E1.9 Estimation
- Syllabus
- 0580–2028–2029
- Topic
- E1.9
- Level
- Extended
Learning objectives
E1.9.1Round values to a specified degree of accuracy.• Round values to a specified degree of accuracy.E1.9.2Make estimates for calculations involving numbers, quantities and• Make estimates for calculations involving numbers, quantities and measurements.E1.9.3Round answers to a reasonable degree of accuracy in the context of a• Round answers to a reasonable degree of accuracy in the context of a given problem. Includes decimal places and significant figures. e.g. write 5764 correct to the nearest thousand. For example, estimate a calculation by rounding each value to one significant figure.
Round to a stated degree of accuracy
Rounding replaces a value with a nearby value at a stated place. Locate the final digit to keep, inspect the next digit, then keep it unchanged for 0–4 or increase it by 1 for 5–9.
| Accuracy request | Where counting starts | Example |
|---|---|---|
| 2 decimal places | first digit after the decimal point | 4.376→4.38 |
| 3 significant figures | first non-zero digit | 0.004376→0.00438 |
| nearest thousand | thousands place | 5764→6000 |
After rounding a whole number, replace removed place-value digits with zeros. For a decimal, remove digits beyond the stated accuracy. Leading zeros only locate the decimal point and are not significant figures.
Trailing zeros may communicate the requested precision: 2.50 has 3 significant figures. For a large value where zeros are ambiguous, standard form can make the accuracy explicit, for example 2.5×104 to 2 significant figures.
Round once from the original value. Repeated rounding can change the result: 2.449 is 2.4 to 2 significant figures, even though rounding first to 2.45 and then again would incorrectly give 2.5.
Estimate a calculation by rounding inputs
An estimate is a deliberately approximate result used to judge size and reasonableness. When an instruction specifies 1 significant figure, round every input to 1 significant figure before calculating.
| Step | Action | Check |
|---|---|---|
| 1 | round each input as instructed | every rounded number has the requested accuracy |
| 2 | copy the operations and brackets unchanged | the structure still matches the original calculation |
| 3 | calculate with the simpler values | write ≈, not =, between the original and estimate |
| 4 | compare scale and sign | the estimate is plausible for the original values |
29.6−9.78.2×3.8≈30−108×4=2032=1.6
If no accuracy is prescribed, choose nearby numbers that simplify the calculation without changing its scale. Avoid choices that make a denominator zero or erase an important small quantity.
Estimating the calculation is different from calculating exactly and merely rounding the final answer. In an estimate, the inputs are simplified first; in an exact calculation, retain full precision until the end.
Choose sensible accuracy for a final answer
A final answer should match both the instruction and what the context can meaningfully represent. Keep full calculator precision during the working, then round once at the end.
| Context or instruction | Usually sensible final form |
|---|---|
| people, trees or complete objects | a whole number, interpreted according to the situation |
| money in ordinary transactions | two decimal places |
| a measured quantity | no more precision than the data justify |
| nearest 1000, stated decimal places or significant figures | exactly the requested accuracy |
Suppose 150 trees each produce 52.4 kg of fruit and one fruit has mass 180 g. Convert kilograms to grams before dividing: 150×52.4×1000÷180=43666.6…. To the nearest thousand, the estimate is 44000 fruits.
A context can also control direction. A count of complete boxes needed may require rounding up, whereas the number of complete groups that can be made may require rounding down. Follow an explicit accuracy instruction when one is given.
Do not round intermediate values unless the question asks for an estimate. Premature rounding accumulates error, and excessive decimal places can claim a precision that the measurements do not support.