A mental strategy is valid if it is mathematically correct and you can carry it out reliably. Two good strategies are often available for the same question, and choosing the shorter one, or using both as a check, is a skill in itself.
This lesson closes non-calculator strategy. It draws on estimating to reject impossible answers and organising working.
How do you decide between strategies?
Look at the numbers before you start.
- Near a round number? Use “round and adjust”. 98 × 15 is close to 100 × 15.
- A percentage built from 10% and 5%? Split it. 15% is 10% plus 5%.
- A simple fraction of the amount? Use the fraction. 15% = 3/20.
- Both look easy? Do both and compare. Agreement is a free check.
Worked example
Find 15% of 240 in two ways.
Method 1, split the percentage: 10% of 240 = 24. Half of that, 5%, is 12. So 15% = 24 + 12 = 36.
Method 2, use the fraction: 15% = 15/100 = 3/20. Then 240 ÷ 20 = 12, and 12 × 3 = 36.
Both methods give 36, so the answer is very likely correct.
A second example for a product: work out 98 × 15.
Method 1, round and adjust: 100 × 15 = 1500. Two lots of 15 too many, so subtract 2 × 15 = 30. This gives 1470.
Method 2, split the number: 98 × 10 = 980 and 98 × 5 = 490. Then 980 + 490 = 1470.
Both give 1470.
The mistake to watch for
A common error is to adjust by the wrong amount after rounding.
Mistaken working: 98 × 15 = 100 × 15 − 15 = 1485
The student rounded 98 up to 100, which is 2 too many, but subtracted only one lot of 15.
The correction is to ask how far the rounded number moved. 100 is 2 more than 98, so subtract 2 lots of 15. The second method, 980 + 490 = 1470, would have exposed the slip straight away.
Check yourself
Try these without a calculator, then open each answer.
1. Work out 99 × 24 using round and adjust.
Show answer
100 × 24 = 2400, then subtract one lot of 24. 2400 − 24 = 2376.
2. Find 35% of 80 by splitting the percentage.
Show answer
10% of 80 = 8, so 30% = 24. 5% = 4. So 35% = 24 + 4 = 28. Check: 35/100 × 80 = 28.
3. Find 7.5% of 640 by any method, and say which you chose.
Show answer
Halving method: 10% = 64, 5% = 32, 2.5% = 16. Then 5% + 2.5% = 32 + 16 = 48. Fraction check: 7.5% = 3/40, and 640 ÷ 40 = 16, so 3 × 16 = 48.
Where this leads next
Put everything in this cluster together in the non-calculator strategy practice set. You can also go back to choosing an efficient fraction method for the fraction side of this cluster. The non-calculator working trainer and the percentage-base explorer let you compare two routes on the same task.
If you tend to rely on a single method, a teacher working with you one-to-one can introduce a second route at the right moment. See online one-to-one Mathematics tuition.