N15–N16NumberFoundation & HigherGrades 4–7

Error Intervals & Bounds

Q: Why is the upper bound not included?

A value exactly on the upper bound would round up to the next value, so it could not have been the original.

Q: Do I always halve the unit?

Yes for rounding. For truncation the interval runs from the value up to one whole unit more.

Q: What does "appropriate degree of accuracy" mean?

The most precise rounding on which the upper and lower bounds agree.

A recipe says 250 g of flour, measured to the nearest 10 g. The flour could weigh anything from 245 g up to just under 255 g. Writing that range as an $\textbf{error interval}$ , and working out how rounding affects the result of a calculation, is what this lesson is about.

45 min AQA GCSE Maths

The big picture

Rounding replaces a whole range of possible values with a single number. The $\textbf{lower bound}$ and $\textbf{upper bound}$ mark the edges of that range: half a unit below and half a unit above the rounded value. The lower bound is included; the upper bound is not, because it would round up.

At Higher tier, AQA asks for bounds of calculations — the greatest possible speed, the smallest possible area — and for answers "to an appropriate degree of accuracy". The trick is to think about each input separately: which bound makes the answer as large, or as small, as possible?

What you'll learn

Write error intervals for values rounded to a given accuracy
Write error intervals for truncated values
Find upper and lower bounds of sums, differences, products and quotients (Higher)
Give a result to an appropriate degree of accuracy using bounds (Higher)

Error intervals for rounded values

Find the unit the value was rounded to, halve it, and go that far either side.

A length of 8.4 cm to 1 decimal place was rounded to the nearest 0.1 cm, so half a unit is 0.05 cm: $8.35 \le l < 8.45$ .

A crowd of 3000 to the nearest 100 people: $2950 \le n < 3050$ .

Example 1The mass of a parcel is 2.6 kg, correct to 1 decimal place. Write down the error interval for the mass

m

1Rounded to the nearest 0.1 kg; half of that is 0.05 kg.

2Lower bound

2.55

, upper bound

2.65

Answer

2.55 \le m < 2.65

Your turnSame method, new numbers — try it before moving on.

A length is 7.4 cm, correct to 1 decimal place. Write down its lower bound.

Your answer

Show worked solution

Solution

1Half of 0.1 is 0.05.
2 $7.4 - 0.05 = 7.35$ cm.

Answer: $7.35$ cm

Tip — Write the interval with $\le$ on the left and $<$ on the right. Using $\le$ at both ends is the most common way to lose this mark.

Truncation

Truncating chops off digits without rounding. A calculator display of 6.27 truncated to 1 decimal place gives 6.2.

So if $x = 6.2$ after truncation to 1 d.p., the true value could be anything from 6.2 up to, but not including, 6.3: $6.2 \le x < 6.3$ .

Why this works

Rounding spreads the interval evenly around the value; truncation puts it entirely above. That difference is why AQA always tells you which one was used.

Bounds of calculations

Adding or multiplying: use both upper bounds for the maximum and both lower bounds for the minimum.

Subtracting $a - b$ : maximum uses upper $a$ and lower $b$ .

Dividing $\dfrac{a}{b}$ : maximum uses upper $a$ and $\textbf{lower}$ $b$ ; minimum uses lower $a$ and upper $b$ .

Example 1A block has mass 250 g to the nearest 10 g and volume 42 cm³ to the nearest cm³. Find the upper and lower bounds of its density.

1Mass:

245 \le m < 255

. Volume:

41.5 \le V < 42.5

2Upper bound:

\dfrac{255}{41.5} = 6.14\dots

3Lower bound:

\dfrac{245}{42.5} = 5.76\dots

AnswerLower bound 5.76 g/cm³, upper bound 6.14 g/cm³

Your turnSame method, new numbers — try it before moving on.

A rectangle measures 8 cm by 5 cm, both to the nearest cm. Work out the upper bound of its area, in cm².

Your answer

Show worked solution

Solution

1Upper bounds: 8.5 cm and 5.5 cm.
2 $8.5 \times 5.5 = 46.75$ cm².

Answer: $46.75$ cm²

Tip — Dividing by a smaller number gives a bigger answer. That is why the lower bound of the denominator gives the upper bound of the result.

Appropriate accuracy

Round the upper and lower bounds to the same accuracy. The most accurate rounding where they agree is the appropriate answer.

For the density above, 6.14 and 5.76 do not agree to 1 decimal place (6.1 and 5.8), but both round to 6 to 1 significant figure.

Example 1Using the density bounds, give the density to an appropriate degree of accuracy, with a reason.

1To 1 d.p.: 5.8 and 6.1 — different.

2To 1 s.f.: 6 and 6 — the same.

Answer6 g/cm³, because both bounds round to 6 to 1 significant figure.

Think like an examiner

Show the bounds of each measurement before calculating.
State which bounds you are combining for the maximum or minimum.
Justify an appropriate degree of accuracy by showing both bounds rounded.

Remember these

\text{value} \pm \tfrac12 \text{ unit}

Bounds of a rounded value.

\max\tfrac{a}{b} = \tfrac{a_{\text{UB}}}{b_{\text{LB}}}

Upper bound of a division.

\max(a - b) = a_{\text{UB}} - b_{\text{LB}}

Upper bound of a subtraction.

Watch out for these

Writing

8.35 \le l \le 8.45

The upper bound is excluded:

8.35 \le l < 8.45

Using the upper bound of the denominator for the maximum of a division.

Divide by the lower bound.

Treating a truncated value like a rounded one.

Truncation gives value

\le x <

value

+ 1

unit.

Rounding bounds to different accuracies.

Compare them at the same accuracy.

Stretch yourself

A sprinter runs 100 m, measured to the nearest metre, in 12.3 seconds, measured to the nearest 0.1 s. Calculate the upper and lower bounds of her average speed, and give the speed to a suitable degree of accuracy.

Hint — Maximum speed: longest distance in the shortest time.

Reveal solution

SolutionDistance:

99.5 \le d < 100.5

. Time:

12.25 \le t < 12.35

. Upper bound of speed

= \dfrac{100.5}{12.25} = 8.204

m/s. Lower bound

= \dfrac{99.5}{12.35} = 8.057

m/s. To 2 s.f. these give 8.2 and 8.1, which differ; to 1 s.f. both give 8. So her speed is 8 m/s to 1 significant figure.

\blacksquare

Questions students ask

Why is the upper bound not included?

A value exactly on the upper bound would round up to the next value, so it could not have been the original.

Do I always halve the unit?

Yes for rounding. For truncation the interval runs from the value up to one whole unit more.

What does "appropriate degree of accuracy" mean?

The most precise rounding on which the upper and lower bounds agree.

Key takeaways

Rounded values have bounds half a unit either side.
Error interval: lower bound $\le x <$ upper bound.
Truncated values: value $\le x <$ value $+$ 1 unit.
For quotients, divide an upper bound by a lower bound to maximise.
Choose accuracy where both bounds agree.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 42 marks

A time is 12.4 seconds, correct to 1 decimal place. Write down the error interval for the time $t$ .[2]
Show mark scheme
- B1 $12.35$ and $12.45$ seen
- B1 $12.35 \le t < 12.45$
Answer: $12.35 \le t < 12.45$

Question 2Grade 63 marks

A runner covers 100 m, correct to the nearest metre, in 12.3 seconds, correct to 1 decimal place.

Work out the upper bound of the runner's average speed. Give your answer to 3 significant figures.[3]
Show mark scheme
- B1Bounds 100.5 m and 12.25 s seen
- M1 $100.5 \div 12.25$
- A1 $8.20$ m/s
Answer: $8.20$ m/s

Question 3Grade 73 marks

$a = 5.3$ and $b = 2.1$ , both correct to 1 decimal place.

Work out the lower bound of $a - b$ .[3]
Show mark scheme
- B1Bounds 5.25 and 2.15 used
- M1Lower bound of $a$ minus upper bound of $b$
- A1 $3.1$
Answer: $3.1$

Examiner note

To make a difference as small as possible, subtract the largest possible $b$ from the smallest possible $a$ .

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

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Answer a few questions, then review anything you missed.

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