Improving rough calculations with percentages

You can get a surprising amount of accuracy in multiplication and division by refining your rough calculation using percentages.

First, some rules. Suppose you want to multiply two numbers A and B, and you round them to a and b.

If the percentage difference between a and A is x%, and between b and B is y%, then

• the percentage difference between a x b and A x B is approximately x+y%
• the percentage difference between a÷b and A÷B is approximately x-y%

This is more accurate the smaller x and y are. You should not rely on it if they are more than 10%.

Example 6224 x 1.93

You could round the first number to 6000, and the second to 2. Your first estimate is 12,000.

The percentage difference between 2 and 1.93 is -3.5% (-0.07÷2).

The percentage difference between 6000 and 6224 is a bit harder but the difference 224÷6 gives about 37 (if you can do a small long division in your head) which is 3.7%.

Therefore your second estimate is 3.7 + -3.5 = +0.2% more than 12,000, which would give you an estimate of 12,024.

Example 378 ÷ 4.12

If you round 4.12 to 4, that suggests rounding 378 to 360 (as 36 is a multiple of 4), giving a first estimate of 90.

The percentage difference between 4 and 4.12 is 3% (0.12÷4).

The percentage difference between 360 and 378 is 18÷360: if you are lucky you can spot that 18 is half of a tenth of 360 which is exactly 5%.

Therefore your second estimate is 5 - 3 = 2% more than 90, which would give you an estimate of 91.8. The exact answer is 91.74757282 which is pretty close, I think you would agree.