Upper and lower bounds GCSE Questions and Worked Answers

Q: Are all bounds questions Higher?

Basic error intervals are not. This guide labels calculations using bounds as Higher.

Q: Why does the lower sign differ from the upper sign?

For positive numbers rounded in the usual way, the lower halfway value rounds to the stated number; the upper halfway value rounds to the next one.

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Start with the meaning

What you need to know about upper and lower bounds

A ruler reading reported as 7.5 cm to the nearest tenth does not tell you the exact length. For example, 7.48 cm and 7.54 cm both round to 7.5 cm. The possible lengths begin halfway towards 7.4 and stop halfway towards 7.6.

See the idea first

The endpoints come from halfway marks

The step between tenths is 0.1, so half a step is 0.05. Subtract and add 0.05 to get 7.45 and 7.55. If x means the true length, 7.45 ≤ x < 7.55 says ‘at least 7.45 but less than 7.55’. This is an error interval.

A value at 7.45 rounds up to 7.5; a value at 7.55 rounds up to 7.6, so the upper endpoint is excluded.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Find an error interval

What the problem asks: A mass is 240 g to the nearest 10 g. What could it really be?

How to solve it: Half the rounding unit is 5 g. The interval is 235 ≤ m < 245 g; the upper endpoint would round to 250 g.

Bound a product

What the problem asks: A rectangle has length 8 cm and width 5 cm, each to the nearest centimetre. Find area bounds.

How to solve it: Both dimensions are positive. Smaller length and smaller width give the lower limit: 7.5 × 4.5. Both upper endpoints give the upper limit: 8.5 × 5.5. This calculation with bounds is Higher-tier work.

Bound a quotient

What the problem asks: A speed is distance ÷ time. Which measurements give its upper bound?

How to solve it: Use the largest distance and the smallest positive time: more distance in less time is faster. Using two upper bounds would not maximise the speed.

A reliable routine

Choose endpoints for a calculation

For Higher-tier calculations with positive, independently varying measurements, each operation tells you which endpoints increase or decrease the answer. This is why there is no single ‘use all upper bounds’ rule.

Write every measurement's interval, using half the rounding unit.
For sums and positive products, pair lower with lower or upper with upper.
For A − B, the lower limit is lower A minus upper B; reverse for the upper limit.
For A ÷ B with B positive, lower A ÷ upper B is the lower limit; upper A ÷ lower B is the upper limit.

Check: An excluded endpoint can still be a bound. Do not call it a largest possible measurement. If a denominator can be zero, the positive-quotient rule is not valid.

Fully worked

Upper and lower bounds GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Nearest whole unit

2 marks

Question

A length L is 18 cm to the nearest centimetre. Write its error interval.

Half a centimetre lies on each side of 18.

18-0.5=17.5

18+0.5=18.5

17.5\le L<18.5\text{ cm}

The upper endpoint is excluded because 18.5 rounds to 19.

Example 2

Significant-figure precision

2 marks

Question

A mass is 0.046 kg to two significant figures. Find its bounds.

The last retained digit is in the thousandths place. Half of 0.001 is 0.0005.

0.0455\le m<0.0465\text{ kg}

Tip: two significant figures does not always mean hundredths.

Example 3

Perimeter limits

Higher only3 marks

Question

A rectangle's dimensions are 12 cm and 7 cm, both to the nearest centimetre. Find bounds for its perimeter.

11.5\le l<12.5,\qquad6.5\le w<7.5

Both terms increase the perimeter.

P_{\rm lower}=2(11.5+6.5)=36

P_{\rm upper}=2(12.5+7.5)=40

Hence $36\le P<40$ cm.

Example 4

Area limits

Higher only3 marks

Question

A rectangle is 8 cm by 5 cm, both to the nearest centimetre. Find its area bounds.

7.5\le l<8.5,\qquad4.5\le w<5.5

A_{\rm lower}=7.5\times4.5=33.75

A_{\rm upper}=8.5\times5.5=46.75

Thus $33.75\le A<46.75$ cm². Both lengths are positive, so increasing either increases area.

Example 5

Difference limits

Higher only3 marks

Question

Two rods measure 20 cm and 12 cm, each to the nearest centimetre. Find bounds for the longer rod minus the shorter.

To minimise the gap, use the shortest long rod and the longest short rod.

d_{\rm lower}=19.5-12.5=7

To maximise it, reverse those choices.

d_{\rm upper}=20.5-11.5=9

The difference satisfies $7<d<9$ cm; neither endpoint is attained.

Example 6

A rate and its accuracy

Higher only4 marks

Question

A journey is 100 m to the nearest metre and takes 8.0 s to the nearest tenth. Find bounds for its average speed. A student calculates 100 ÷ 8 and rounds to 13 m/s. Is that whole-number speed guaranteed?

99.5\le d<100.5,\quad7.95\le t<8.05

v_{\rm lower}=\frac{99.5}{8.05}=12.3602\ldots

v_{\rm upper}=\frac{100.5}{7.95}=12.6415\ldots

The possible speeds cross the 12.5 rounding boundary. Some round to 12 and others to 13, so 13 m/s is not guaranteed. Keep the endpoint quotients before rounding.

10 original questions · total 23 marks

Upper and lower bounds GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 28 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

1

Nearest ten

2 marks

x is 370 to the nearest ten. Write an error interval.

Show worked answer

Half of 10 is 5.

365\le x<375

2

Two decimal places

2 marks

y is 3.62 to two decimal places. Write its error interval.

Show worked answer

The unit is 0.01 and half is 0.005.

3.615\le y<3.625

3

Truncated value

2 marks

A positive number x is truncated to one decimal place, giving 4.7. Write its interval.

Show worked answer

Truncation cuts off later digits instead of choosing a nearest value.

4.7\le x<4.8

Do not use half a tenth.

4

An endpoint

1 mark

Why is 6.25 excluded when a number rounds to 6.2 to one decimal place?

Show worked answer

At 6.25 the ordinary positive-number tie rule rounds up to 6.3, not 6.2.

5

Sum

Higher only3 marks

a = 6 and b = 9, both to the nearest whole number. Find bounds for a + b.

Show worked answer

5.5\le a<6.5,\quad8.5\le b<9.5

5.5+8.5=14

6.5+9.5=16

So $14\le a+b<16$ .

6

Product

Higher only3 marks

A rectangle has sides 4.0 cm and 3.0 cm, each to one decimal place. Find its area bounds.

Show worked answer

The half-unit is 0.05 cm.

A_{\rm lower}=3.95\times2.95=11.6525

A_{\rm upper}=4.05\times3.05=12.3525

So $11.6525\le A<12.3525$ cm².

7

Quotient

Higher only3 marks

a = 30 and b = 5, both to the nearest integer. Find the lower and upper bounds of a ÷ b as fractions.

Show worked answer

29.5\le a<30.5,\quad4.5\le b<5.5

\text{Lower}=\frac{29.5}{5.5}=\frac{59}{11}

\text{Upper}=\frac{30.5}{4.5}=\frac{61}{9}

A larger denominator reduces a positive quotient.

8

Difference

Higher only3 marks

p = 15 and q = 8, both to the nearest whole number. Find bounds for p − q.

Show worked answer

d_{\rm lower}=14.5-8.5=6

d_{\rm upper}=15.5-7.5=8

Therefore $6<p-q<8$ .

9

Guaranteed rounding

Higher only2 marks

A calculated value lies between 5.21 and 5.24. What is its guaranteed value to one decimal place?

Show worked answer

Every possible value is below 5.25 and above 5.15, so all round to 5.2. The answer to one decimal place is guaranteed.

10

Do not round a bound inward

Higher only2 marks

An upper limit is exactly 7 ÷ 3. Can 2.33 be used as a safe upper bound? Explain.

Show worked answer

\frac73=2.3333\ldots>2.33

Some allowed values could exceed 2.33. Keep $7/3$ , or use an explicitly outward bound such as 2.34.

Examiner-style feedback

Common upper and lower bounds mistakes

Always pairing upper endpoints

For a positive quotient, the upper numerator goes with the lower denominator.

Confusing decimal places with a unit

Find the value of the final retained place, then halve it.

Calling a limit a maximum

The usual upper endpoint of a rounded measurement is excluded; values may approach it without reaching it.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Find each input interval.
Track whether each input increases or decreases the result.
Keep exact endpoint calculations.
Check the whole output interval before claiming a rounded answer.

Quick answers

Upper and lower bounds FAQ

Are all bounds questions Higher?

Basic error intervals are not. This guide labels calculations using bounds as Higher.

Why does the lower sign differ from the upper sign?

For positive numbers rounded in the usual way, the lower halfway value rounds to the stated number; the upper halfway value rounds to the next one.

Build connected skills

What to revise next

Prerequisite

Rounding

Understand the retained place and half-unit.

Revise rounding

Rates

Speed, distance and time

Understand the rate before bounding it.

Revise speed, distance and time

Content standards

Curriculum and rights review

N15 error intervals across tiers; N16 limits of calculations labelled Higher. Positive measurements and ordinary nearest rounding unless specified. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content