Significant Figures
Estimation
In this note
- 1.9Estimate a calculation by rounding every number to 1 significant figure, writing the rounded calculation.
- 1.9Estimate with formulas, areas, costs and square roots.
- 1.9Say whether an estimate is an overestimate or an underestimate, and why.
Before you start: rounding.
Formula bank
The estimation method
An estimate is a quick answer you can do in your head, to check a calculator answer. The exam rule is fixed: round every number to 1 s.f., WRITE that calculation (the method mark), then work it out exactly.
General methodHow to estimate any calculation
- Round EACH number to 1 s.f. - including numbers inside powers and roots, and each term of a sum.
- Write the rounded calculation down.
- Work it out without a calculator; don't round the answer again.
Example Estimate \(\dfrac{38.6-9.17}{0.512}\).
Round each number: 40, 9, 0.5.
- \(\dfrac{40-9}{0.5}=\dfrac{31}{0.5}\)
Answer 62
Try it yourself
Show answer
Answer 10
Hint
Show answer
Answer 80
Hint
Show answer
Answer 2
Formulas, areas and costs
The same rule works in any setting: round every given number to 1 s.f., then do the calculation. A patio 28 m by 13 m with tiles at $4.85 per m² costs about \(30\times10\times5=\$1500\).
Try it yourself
Show answer
Answer 50
Show answer
Answer 1200
Overestimate or underestimate?
General methodHow to decide whether an estimate is too big or too small
- Note the direction each number was rounded (up or down).
- Rounding UP something you multiply by, or the top → estimate too big. Rounding UP the bottom (divisor) → too small.
- Mixed directions? Work out the exact value and compare.
Example Is \(60\times20=1200\) an over- or underestimate of \(63.8\times18.4\)?
One rounded down, one up: compare with the exact value.
- exact \(=1173.92\)
- 1200 > 1173.92
Answer an overestimate
Try it yourself
Show answer
Answer too small
Ways they test it
Type 1 Estimate a fraction
"By writing each number correct to 1 s.f., estimate \(\dfrac{38.6-9.17}{0.512}\)." (2 marks)
Spot it the exact words "1 significant figure".
- Rounded calculation written (M1).
- Its value (A1).
Type 2 Powers and roots
"Estimate \(\dfrac{3.92\times2.18^{2}}{0.512}\)."
Spot it a power or root.
- Round the number inside first (2.18 → 2).
- Then square or root.
For example \(\dfrac{3.92\times2.18^{2}}{0.512}\)
- \(\dfrac{4\times2^{2}}{0.5}=\dfrac{16}{0.5}\)
Answer 32
Type 3 Cost or area, then over/under
"Estimate the cost of tiling 28 m by 13 m at $4.85 per m². Is it more or less than the actual cost?"
Spot it a context + "explain".
- Estimate.
- Exact value; compare; explain with the biggest rounding.
For example 28 × 13 × 4.85
- \(30\times10\times5=1500\)
- exact 1765.40
Answer $1500, an underestimate
Type 4 Check a calculator answer
"Show that your answer is reasonable."
Spot it after a long calculation.
- Estimate it.
- Compare sizes.
Mistakes to avoid
- ✗ \(\dfrac{38.6-9.17}{0.512}\) → \(\dfrac{30}{0.5}\)✓ \(\dfrac{40-9}{0.5}\)Round EACH number, not the total.
- ✗ 39 for 38.6✓ 4039 is 2 s.f.
- ✗ Calculator answer rounded at the end✓ rounded calculation writtenThe method earns the marks.
- ✗ Rounding the divisor up gives an overestimate✓ an underestimateYou divide by more.
Summary
- Every number to 1 s.f.; write the rounded calculation.
- Round inside roots and powers too.
- ÷ by a small decimal makes it bigger.
- Divisor rounded up → underestimate; top rounded up → overestimate.
If you see …
- "by writing each number to 1 s.f."show the rounded calculation.
- "more or less than"look at which way each number was rounded.
Practice
Practice: easy, medium, hard
Easy
Show working
- \(\tfrac{600\times0.2}{3}\)
Answer 40
Show working
- \(70\div7\)
Answer 10
Show working
- \(\tfrac{9+20}{0.5}\)
Answer 58
Medium
Guide me
What is it asking? an estimate
What do we know? a square
Plan round 2.18 first
Show working
- \(\tfrac{4\times4}{0.5}\)
Answer 32
Guide me
What is it asking? an estimate
What do we know? ÷ 0.5
Plan 2 × 50 ÷ 0.5
Show working
- \(\tfrac{100}{0.5}\)
Answer 200
Guide me
What is it asking? an estimate
What do we know? 387 → 400
Plan \(\sqrt{400}=20\)
Show working
- \(\tfrac{20\times6}{0.3}=\tfrac{120}{0.3}\)
Answer 400
Hard
Guide me
What is it asking? an estimate
What do we know? a small decimal on the bottom
Plan 800 ÷ (0.04 × 20)
Show working
- \(0.04\times20=0.8\)
- \(800\div0.8\)
Answer 1000
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\dfrac{800}{0.04\times20}\) | M1 | ||
| 1000 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? area in hectares
What do we know? 312 → 300, 18.7 → 20
Plan estimate m², then ÷ 10 000
Show working
- \(300\times20=6000\) m²
- \(\div10\,000\)
Answer 0.6
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 300 × 20 | M1 | ||
| 0.6 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? an estimated cost
What do we know? three numbers
Plan 30 × 10 × 5
Show working
- \(30\times10\times5\)
Answer $1500
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(30\times10\times5\) | M1 | ||
| 1500 | A1 | ||
| Total: 2 marks | |||
Can I do this?
- I can estimate by rounding every number to 1 s.f. and showing the rounded calculation.
- I can estimate with roots, powers, formulas, areas and costs.
- I can say whether an estimate is too big or too small, and why.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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