Limits Of Accuracy
Upper and lower bounds
In this note
- 1.10Write the upper bound and lower bound of a rounded value (nearest 10, 5, 0.1, mm, s.f., d.p.).
- 1.10Write the error interval, e.g. \(61.5\le L<62.5\).
- 1.10Find the bounds of a total of several equal rounded measurements.
Before you start: rounding. Next: bounds in calculations.
Formula bank
A rounded number is a range
"62 m to the nearest metre" could be anything from 61.5 m (which rounds up to 62) to just under 62.5 m. So the true value lies in an interval, half a unit each side.
General methodHow to find the bounds of any rounded value
- Find the unit of rounding from the wording: nearest 5 → 5; nearest mm (in cm) → 0.1; to 3 s.f. → the place of the 3rd figure.
- Halve it.
- Lower bound = value − half; upper bound = value + half.
- Error interval: \(\text{LB}\le x<\text{UB}\).
Example A mass m is 600 kg to the nearest 50 kg. Complete … ≤ m < …
Unit 50, half is 25.
- 600 − 25 = 575
- 600 + 25 = 625
Answer \(575\le m<625\)
Try it yourself
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Answer 65
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Answer 177.5
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Answer 0.365
Hint
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Answer 18.25
| Measurement | Unit | LB | UB |
|---|---|---|---|
| 2.5 kg, nearest 0.1 kg | 0.1 | 2.45 | 2.55 |
| 1240 cm³, nearest 20 | 20 | 1230 | 1250 |
| 3.40, to 3 s.f. | 0.01 | 3.395 | 3.405 |
| 4000, to 1 s.f. | 1000 | 3500 | 4500 |
| 4000, to 2 s.f. | 100 | 3950 | 4050 |
Totals of equal measurements
General methodHow to find the bounds of a total of n equal items
- Bounds of ONE item.
- Multiply each bound by n.
5 sides each ± 0.5 means the total is ± 2.5, not ± 0.5.
Example A regular pentagon has sides of 64 mm to the nearest mm. Bounds of its perimeter?
One side 63.5 to 64.5, then × 5.
- \(5\times63.5=317.5\)
- \(5\times64.5=322.5\)
Answer 317.5 mm and 322.5 mm
Try it yourself
Show answer
Answer 498
Ways they test it
Type 1 Write down a bound
"Amir's height is 175 cm to the nearest 5 cm. Write down the upper bound." (1 mark)
Spot it "upper bound" / "lower bound".
- Unit, half, add or subtract.
For example 175 cm to the nearest 5
- half of 5 = 2.5
Answer 177.5 cm
Type 2 Error interval
"Complete the statement … ≤ m < …" (2 marks)
Spot it an inequality to fill in.
- Both bounds.
- ≤ at the bottom, < at the top.
Type 3 Awkward units
"18.3 cm to the nearest millimetre" / "4.70 to 3 s.f."
Spot it mm in cm, or s.f./d.p.
- Work out the unit first (0.1 cm; 0.01).
Type 4 Totals
"An equilateral triangle has sides of 15 cm to the nearest cm. Bounds of the perimeter?"
Spot it several equal rounded lengths.
- Bounds of one side.
- × number of sides.
For example 15 cm, 3 sides
- \(3\times14.5=43.5\)
- \(3\times15.5=46.5\)
Answer 43.5 cm and 46.5 cm
Mistakes to avoid
- ✗ Nearest 5: ± 0.5✓ ± 2.5Half of the unit.
- ✗ 18.3 cm to the nearest mm: 17.8 and 18.8✓ 18.25 and 18.351 mm = 0.1 cm.
- ✗ Upper bound written as 62.49✓ 62.5By convention write the bound itself.
- ✗ Perimeter = 5 × 64 ± 0.5✓ 5 × 63.5 to 5 × 64.5Every side is uncertain.
Summary
- Bounds = value ± half the unit of rounding.
- Find the unit from the wording first (nearest 5, nearest mm, s.f., d.p.).
- Error interval: \(\text{LB}\le x<\text{UB}\).
- Totals of n equal items: n × each bound.
If you see …
- "nearest 5"± 2.5.
- "to 3 s.f."the unit is the place of the 3rd figure.
- several equal piecesmultiply the bounds.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 4.5
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Answer 275
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Answer 22.5
Medium
Guide me
What is it asking? the LB
What do we know? unit 0.1 cm
Plan 7.4 − 0.05
Show working
- 7.4 − 0.05
Answer 7.35
Guide me
What is it asking? the LB
What do we know? unit 100
Plan 2400 − 50
Show working
- 2400 − 50
Answer 2350
Guide me
What is it asking? the UB
What do we know? unit 0.1
Plan 1.6 + 0.05
Show working
- 1.6 + 0.05
Answer 1.65
Hard
Guide me
What is it asking? the error interval
What do we know? 3 s.f.: last figure in the hundredths
Plan ± 0.005
Show working
- unit 0.01, half 0.005
Answer \(4.695\le m<4.705\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 4.695 | B1 | ||
| 4.705 | B1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? the LB of the perimeter
What do we know? 6 sides, each ≥ 79.5
Plan 6 × 79.5
Show working
- \(6\times79.5\)
Answer 477
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(6\times79.5\) | M1 | ||
| 477 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? the LB
What do we know? the 0 is the 3rd s.f.
Plan unit 0.1, half 0.05
Show working
- 12.0 − 0.05
Answer 11.95
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 11.95 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can explain why a rounded value stands for an interval.
- I can write bounds for any rounding unit, including s.f. and d.p.
- I can write an error interval with ≤ and <.
- I can find the bounds of totals of equal measurements.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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