GCSE · Maths · Edexcel · Spec 1MA1 · Foundation+Higher

Limits of accuracy and bounds

Your ruler says the pencil is 8 cm. But that's only to the nearest centimetre, so how long could it really be? Drag the ends and find out.

Limits of accuracy

How long could the pencil really be?

A pencil is 8 cm long, correct to the nearest centimetre. Drag the two ends to cover every length x it could really be, then decide whether each end is included.

Lower endUpper end

Include the lower end (open circle)Include the upper end (open circle)

Exam line: Write an error interval as lower bound ≤ x < upper bound.

Why not 12.4?

Where does the upper end really sit?

A ribbon is measured as 12 cm, correct to the nearest centimetre.

What is the upper bound of its length? Pick the answer closest to what you think right now.

12.4 cm. Anything longer than that would round up to 13 cm.12.49 cm. That's the last length that still rounds down to 12.12.5 cm, even though 12.5 itself would round up to 13.13 cm. It's the next whole centimetre up.

How sure are you?

I'm sureI think soI'm guessing

12.4 cm. Anything longer than that would round up to 13 cm.

This is a common misconception

Try 12.45 cm. It's longer than 12.4 cm, and it still rounds to 12. So do 12.48 and 12.499. A length isn't stuck at one decimal place: it can take any value at all, so the true length can creep right up towards 12.5 cm. Stopping at 12.4 leaves a whole chunk of possible lengths out.

12.49 cm. That's the last length that still rounds down to 12.

This is a common misconception

There is no 'last' length that rounds down. After 12.49 comes 12.499, then 12.4999, and you can keep adding 9s for ever. Every one of them rounds to 12. So instead of chasing a last value that doesn't exist, we use the number they're all heading towards: 12.5 cm.

12.5 cm, even though 12.5 itself would round up to 13.

This matches the evidence

Yes. 12.5 isn't in the interval (it rounds up to 13), which is why the interval is written 11.5 ≤ x < 12.5. But 12.4999…, with the 9s going on for ever, is exactly 12.5. Call it x. Then 10x = 124.999…, so 10x − x = 124.999… − 12.4999… = 112.5. That means 9x = 112.5, so x = 12.5. Same number, so 12.5 is the upper bound you use.

13 cm. It's the next whole centimetre up.

This is a common misconception

13 cm is the next rounded value, not the edge of this interval. A ribbon of 12.8 cm would already round to 13, not 12. The edge sits halfway between 12 and 13: 12.5 cm.

The upper bound of 12 cm (to the nearest centimetre) is 12.5 cm. It isn't in the interval, but every length up to it is, and 12.4999… recurring is exactly 12.5.

Improving accuracy

One reading, three syringes

A dose of medicine reads 3.0 ml on three syringes, each read to the nearest mark. One is marked every 1 ml, one every 0.2 ml and one every 0.1 ml. Place the lower (lo) and upper (hi) bound of the dose for each.

1 lo — Marked every 1 ml: lower bound1 hi — Marked every 1 ml: upper bound0.2 lo — Marked every 0.2 ml: lower bound0.2 hi — Marked every 0.2 ml: upper bound0.1 lo — Marked every 0.1 ml: lower bound0.1 hi — Marked every 0.1 ml: upper bound

Spot the slip

Where does this density answer go wrong?

A block has mass 3.42 kg, correct to 2 decimal places, and volume 5 cm³, correct to 1 significant figure. Density = mass ÷ volume. Find the upper bound of the density.

A student's answer — which line goes wrong?

Mass: 3.42 to 2 d.p. is to the nearest 0.01, so 3.415 ≤ m < 3.425.
Volume: 5 to 1 s.f. is to the nearest 1, so 4.5 ≤ V < 5.5.
For the biggest density, use the biggest mass: 3.425 kg.
Divide by the biggest volume: 3.425 ÷ 5.5
= 0.6227…
Upper bound of the density = 0.623 kg/cm³ (3 s.f.)

Combined calculations

Push every part the same way

a = (v − u) ÷ t. v = 34.4 to 3 significant figures, u = 24 to 2 significant figures and t = 12.5 to 1 decimal place. Find the upper bound of a, to 3 significant figures.

Degrees of accuracy: v = 34.4 to 3 s.f. is to the nearest 0.1, u = 24 to 2 s.f. is to the nearest 1, and t = 12.5 to 1 d.p. is to the nearest 0.1.With significant figures, the degree of accuracy is the place value of the last significant figure.
Bounds: 34.35 ≤ v < 34.45, 23.5 ≤ u < 24.5 and 12.45 ≤ t < 12.55.
To make a as big as possible, make the top, v − u, as big as possible and the bottom, t, as small as possible.
missing step

Which line is step 4?

Biggest top: 34.45 − 23.5 = 10.95Biggest top: 34.45 − 24.5 = 9.95Biggest top: 34.35 − 23.5 = 10.85

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Find the two ends of that range, and you can find the biggest and smallest answer any calculation could really give.

What you need to know

Any number given to a degree of accuracy has an error interval, written lower bound ≤ x < upper bound. The lower bound is included; the upper bound is not.
The lower bound is the smallest value the number could have been before rounding. The upper bound is the smallest value that would round up to the next rounded value. Together they are the number's limits of accuracy.
The bounds sit halfway to the neighbouring rounded values, so the degree of accuracy decides them. For significant figures, the degree of accuracy is the place value of the last significant figure.
The upper bound is used in calculations even though it isn't in the interval, because a recurring decimal such as 12.4999… is equal to 12.5.
For a calculation, give each value the bound that pushes the answer the way you want. For differences and quotients, the second value takes the opposite bound to the first.
Measuring to a finer degree of accuracy narrows the interval, which makes the measurement more accurate.

The big picture

A rounded number stands for a whole interval of values. Its bounds, the limits of accuracy, are half the degree of accuracy either side of it, and the error interval is written lower bound ≤ x < upper bound: the lower bound is included and the upper bound is not, although the upper bound is still the value you calculate with. To find the bounds of a calculation, give each value whichever bound pushes the answer the way you want. That means biggest with biggest for sums and products, but for differences and quotients the second value takes the opposite bound.

Key points

18 to the nearest whole number: 7.5 ≤ x < 8.5. 4.6 to 1 decimal place: 4.55 ≤ x < 4.65. 70 to the nearest 10: 65 ≤ x < 75. 45 to the nearest 5: 42.5 ≤ x < 47.5.

2Significant figures: 9 to 1 s.f. gives 8.5 ≤ x < 9.5, and 56 to 2 s.f. gives 55.5 ≤ x < 56.5.

3Sums and products: upper bound = upper with upper; lower bound = lower with lower.

4Differences: upper bound = upper bound of the larger − lower bound of the smaller; lower bound = lower bound of the larger − upper bound of the smaller.

5Quotients: upper bound = upper bound of the dividend ÷ lower bound of the divisor; lower bound = lower bound of the dividend ÷ upper bound of the divisor.

6In context, first decide which extreme answers the question, then make the answer sensible. A 160 ml jug (nearest 20 ml) and 40 ml cups (nearest 5 ml): the fewest full cups is 150 ÷ 42.5 = 3.52…, so 3 full cups. Round down, because the fourth cup isn't full.

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

A bookcase can hold 100 kg, to the nearest kilogram. It already holds 95 kg of books, to the nearest kilogram. Three more books are added, weighing 1.8 kg and 1.5 kg (both to 1 decimal place) and 2 kg (to 1 significant figure). Could the bookcase end up over its capacity?

⚠ Watch out

Using the biggest value for everything. Biggest with biggest gives the upper bound of a sum or a product, but for a quotient you divide by the LOWER bound of the divisor, and for a difference you subtract the LOWER bound of the smaller value.

🧠

Memory hook

Share a pizza between fewer people and everyone gets a bigger slice. Dividing by a smaller number gives a bigger answer, so the biggest quotient uses the smallest divisor, and the smallest quotient uses the biggest divisor.

✓

Check yourself

k = 18 and w = 6, both to the nearest whole number. Which bounds give the upper bound of k ÷ w, and why? Work it out to 3 significant figures.

Flashcards

(12)

What is an error interval?

The range of values a rounded number could have had before it was rounded, written lower bound ≤ x < upper bound.

What are the lower and upper bounds of a rounded number?

Lower bound: the smallest value it could have been before rounding (included, ≤). Upper bound: the smallest value that would round up to the next rounded value (not included, <).

How do you find the bounds of a rounded value?

Add and subtract half the degree of accuracy. 70 to the nearest 10 has bounds 65 and 75.

What is the degree of accuracy of a number rounded to significant figures?

The place value of its last significant figure. 56 to 2 s.f. is to the nearest 1, so 55.5 ≤ x < 56.5.

Do 30 to the nearest whole number and 30 to the nearest 10 have the same bounds?

No. To the nearest whole number: 29.5 and 30.5. To the nearest 10: 25 and 35.

Why is 12.5 cm the upper bound of 12 cm (to the nearest cm), not 12.4 cm?

Length can take any value, so 12.45, 12.49, 12.499… all round to 12. They head towards 12.5, and 12.4999… recurring equals 12.5.

Bounds of a + b or a × b?

Upper bound: upper bound of a with upper bound of b. Lower bound: lower with lower.

Bounds of a − b?

Upper bound: upper bound of a − lower bound of b. Lower bound: lower bound of a − upper bound of b.

Bounds of a ÷ b?

Upper bound: upper bound of a ÷ lower bound of b. Lower bound: lower bound of a ÷ upper bound of b.

How do you find a bound of a combined calculation such as (v − u) ÷ t?

Give each part the bound that pushes the whole answer the way you want. For the upper bound: the biggest top and the smallest bottom.

How can a measurement be made more accurate?

Measure to a finer degree of accuracy. The bounds move closer together, so the interval is narrower.

In a context question, how do you choose which bounds to use?

Work out which extreme the question is about (the fewest, the most, whether something might happen), pick the bounds that give it, then make the answer sensible, e.g. round down for full containers.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

Limits of accuracy and bounds

How long could the pencil really be?

Where does the upper end really sit?

One reading, three syringes

Which end does each value take?

Where does this density answer go wrong?

Push every part the same way

More Edexcel GCSE Maths topics