Paper Together: sit a paper in a room with your friends.See how it works
This page sets out the method for non-calculator arithmetic tricks in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Non-calculator arithmetic tricks topic page.
When you see 'Given that , work out …', never redo the multiplication — shift the decimal place.
When a price or number is close to a round number (3.95, 49, 1.99), use the round number and fix the difference.
After any Paper 1 calculation, spend ten seconds checking before you move on.
19 × 4 = 20 × 4 − 4.
You subtract one group, not 1.
16 × 25 = 8 × 50 = 4 × 100.
Move a factor of 2 from one side to the other.
7 × 13 = 7 × 10 + 7 × 3.
The distributive law is the non-calculator method.
21² = (20 + 1)² = 400 + 40 + 1.
Use the square you know, then the two extra pieces.
96 ÷ 6 is 60 ÷ 6 plus 36 ÷ 6.
Split the dividend into multiples you can divide.
19 × 4 should be a little under 80.
If the adjustment goes the wrong way, the answer jumps the wrong side of the easy product.
Start from a product you are given, not one you recompute. 34 × 26 = 884. To get 3.4 × 2.6, divide 34 by 10 and divide 26 by 10. The product is divided by 10 × 10 = 100. So 884 ÷ 100 = 8.84. Both decimals moved one place, so the answer moves two places.
If only one factor changes, the product moves one place. 340 is 34 × 10, and the other factor stays 26, so 340 × 26 = 884 × 10 = 8840. In the same way, 3.4 × 26 = 884 ÷ 10 = 88.4. Count how many places the factors moved, and move the product by that total.
A size check catches a missed place. 3.4 × 2.6 should be close to 3 × 3 = 9, and 8.84 agrees. 88.4 is about ten times too big, which is what you get if you divide 884 by 10 instead of by 100. 8840 is the right size for 340 × 26, a bit under 300 × 30 = 9000.
Method: 1. Start from the product you were given. 2. If a factor becomes 10 times bigger, the product becomes 10 times bigger. 3. If a factor becomes 10 times smaller, divide the product by 10.
Multiplying by 5 is the same as multiplying by 10 and then halving, because 5 = 10/2. For 48 × 5, first 48 × 10 = 480, then 480 ÷ 2 = 240. Halving first also works: 48 ÷ 2 = 24, then 24 × 10 = 240. Use the order that keeps whole numbers.
Multiplying by 25 is multiplying by 100 and dividing by 4, because 25 = 100/4. For 36 × 25, first 36 × 100 = 3600, then 3600 ÷ 4 = 900. Dividing by 2 instead of by 4 leaves 1800, which is 36 × 50, not 36 × 25. The 4 is essential.
Check with a split. 36 × 20 = 720 and 36 × 5 = 180, so 720 + 180 = 900. The same answer from two routes is a strong check. A rough size check also helps: 36 × 25 should be near 40 × 25 = 1000, and 900 is just under that.
Method: 1. Multiplying by 5 is multiply by 10, then halve. 2. Multiplying by 25 is multiply by 100, then divide by 4. 3. Use the easy product, then adjust.
3.95 is 0.05 below 4. So 3.95 × 6 = 4 × 6 - 0.05 × 6. Four groups of 6 make 24. The extra 0.05 in each group is 0.05 × 6 = 0.3. Subtract it, because the start was too high: 24 - 0.3 = 23.7. Adding the 0.3 would be the product for 4.05, not for 3.95.
The adjustment is one whole group, not a single unit. 19 × 4 = 20 × 4 - 1 × 4 = 80 - 4 = 76. Subtracting 1 instead of 4 gives 79, which removes one unit rather than one group of 4. If the easy number is above the real one, subtract. If it is below, add.
Write the easy product and the adjustment on the same line so the sign is visible. 3.95 × 6 is a little under 24, and 23.7 agrees. If the answer lands above 24, the adjustment was added the wrong way. The trick is only finished when the direction matches the rounding.
Method: 1. Round the awkward number to a nearby whole number. 2. Work out that easier product. 3. Add or subtract the small adjustment you rounded away.
Match the trick to the number. A factor of 5 wants × 10 then ÷ 2. A factor of 25 wants × 100 then ÷ 4. A number just under a whole wants round-and-subtract. A question that gives 34 × 26 = 884 wants a place shift, not a fresh long multiplication. Using the given product is the method.
After the answer, compare it with a one-significant-figure estimate. 3.95 × 6 is about 4 × 6 = 24, and 23.7 is just under 24. 36 × 25 is about 40 × 30 = 1200, and 900 is a little below that, as expected, because both numbers were rounded up. 3.4 × 2.6 is about 3 × 3 = 9, and 8.84 is just under 9.
If the estimate and the answer disagree by a factor of 10, a decimal point has moved too far or not far enough. Fix the places before you change the digits. The digits of 884 should still be visible in 8.84, 88.4 and 8840. A completely different string of digits means the known product was not used.
Method: 1. Pick the trick that matches the number you were given. 2. A known product wants a digit shift, and a near-whole number wants an adjustment. 3. Check the size of the answer against a rough estimate.
Real non-calculator papers often print one multiplication and then ask three or four related questions. Besides moving the decimal point, a favourite step is to change one of the numbers by one. If 27 × 46 = 1242, then 27 × 47 is one more lot of 27, so it equals 1242 + 27 = 1269. Likewise 27 × 45 = 1242 - 27 = 1215.
Notice which number you add or subtract. It is the number that stayed the same, not the one that changed. Doubling one number doubles the answer, so 54 × 46 = 2484. A division uses the same fact backwards: 1242 ÷ 46 = 27, because division undoes multiplication. Always write the given fact at the start of the line so it is clear which calculation you are using.
Method: 1. Write the given fact and compare each new question with it. 2. Decide what changed: a decimal point, one number by one, or one number doubled. 3. Add or subtract the number that stayed the same, or scale the answer by the same factor. 4. Check the size of your answer with a rough estimate.
Tip: When the question says "use the calculation to help", a long method from scratch usually scores no marks.
When no fact is given, a two-digit by two-digit product is done by splitting one number into tens and units. For 68 × 35, work out 68 × 30 = 2040 and 68 × 5 = 340. Add the parts to get 2380. A grid or column method does exactly the same thing, so use the layout you trust.
A neat variation is to round up and then subtract. 47 × 38 is 47 × 40 - 47 × 2 = 1880 - 94 = 1786. Each partial product is small enough to do in your head, and the working is easy to follow. Method marks are given for correct partial products, so show them even when your final answer has a slip.
Method: 1. Split one number into tens and units, or into a round number and a small difference. 2. Work out each partial product separately. 3. Add or subtract the partial products. 4. Estimate with rounded numbers to check the size of the result.
To multiply decimals, ignore the decimal points, multiply the whole numbers, then put the point back. Count the total number of decimal places in the question and give the answer that many. For 0.006 × 0.35, work out 6 × 35 = 210. There are 3 + 2 = 5 decimal places, so the answer is 0.00210 = 0.0021.
To divide by a decimal, multiply both numbers by the same power of ten so that you divide by a whole number. 54 ÷ 0.9 becomes 540 ÷ 9 = 60. Dividing by a number less than 1 makes the answer bigger, so 54 ÷ 0.9 cannot be 5.4. Use this size check on every decimal division.
Method: 1. Multiply: remove the decimal points and multiply the whole numbers. 2. Count the decimal places in both numbers and put that many in the answer. 3. Divide: multiply both numbers by the same power of ten to make the divisor whole. 4. Check: dividing by less than 1 must make the answer bigger.
Doubling one number and halving the other keeps a product the same. 15 × 16 = 30 × 8 = 240, and 2.5 × 0.48 × 40 becomes 2.5 × 40 = 100 first, then 100 × 0.48 = 48. Look for pairs of numbers that make ten, a hundred or a thousand and multiply those together first.
Numbers that are just under a round number are done by rounding up and subtracting. Five tickets at £19.98 cost 5 × 20 - 5 × 0.02 = 100 - 0.10 = £99.90. A difference of squares, such as 52^2 - 48^2, is (52 - 48)(52 + 48) = 4 × 100 = 400. Choose the shortcut that gives the smallest numbers to work with.
Method: 1. Scan the calculation for factors that multiply to a round number. 2. Reorder the multiplication so the easy pair goes first. 3. For a near-round number, round up and then subtract the difference multiplied out. 4. Estimate the answer to check the final result is the right size.
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