Estimation
Section: Number 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Rounding to Decimal Places and Significant Figures
Rounding replaces a number with a simpler one that is close to its actual value. Decimal places count digits after the decimal point; significant figures count the "meaningful" digits of a number, wherever they fall.
- To round to a number of decimal places, count that many digits after the decimal point, then look at the next digit to decide whether to round up or down
- The first significant figure of a number is its first non-zero digit, reading from the left
- Leading zeros are never significant, e.g. in 0.0567, the first significant figure is 5
- Zeros between or after significant digits usually are significant, e.g. in 3.05, all three digits are significant
- When rounding a whole number to a given number of significant figures, placeholder zeros are needed to keep the digits in their correct place value columns
Worked Example: Rounding to a Given Accuracy
- Question: Round 3826 to the nearest thousand.
- Step 1: The thousands digit is 3; the next digit (hundreds) is 8, which is 5 or more, so round up
- Answer: 4000
- Question: Round 0.05672 to (i) 2 decimal places, (ii) 3 significant figures.
- Step 1 (i): Keep 2 digits after the point (0.05...) and look at the 3rd digit, 6 - round up: 0.05672 ≈ 0.06
- Step 1 (ii): The first 3 significant figures are 5, 6, 7; the next digit is 2, so round down: 0.05672 ≈ 0.0567
Common Mistakes
MistakeCounting leading zeros as significant figures, e.g. saying 0.0567 has 4 significant figures
Fixleading zeros are never significant - counting starts from the first non-zero digit, so 0.0567 has 3 significant figures (5, 6, 7)
MistakeDropping place value when rounding a whole number to significant figures, e.g. rounding 3826 to 2 s.f. as 38
Fixplaceholder zeros must be kept to preserve the number's size: 3826 to 2 s.f. is 3800, not 38
Estimating by Rounding to 1 Significant Figure
Rounding every number in a calculation to 1 significant figure first gives a quick, sensible estimate of the answer, without needing a calculator.
- Round every value in the calculation to 1 significant figure before working anything out
- Then carry out the calculation using the rounded values - this is the estimate
Worked Example: Estimating a Calculation
- Question: By rounding each number to 1 significant figure, estimate the value of (0.62 × 38.4/19.3).
- Step 1: Round each number to 1 s.f.: 0.62 → 0.6, 38.4 → 40, 19.3 → 20
- Step 2: Substitute the rounded values: (0.6 × 40/20)
- Step 3: Evaluate: (24/20) = 1.2
- Answer: 1.2
Common Mistakes
MistakeRounding only some of the numbers in the calculation, leaving others exact
Fixevery number in the calculation must be rounded to 1 significant figure first, before any working is carried out
MistakeWorking out the exact answer first, then rounding that final answer to 1 significant figure
Fixestimation means rounding the inputs before calculating, not rounding the final exact answer afterwards - the two methods can give different results
Rounding Appropriately in Context
The correct way to round an answer often depends on what it represents. Money is naturally rounded to 2 decimal places, and some real-life quantities must always be rounded up, even when the decimal part is small.
- Amounts of money are normally rounded to 2 decimal places, since currencies have a smallest unit (e.g. cents or pence)
- When a quantity must be a whole number of items that are enough for everyone or everything (e.g. buses, boxes, tables), the answer must be rounded up, even if the decimal part is less than 0.5
Worked Example: Rounding Up in Context
- Question: 134 students are going on a trip. Each coach holds 25 students. How many coaches are needed?
- Step 1: Divide: 134 ÷ 25 = 5.36
- Step 2: Normal rounding would give 5, but 5 coaches only seat 125 students - not enough for all 134
- Answer: 6 coaches are needed (rounded up, regardless of the decimal part)
Common Mistakes
MistakeRounding 5.36 coaches to the nearest whole number (5), following the usual rounding rule
Fixwhen the context needs enough whole items for everyone, always round up, even if the decimal part is below 0.5 - 5 coaches would leave students behind
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