Rounding and estimating

To round, look at the digit just after the one you're keeping.

1. Find the last digit you need to keep.
2. Look at the next digit.
3. If it is 5 or more, round up. If it is 4 or less, leave the kept digit as it is.
4. Drop the digits after it (or, before the decimal point, replace them with zeros).

Example: round 7.4682 to 2 decimal places, 1 decimal place and the nearest whole number.

1. 2 d.p.: keep 7.46, next digit 8, round up: 7.47
2. 1 d.p.: keep 7.4, next digit 6, round up: 7.5
3. Whole number: keep 7, next digit 4, leave it: 7

Example: round 3468 to the nearest 100.

1. The hundreds digit is 4; the next digit is 6, so round up.
2. 3500

The first significant figure is the first digit that isn't zero. Count on from there, including any zeros in between.

1. Find the first non-zero digit and count the figures you need.
2. Round using the next digit, then fill with zeros up to the decimal point.

Example: round 0.004 56 to 2 significant figures.

1. The first significant figure is 4; keep 4 and 5.
2. The next digit is 6, so round up: 0.0046

Example: round 34 567 to 2 s.f. and 20.395 to 3 s.f.

1. 34 567: keep 3 and 4, next digit 5, round up: 35 000
2. 20.395: the zero between 2 and 3 counts, so keep 2, 0, 3; next digit 9, round up: 20.4

To estimate an answer, round every number to 1 significant figure, then work it out in your head.

1. Round each number to 1 s.f.
2. Work out the calculation with the rounded numbers.

If you round numbers up in a multiplication, the estimate is an overestimate (too big). If you round the number you divide by up, the estimate is an underestimate (too small).

Example: estimate 48.7 × 3.12 ÷ 0.495.

1. 48.7 ≈ 50, 3.12 ≈ 3 and 0.495 ≈ 0.5
2. 50 × 3 = 150
3. 150 ÷ 0.5 = 300

Example: estimate 3.8 × 6.7. Is it an overestimate or an underestimate?

1. 3.8 ≈ 4 and 6.7 ≈ 7
2. 4 × 7 = 28
3. Both numbers were rounded up, so 28 is an overestimate.

To estimate a square root without a calculator, find the square numbers either side.

1. Find the square numbers just below and just above the number.
2. The square root is between their roots.
3. Judge how far along it is: closer to the lower square means closer to the lower root.

Example: estimate √70 to 1 decimal place.

1. 64 = 8² and 81 = 9², so √70 is between 8 and 9.
2. 70 is 6 of the 17 steps from 64 to 81: about a third of the way.
3. √70 ≈ 8.4

A calculator gives √70 = 8.366…, so 8.4 is a good estimate.

Example: between which two whole numbers is √40?

1. 36 = 6² and 49 = 7²
2. 6 and 7