Upper and Lower Bounds – Grade 9 IGCSE Maths

Welcome to Bounds!

When we measure something, our result is always an approximation. Understanding upper and lower bounds tells us the range of values that a rounded number could actually represent. This is fundamental in science, engineering, and everyday measurement.

Upper Bound = value + ½ unit Lower Bound = value − ½ unit

Rounding Review

Nearest unit, dp, sf

Upper & Lower Bounds

UB = value + ½ unit

Error Interval

a ≤ x < b notation

Products & Quotients

Max/min of combined measurements

1. Rounding and Units

The degree of accuracy tells you the precision of a measurement. Common examples:

Nearest whole number: unit = 1
Nearest 10: unit = 10
1 decimal place (1 d.p.): unit = 0.1
2 decimal places (2 d.p.): unit = 0.01
1 significant figure (1 s.f.): unit depends on the magnitude

The "unit" is the smallest step in the rounding. For 1 d.p., the unit is 0.1. For nearest integer, the unit is 1.

2. Upper and Lower Bounds

When a value is rounded, the true value could be anywhere within a range. The boundaries of this range are the upper and lower bounds.

Upper Bound (UB) = rounded value + ½ × unit
Lower Bound (LB) = rounded value − ½ × unit

Example: A length is measured as 7 cm to the nearest cm.
Unit = 1 cm → UB = 7 + 0.5 = 7.5 cm
LB = 7 − 0.5 = 6.5 cm

Example: A mass is 3.4 kg to 1 d.p.
Unit = 0.1 kg → UB = 3.4 + 0.05 = 3.45 kg
LB = 3.4 − 0.05 = 3.35 kg

3. Error Interval Notation

The error interval is written using inequality notation. Note that the upper bound is not included (strict inequality), because at exactly the upper bound we would round up to the next value.

LB ≤ x < UB

Length = 7 cm (nearest cm): 6.5 ≤ x < 7.5
Mass = 3.4 kg (1 d.p.): 3.35 ≤ x < 3.45

The lower bound uses ≤ (the value could be exactly the lower bound) but the upper bound uses < (the value is strictly less than the upper bound).

4. Bounds of Products and Quotients

When combining measurements, use the appropriate bounds to find maximum or minimum results.

Maximum of (a × b) = UB(a) × UB(b)
Minimum of (a × b) = LB(a) × LB(b)
Maximum of (a ÷ b) = UB(a) ÷ LB(b)
Minimum of (a ÷ b) = LB(a) ÷ UB(b)

Example: a = 5 (nearest whole), b = 3 (nearest whole).
UB(a)=5.5, LB(a)=4.5, UB(b)=3.5, LB(b)=2.5
Max of a × b = 5.5 × 3.5 = 19.25
Min of a ÷ b = 4.5 ÷ 3.5 = 1.29 (2 d.p.)

Rule for max/min of a quotient:
Maximum → big numerator ÷ small denominator (UB ÷ LB)
Minimum → small numerator ÷ big denominator (LB ÷ UB)

5. Bounds and Significant Figures

1 s.f.: 500 (1 s.f.) → unit = 100 → UB = 550, LB = 450
2 s.f.: 4800 (2 s.f.) → unit = 100 → UB = 4850, LB = 4750
3 s.f.: 6.45 (3 s.f.) → unit = 0.01 → UB = 6.455, LB = 6.445

For significant figures, identify the place value of the last significant figure — that is your unit for finding bounds.

Example 1 — Bounds of 6 (nearest whole)

Unit = 1. UB = 6 + 0.5 = 6.5. LB = 6 − 0.5 = 5.5

Error interval: 5.5 ≤ x < 6.5

Example 2 — Bounds of 4.8 (1 d.p.)

Unit = 0.1. UB = 4.8 + 0.05 = 4.85. LB = 4.8 − 0.05 = 4.75

Example 3 — Max area of rectangle

Length = 8 cm (nearest cm), Width = 5 cm (nearest cm)

UB(L) = 8.5, UB(W) = 5.5

Max area = 8.5 × 5.5 = 46.75 cm²

Example 4 — Min of quotient

a = 10 (nearest whole), b = 3 (nearest whole)

LB(a) = 9.5, UB(b) = 3.5

Min of a/b = 9.5 ÷ 3.5 = 2.71 (2 d.p.)

Example 5 — Upper bound of 0.3 (1 d.p.)

Unit = 0.1. UB = 0.3 + 0.05 = 0.35

Example 6 — Bounds with significant figures

800 rounded to 1 s.f.: unit = 100. UB = 850, LB = 750.

Error interval: 750 ≤ x < 850

Bounds Calculator

Enter a measured value and its rounding unit to find upper bound, lower bound, and error interval.

Value: Rounding unit:

Results will appear here.

Product/Quotient Bounds

Find the max/min of a × b and a ÷ b given rounded values.

a = unit a: b = unit b:

Results will appear here.

Exercise 1 — Finding Upper Bounds

Find the upper bound of each rounded measurement.

1. 3 cm, measured to the nearest cm. Upper bound?

2. 7 kg, measured to the nearest kg. Upper bound?

3. 12 m, measured to the nearest m. Upper bound?

4. 45 seconds, measured to the nearest second. Upper bound?

5. 100 ml, measured to the nearest ml. Upper bound?

6. 0.6 cm, measured to 1 d.p. Upper bound?

7. 2.3 kg, measured to 1 d.p. Upper bound?

8. 8.0 m, measured to 1 d.p. Upper bound?

9. 14 litres, measured to nearest litre. Upper bound?

10. 99 kg, measured to nearest kg. Upper bound?

Exercise 2 — Finding Lower Bounds

Find the lower bound of each rounded measurement.

1. 3 cm, nearest cm. Lower bound?

2. 7 kg, nearest kg. Lower bound?

3. 12 m, nearest m. Lower bound?

4. 45 s, nearest second. Lower bound?

5. 100 ml, nearest ml. Lower bound?

6. 0.6 cm (1 d.p.). Lower bound?

7. 2.3 kg (1 d.p.). Lower bound?

8. 8.0 m (1 d.p.). Lower bound?

9. 14 litres (nearest litre). Lower bound?

10. 99 kg (nearest kg). Lower bound?

Exercise 3 — Maximum of a Product

Both values are measured to the nearest whole unit. Find the maximum possible value of a × b. Give answer to 2 d.p.

1. a = 3, b = 4 (both nearest whole). Max of a × b?

2. a = 4, b = 5 (both nearest whole). Max of a × b?

3. a = 5, b = 6 (both nearest whole). Max of a × b?

4. a = 6, b = 7 (both nearest whole). Max of a × b?

5. a = 7, b = 8 (both nearest whole). Max of a × b?

6. a = 2, b = 3 (both nearest whole). Max of a × b?

7. a = 1, b = 2 (both nearest whole). Max of a × b?

8. a = 8, b = 9 (both nearest whole). Max of a × b?

9. a = 9, b = 10 (both nearest whole). Max of a × b?

10. a = 10, b = 11 (both nearest whole). Max of a × b?

Exercise 4 — Minimum of a Quotient

Both values nearest whole. Find minimum possible a ÷ b. Give to 2 d.p.

1. a = 3, b = 4. Min of a ÷ b?

2. a = 4, b = 5. Min of a ÷ b?

3. a = 5, b = 6. Min of a ÷ b?

4. a = 6, b = 7. Min of a ÷ b?

5. a = 3, b = 6. Min of a ÷ b?

6. a = 3, b = 7. Min of a ÷ b?

7. a = 2, b = 6. Min of a ÷ b?

8. a = 4, b = 7. Min of a ÷ b? (to 2 d.p.)

9. a = 5, b = 7. Min of a ÷ b? (to 2 d.p.)

10. a = 2, b = 8. Min of a ÷ b?

Exercise 5 — Upper Bounds Mixed

Find the upper bound for each measurement (unit given in brackets).

1. 5 m (nearest m). UB?

2. 10 kg (nearest kg). UB?

3. 25 s (nearest s). UB?

4. 0.3 cm (1 d.p.). UB?

5. 1.0 kg (1 d.p.). UB?

6. 4.1 cm (1 d.p.). UB?

7. 8.7 m (1 d.p.). UB?

8. 19 litres (nearest litre). UB?

9. 0.04 g (2 d.p.). UB?

10. 6 km (nearest km). UB?

Practice — 20 Questions

Mixed practice on bounds. All values measured to nearest whole unit unless stated.

1. Upper bound of 3 (nearest whole).

2. Upper bound of 7 (nearest whole).

3. Upper bound of 12 (nearest whole).

4. Upper bound of 45 (nearest whole).

5. Upper bound of 100 (nearest whole).

6. Upper bound of 0.6 (1 d.p.).

7. Upper bound of 2.3 (1 d.p.).

8. Upper bound of 8.0 (1 d.p.).

9. Upper bound of 14 (nearest whole).

10. Upper bound of 99 (nearest whole).

11. Lower bound of 3 (nearest whole).

12. Lower bound of 7 (nearest whole).

13. Lower bound of 12 (nearest whole).

14. Lower bound of 45 (nearest whole).

15. Lower bound of 100 (nearest whole).

16. Lower bound of 0.6 (1 d.p.).

17. Lower bound of 2.3 (1 d.p.).

18. Lower bound of 8.0 (1 d.p.).

19. Lower bound of 14 (nearest whole).

20. Lower bound of 99 (nearest whole).

Challenge — 8 Questions

Find upper bounds for these measurements.

1. 5 m (nearest m). Upper bound?

2. 10 kg (nearest kg). Upper bound?

3. 25 s (nearest s). Upper bound?

4. 0.3 cm (1 d.p.). Upper bound?

5. 1.0 kg (1 d.p.). Upper bound?

6. 4.1 cm (1 d.p.). Upper bound?

7. 8.7 m (1 d.p.). Upper bound?

8. 19 litres (nearest litre). Upper bound?