Approximation and estimation Revision Notes

Basic Rules of Rounding

When rounding a number, first identify the specific place value you need to round to. This could be to the nearest whole number, ten, hundred, thousand, or to a certain number of decimal places. Once you've identified the target digit, look at the digit immediately to its right. This 'deciding digit' determines whether you round up or down. For example, if rounding 345 to the nearest ten, the target digit is 4 (in the tens place). The deciding digit is 5. If rounding 1287 to the nearest hundred, the target digit is 2 (in the hundreds place). The deciding digit is 8.

Rounding Up: Rule of 5 or Greater

If the deciding digit (the digit to the right of your target place value) is 5, 6, 7, 8, or 9, you round up. To round up, you increase the digit in the target place value by one. All digits to the right of the target place value then become zeros if they are before the decimal point, or they are simply dropped if they are after the decimal point. For instance, rounding 6.78 to one decimal place: the target is 7, the deciding digit is 8. Since 8 is '5 or greater', the 7 becomes 8. The result is 6.8. Rounding 145 to the nearest ten: the target is 4, the deciding digit is 5. Since 5 is '5 or greater', the 4 becomes 5 and the 5 becomes 0. The result is 150.

Rounding Down: Rule of Less Than 5

If the deciding digit (the digit to the right of your target place value) is 0, 1, 2, 3, or 4, you round down. To round down, the digit in the target place value remains unchanged. All digits to the right of the target place value become zeros if they are before the decimal point, or they are simply dropped if they are after the decimal point. For example, rounding 12.34 to one decimal place: the target is 3, the deciding digit is 4. Since 4 is 'less than 5', the 3 remains 3. The result is 12.3. Rounding 721 to the nearest hundred: the target is 7, the deciding digit is 2. Since 2 is 'less than 5', the 7 remains 7, and the 2 and 1 become zeros. The result is 700.

Rounding to the Nearest Whole Number/Integer

To round a number to the nearest whole number (or integer), you need to look at the first digit after the decimal point. If this digit is 5 or more (5, 6, 7, 8, 9), you round the whole number part up. If it is less than 5 (0, 1, 2, 3, 4), you keep the whole number part as it is. For example, 12.6 rounds to 13, and 12.3 rounds to 12. The decimal part is then dropped.

Rounding to a Specific Number of Decimal Places

When rounding to a certain number of decimal places (d.p.), you count from the decimal point to find the digit in that position. Then, look at the digit immediately to its right. If this next digit is 5 or more, you round up the digit in the d.p. position. If it is less than 5, the digit in the d.p. position stays the same. All digits after the specified decimal place are removed. For instance, 4.372 rounded to 2 d.p. is 4.37 (because 2 is less than 5). And 4.378 rounded to 2 d.p. is 4.38 (because 8 is 5 or more).

Rounding to the Nearest 10, 100, or 1000

To round to the nearest 10, 100, or 1000, identify the place value you are rounding to. Then, look at the digit immediately to its right. If this digit is 5 or more, round up the digit in the target place value. If it is less than 5, keep the digit in the target place value the same. All digits to the right of the rounded digit become zero. For example, 345 rounded to the nearest 10 is 350 (because the '5' in the units place means we round up the '4' in the tens place). 342 rounded to the nearest 100 is 300 (because the '4' in the tens place means we don't round up the '3' in the hundreds place).

What is Estimation?

Estimation involves finding an approximate value for a calculation or quantity. It's used when an exact answer isn't needed or when you want to quickly check if a precise calculation is reasonable. The key idea is to simplify the numbers involved before performing the operation. This often means rounding numbers to one significant figure to make mental arithmetic easier and quicker. For instance, estimating 47 x 23 would involve rounding 47 to 50 and 23 to 20, then calculating 50 x 20 = 1000.

Rounding for Estimation

The most common method for estimation is rounding numbers to one significant figure. This simplifies calculations significantly. To round to one significant figure: identify the first non-zero digit. If the digit to its right is 5 or more, round up the first significant figure. If it is less than 5, keep the first significant figure the same. Replace all other digits to the right of the first significant figure with zeros. For example, 345 rounded to one significant figure is 300, and 0.087 rounded to one significant figure is 0.09. Apply this to all numbers in a calculation, then perform the operation.

Estimating Products and Quotients

When estimating products (multiplication) or quotients (division), rounding each number to one significant figure is typically the best approach. For example, to estimate 48.7  9.2: Round 48.7 to 50 (1 s.f.). Round 9.2 to 9 (1 s.f.). Estimated product = 50  9 = 450. For division, to estimate 623  29: Round 623 to 600 (1 s.f.). Round 29 to 30 (1 s.f.). Estimated quotient = 600  30 = 20. Always consider if the rounded numbers make the calculation easier before finalising your estimate.

Estimating Sums and Differences

For sums (addition) and differences (subtraction), rounding to one significant figure is also a good strategy. However, sometimes rounding to a specific place value (e.g., to the nearest 10 or 100) might give a more accurate or more sensible estimate depending on the context. For example, to estimate 783 + 121: Round 783 to 800 (1 s.f.). Round 121 to 100 (1 s.f.). Estimated sum = 800 + 100 = 900. Or, if rounding to the nearest 100 for both numbers makes sense, it's 800 + 100 = 900. The key is to simplify the numbers to make the mental calculation straightforward while keeping the estimate reasonable.

What is Estimation?

Estimation means finding an approximate value for a calculation, rather than an exact one. This is useful for checking calculations, making quick decisions, or when exact numbers are not necessary. The main principle is to simplify the numbers involved to make the calculation easier, typically by rounding. A good estimate should be close enough to the actual answer to be useful, but not so precise that it becomes difficult to calculate mentally or quickly.

Rounding to One Significant Figure

The most common method for estimation at Foundation tier is rounding each number in a calculation to one significant figure (1 sf). To round to 1 sf: find the first non-zero digit from the left. Look at the digit immediately to its right. If this digit is 5 or more, round up the first significant figure. If it's less than 5, keep the first significant figure as it is. Replace all other digits to the right of the first significant figure with zeros (for whole numbers) or discard them (for decimals). For example, 37.8 rounds to 40 (1 sf), 0.052 rounds to 0.05 (1 sf), and 2430 rounds to 2000 (1 sf).

Applying Estimation to Calculations

Once you have rounded each number to one significant figure, perform the calculation using these rounded numbers. For example, to estimate 47.9 × 2.1: round 47.9 to 50 and 2.1 to 2. Then calculate 50 × 2 = 100. This is the estimated answer. Similarly, for division like 812 ÷ 4.3: round 812 to 800 and 4.3 to 4. Then calculate 800 ÷ 4 = 200. Always clearly show your rounding steps before giving your final estimated answer.