Estimation

An estimate is an answer that is close to the real answer and quick to work out. You replace each awkward number with a nearby number that is easy to use, then do the easier calculation.

The sign \approx means "is approximately equal to". For example, 38\times62\approx40\times60

In the exam you are usually asked to "Work out an estimate for" a calculation, for 2 or 3 marks, on the non-calculator paper. Round each number to one significant figure, write down the easier calculation, then work it out. Working out the exact answer usually gets no marks, even if you round it at the end.

Some questions then ask "Is your answer an underestimate or an overestimate?" and want a reason, for 1 more mark.

The first significant figure is the first digit that is not zero. In 382 it is the 3, worth 300.

To round to one significant figure, look at the next digit. If it is 5 or more, round up. If it is less than 5, keep the first digit as it is. Then write zeros in the places before the decimal point, so the number stays the same size.

• 382\approx400 because the next digit is 8
• 6.7\approx7
• 2850\approx3000
• 19.6\approx20, because the 1 is worth 10 and the next digit is 9

Zeros at the start of a decimal are not significant. In 0.047 the first significant figure is the 4, in the hundredths column.

The next digit is 7, so round up: 0.047\approx0.05 to one significant figure. Keep the zeros in front, so the size does not change. It is not 0.5 or 5.

In the same way, 0.31\approx0.3 and 0.19\approx0.2

When you multiply, multiply the first digits, then put back all the zeros. For 700\times300, work out 7\times3=21, then add the four zeros: 210000

When you divide by a number ending in zero, take the same number of zeros off both numbers. For 4200\div60, work out 420\div6=70

When you divide by a decimal, multiply both numbers by 10 until you are dividing by a whole number. For 8\div0.4, work out 80\div4=20

Dividing by a number less than 1 makes the answer bigger. That is a quick check that you have not gone wrong.

Rounding changes the numbers, not the calculation. Keep every bracket, fraction line and operation where it was.

For a fraction, work out the top and the bottom first, then divide the top by the bottom.

An overestimate is bigger than the real answer. An underestimate is smaller than the real answer.

For adding or multiplying, rounding the numbers up gives an overestimate, and rounding them down gives an underestimate.

Dividing needs more care. Rounding up the number being divided makes the answer bigger. Rounding up the number you divide by makes the answer smaller: 12\div4=3, but 12\div6=2

Your reason must be about how you rounded the numbers in the calculation, not about rounding the answer. If one change makes the answer bigger and another makes it smaller, you can say you cannot tell, as long as you give that reason.

• Working out the exact answer. Round every number to one significant figure first. An exact answer usually gets no marks, even if you round it afterwards.
• Not rounding to one significant figure. Look at the first digit and the one after it: 486 becomes 500, not 490, and 7.8 becomes 8, not 7.
• Losing zeros. Count the zeros: 700\times300=210000, and 0.047 rounds to 0.05, not 0.5.
• Dividing the wrong way round. Keep the numbers in the order of the calculation: 3\div30=0.1, not 10.
• Giving a reason about rounding the answer. Say how the numbers in the calculation were rounded, up or down.
• Looking at only one number in a division. Think about both numbers, because rounding the number you divide by up makes the answer smaller.