C1.9 Estimation
- Syllabus
- 0580–2028–2029
- Topic
- C1.9
- Level
- Core
Learning objectives
C1.9.1Round values to a specified degree of accuracy.• Round values to a specified degree of accuracy.C1.9.2Make estimates for calculations involving numbers, quantities and• Make estimates for calculations involving numbers, quantities and measurements.C1.9.3Round 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 place or number of significant figures
Rounding replaces a value with the nearest value at a stated accuracy. Identify the final digit to keep, inspect the next digit, and increase the kept digit by 1 only when the next digit is 5 or more.
| Accuracy instruction | Where counting starts | Example |
|---|---|---|
| nearest 10, 100, 1000, … | named place in the whole-number part | 11678→11700 to nearest 100 |
| decimal places (dp) | first digit after the decimal point | 3.72194→3.722 to 3 dp |
| significant figures (sf) | first non-zero digit | 0.03682→0.037 to 2 sf |
Keep all digits before the rounding position unchanged. If rounding up creates a 10, carry left through place values: 9876 to the nearest thousand is 10000.
Zeros can communicate accuracy. The value 57.3997 to 4 significant figures is 57.40: the final zero must remain because it is the fourth significant figure. Similarly, 0.0050 has 2 significant figures.
The rounded result alone may not reveal the instruction: 4896→4900 could be rounding to the nearest hundred or nearest ten. State the requested accuracy with the result when context does not already specify it.
Estimate a calculation with easy nearby values
An estimate replaces input values with nearby values that make the calculation quick while preserving its original structure. When instructed, round every input to 1 significant figure before calculating.
Use this order: round each input separately; rewrite the complete expression with the rounded values and the same brackets, powers and operations; calculate that simpler expression; then compare its scale with the original values.
rac{41.3}{9.79 imes0.765}pprox rac{40}{10 imes0.8}=5
For 23.5423.8−78.4, rounding to 1 significant figure gives 20400−80=16. The subtraction remains in the numerator; removing its grouping would estimate a different calculation.
An estimate is not always above or always below the exact answer. Its direction depends on how each rounded input affects the operation: rounding both positive factors down makes their estimated product smaller, but rounding a denominator down can make a quotient larger.
Do not round an intermediate result again unless instructed. Estimation simplifies inputs before calculation; it is different from calculating accurately and rounding only the final answer.
Choose a sensible accuracy for a final answer
A reasonable final accuracy communicates what the context and input data can support. Keep guard digits during working, then round once at the end using any explicit instruction or the practical meaning of the result.
| Context | Sensible final form | Reason |
|---|---|---|
| counted objects or whole items | whole number, with direction chosen by context | partial people, buses or packs may be impossible |
| money in ordinary currency units | usually 2 decimal places | records the smallest common currency unit |
| measured quantity | usually no more significant figures than the least precise input | avoids claiming unsupported measurement precision |
| specified dp or sf | follow the stated instruction exactly | the requested accuracy controls the final digit |
Context can control rounding direction rather than nearest rounding. If 5.25 bags are needed and only whole bags can be bought, the answer is 6 bags; rounding to 5 would leave too little material.
For a calculator value 4.285714… requested to 4 significant figures, retain the full value until the end and report 4.286. Early rounding of intermediate values can change the final digit.
Do not add decimal places merely because a calculator displays them. A result such as 12 people, 8.40,or3.7$ cm can each be appropriately precise in its own context; one universal number of decimal places is not sensible.