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This lesson covers upper and lower bounds, error intervals, and using bounds in calculations. Bounds is a Higher-tier topic in the Edexcel GCSE Mathematics (1MA1) specification and is commonly worth 3-5 marks on the exam.
| Term | Meaning |
|---|---|
| Lower bound | The smallest value that rounds (or truncates) to the given measurement |
| Upper bound | The smallest value that would round UP to the next measurement |
| Error interval | The range of possible values, written using inequalities |
| Degree of accuracy | How precisely a measurement has been recorded (e.g. to 1 d.p., to the nearest 10) |
| Continuous data | Data that can take any value within a range (e.g. length, mass, time) |
| Discrete data | Data that can only take specific values (e.g. number of people) |
| Truncation | Cutting off digits without rounding |
When a value has been rounded, the true value could be anything within half a unit either side.
Formula:
A length is measured as 8.3 cm, correct to 1 decimal place. Find the error interval.
Degree of accuracy = 0.1 cm, so half = 0.05 cm.
Error interval: 8.25≤ length < 8.35
Important: Use ≤ for the lower bound (it IS included) and < for the upper bound (it is NOT included, because it would round up to 8.4).
A mass is 750 g, rounded to the nearest 10 g. Write the error interval.
Degree of accuracy = 10 g, half = 5 g.
Error interval: 745≤mass<755
A crowd at a concert is 24,000, rounded to the nearest 1,000. Write the error interval.
Degree of accuracy = 1000, half = 500.
Error interval: 23,500≤ crowd < 24,500
When a value has been truncated (not rounded), the lower bound is the truncated value itself, and the upper bound is one unit of accuracy above.
A length is truncated to 4.7 cm (to 1 decimal place). Write the error interval.
Error interval: 4.7≤ length < 4.8
Key Difference from Rounding: When rounding to 1 d.p., 4.7 has bounds 4.65 to 4.75. When truncating to 1 d.p., 4.7 has bounds 4.7 to 4.8. Edexcel specifically tests whether students know this difference.
When performing calculations with rounded values, you need to decide which combination of upper and lower bounds gives the maximum or minimum result.
| Operation | Maximum result | Minimum result |
|---|---|---|
| A + B | Upper A + Upper B | Lower A + Lower B |
| A - B | Upper A - Lower B | Lower A - Upper B |
| A×B | Upper A× Upper B | Lower A× Lower B |
| A÷B | Upper A÷ Lower B | Lower A÷ Upper B |
Memory aid: For subtraction and division, to MAXIMISE the result, make the first number as BIG as possible and the second as SMALL as possible (and vice versa for minimum).
p = 6.4 (correct to 1 d.p.) and q = 3.8 (correct to 1 d.p.). Calculate the upper and lower bounds of p + q.
Bounds of p: 6.35≤p<6.45 Bounds of q: 3.75≤q<3.85
a = 12.5 cm (correct to 1 d.p.) and b = 4.3 cm (correct to 1 d.p.). Calculate the upper and lower bounds of a - b.
Bounds of a: 12.45≤a<12.55 Bounds of b: 4.25≤b<4.35
The base of a rectangle is 8.4 cm and the height is 3.6 cm, both correct to 1 decimal place. Calculate the upper and lower bounds of the area.
Bounds of base: 8.35≤base<8.45 Bounds of height: 3.55≤ height < 3.65
A car travels 148 km (to the nearest km) in 2.4 hours (to 1 d.p.). Calculate the upper and lower bounds of the average speed.
Bounds of distance: 147.5≤d<148.5 Bounds of time: 2.35≤t<2.45
Speed = distance ÷ time
A common exam question asks: "To what degree of accuracy can you give the answer?"
Calculate the upper and lower bounds of the answer, then find the level of accuracy to which they both round to the same value.
A square has side length 8.3 cm, correct to 1 d.p. Calculate the perimeter and give your answer to a suitable degree of accuracy.
Bounds of side length: 8.25≤s<8.35
Perimeter = 4s.
To 2 s.f.: 33.0 rounds to 33, and 33.4 rounds to 33. Both agree.
Answer: The perimeter is 33 cm (to 2 significant figures), since both bounds round to 33 at this level of accuracy.
Edexcel Exam Tip: In exam questions asking for a suitable degree of accuracy, the bounds will always agree at some level. Check 1 s.f. first, then 2 s.f., etc. State the degree of accuracy and justify it.
A runner completes a 100 m track (measured to the nearest metre) in 12.3 seconds (measured to 1 d.p.). Calculate the lower bound of the runner's speed.
Bounds of distance: 99.5≤d<100.5 Bounds of time: 12.25≤t<12.35
For LOWER bound of speed (speed = distance ÷ time):
Answer: 8.06 m/s (to 3 s.f.)
Error intervals are always written using inequality notation:
h = 12 cm, correct to the nearest centimetre. Write the error interval for h.
11.5≤h<12.5
| Mistake | Correction |
|---|---|
| Using ≤ for both bounds | Upper bound uses < (strictly less than) |
| For maximum of A - B, using lower A - upper B | To MAXIMISE a subtraction, use UPPER A - LOWER B |
| Confusing truncation bounds with rounding bounds | Truncation: lower bound = given value; Rounding: lower bound = given value - half precision |
| Forgetting to state the degree of accuracy | Always give the appropriate degree of accuracy when asked |
| Using rounded values instead of bounds in calculations | Use the actual upper/lower bounds, not the rounded values |
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