب الطويل: اضرب في رقم الآحاد الأول، وبعدين لما تضرب في رقم العشرات لازم تكتب صفر في خانة الآحاد الأول، لأنك بتضرب في 30 مش في 3. في الآخر اجمع السطرين وقارن الناتج بالتقدير.
6. Check your answer
- Size check: is the answer close to your estimate? 158 582 vs 160 000 ✓
- Last digit check: the ones digit of the answer must match the ones digit of (ones × ones). 4 286 × 37: 6 × 7 = 42, so the answer must end in 2 ✓
- Even/odd check: even × any whole number = even. 4 286 is even, so the product must be even ✓
- Inverse: divide back — 9 342 ÷ 27 should give 346 (useful on Paper 2 with a calculator).
- A different method: do a quick grid to check a column answer.
Worked examples
Worked example 1 (1 mark) — Ring the best estimate for 6 912 × 8. 5 600 48 000 56 000 560 000
Round 6 912 to 7 000. 7 000 × 8 = 56 000. Answer: 56 000. (The exact answer is 55 296.) Common mistake: choosing 5 600 or 560 000 because of a wrong number of zeros — count the zeros carefully: 7 000 has 3 zeros, so the answer has 3 zeros after 56.
Worked example 2 (1 mark) — Calculate 3 058 × 7. Show your working.
- Estimate: 3 000 × 7 = 21 000.
- 8 × 7 = 56 → write 6, carry 5.
- 5 × 7 = 35, + 5 = 40 → write 0, carry 4.
- 0 × 7 = 0, + 4 = 4 → write 4 (no carry).
- 3 × 7 = 21 → write 21.
- Answer: 21 406 ✓ close to 21 000.
Common mistake: in the hundreds column 0 × 7 = 0, but you must still add the carry 4. Writing 0 there gives the wrong answer 21 006.
Worked example 3 (2 marks) — Calculate 2 645 × 48.
| Step | Working |
|---|---|
| Estimate | 3 000 × 50 = 150 000 (both rounded up, so the answer is less than this) |
| 2 645 × 8 | 5×8 = 40 → 0 carry 4; 4×8 = 32 + 4 = 36 → 6 carry 3; 6×8 = 48 + 3 = 51 → 1 carry 5; 2×8 = 16 + 5 = 21 → 21 160 |
| 2 645 × 40 | write 0, then 2 645 × 4 = 10 580 → 105 800 |
| Add | 21 160 + 105 800 = 126 960 |
| Check | less than 150 000 ✓; last digit: 5 × 8 = 40 ends in 0 ✓ |
In the mark scheme a correct method with one small arithmetic slip usually still earns 1 of the 2 marks — another reason to show every row.
Worked example 4 (2 marks) — A theatre has 1 275 seats. Every seat is sold on each of 24 nights. How many tickets are sold altogether?
- Key words "each of 24 nights" and "altogether" → multiply: 1 275 × 24.
- Estimate: 1 000 × 20 = 20 000 (or closer: 1 300 × 25 = 32 500).
- 1 275 × 4 = 5 100.
- 1 275 × 20 = 25 500 (write 0, then 1 275 × 2 = 2 550).
- 5 100 + 25 500 = 30 600.
- Answer: 30 600 tickets.
Worked example 5 (1 mark) — Omar says 487 × 52 = 3 409. Without working out the exact answer, explain how you know he is wrong.
Estimate: 500 × 50 = 25 000. His answer 3 409 is far too small. Answer: 487 × 52 is about 25 000, so 3 409 cannot be correct. (Omar forgot the place-holder zero: he added 974 + 2 435 instead of 974 + 24 350. The correct answer is 25 324.)
Worked example 6 (2 marks) — 37 × 24 = 888. Use this fact to work out (a) 370 × 24 and (b) 37 × 25.
- (a) 370 is ten times 37, so the product is ten times bigger: 888 × 10 = 8 880.
- (b) 37 × 25 is one more lot of 37 than 37 × 24: 888 + 37 = 925.
- Common mistake: in (b) adding 1 to get 889. You add one more 37, not one more 1.
- Forgetting the place-holder zero in the second row of long multiplication (multiplying by 3 instead of 30).
- Forgetting to add the carry, especially after a 0 digit (0 × 7 + carry 4 = 4, not 0).
- Adding the carry before multiplying (the rule is multiply first, then add the carry).
- Digits not lined up in columns, so the final addition goes wrong — write neatly, one digit per square.
- Wrong number of zeros in estimates: 3 000 × 50 = 150 000 (4 zeros), not 15 000.
- Mixing up row 1 carries with row 2 carries — cross them out when row 1 is finished.
- Never estimating, so a silly answer is not noticed.
- "Ring the best estimate" or "Estimate the answer" — round to one non-zero digit and show the rounded calculation.
- "Calculate … Show your working." — usually 1 or 2 marks. Working can earn a method mark even if the final answer is wrong.
- Word problems: "each", "every", "altogether", "total cost" often mean multiply. Write the calculation first (1 275 × 24), then solve.
- Missing-digit calculations: a written method with some digits replaced by boxes. Work column by column from the right, using the carries.
- "Use the fact that … to work out …" — do not start again; change the given answer (× 10, double, add one more lot).
- "Explain how you know the answer is wrong" — an estimate is the quickest convincing reason.
- Paper 1 has no calculator: practise written methods. On Paper 2 you may use a calculator, but still estimate to catch typing mistakes.
بعد ما تخلص: (1) قارن الإجابة بالتقدير — لازم تكون قريبة منه. (2) رقم الآحاد في الإجابة لازم يساوي رقم آحاد (آحاد × آحاد). (3) لو أي رقم فيهم زوجي، الإجابة لازم تكون زوجية. ولو الإجابة بعيدة جداً عن التقدير، غالباً نسيت الصفر أو نسيت تضيف الحمل.
Practice
- Always estimate first: round to one non-zero digit (4 286 × 37 ≈ 4 000 × 40 = 160 000).
- Short multiplication: start at the ones, multiply, then add the carry.
- Long multiplication = (× ones digit) + (× tens digit with a place-holder zero), then add.
- Check with the estimate, the last digit, and even/odd.
- Use known facts: 10 times one number → 10 times the product; one more lot → add one more of the number.
All pages in this course
Start here
Number: integers and calculation
- Counting On and Back and Negative Numbers
- Sequences, Letters and Position-to-Term Rules
- Order of Operations, Brackets and Laws of Arithmetic
- Multiplying Whole Numbers (up to 10 000 × 2-digit)
- Dividing Whole Numbers (up to 1000 ÷ 2-digit)
- Factors, Multiples and Tests of Divisibility
- Square Numbers and Cube Numbers
Number: place value, decimals, fractions, percentages and ratio
- Place Value and Decimals to Thousandths
- Rounding Decimals
- Fractions: Division, Operators and Simplest Form
- Adding and Subtracting Fractions
- Multiplying and Dividing Fractions by Whole Numbers
- Fractions, Decimals and Percentages: Equivalence and Ordering
- Adding, Subtracting, Multiplying and Dividing Decimals
- Ratio and Direct Proportion
Geometry and measure
- Time: Decimals and Mixed Units
- Quadrilaterals: Properties and Classification
- Circles: Parts and Constructing Circles
- Angles: Measuring, Drawing and Angles in a Triangle
- Area of Rectangles and Right-angled Triangles
- 3D Shapes, Compound Shapes and Nets
- Capacity, Volume and Surface Area
- Coordinates in Four Quadrants
- Rotational Symmetry, Translation, Reflection and Rotation
Statistics and probability
Practice tests