GCSE · Maths · AQA · Spec 8300 · Foundation

Rounding to decimal places and significant figures

A parcel label says 2.6 kg. Could it really weigh 2.64 kg? What about 2.65 kg? One zoomed-in piece of number line answers both.

Rounding · the half-way points

Which masses round to 2.6 kg?

A parcel's mass m is recorded as 2.6 kg, rounded to 1 decimal place. At 1 decimal place the only values near it are 2.5, 2.6 and 2.7, and every mass is rounded to whichever of these it is nearest. Drag the two ends so the bar covers every mass that rounds to 2.6. At each end, ask of the value under the slider: is it nearer 2.6, or nearer the neighbour? Then decide whether each end is included.

Lower endUpper end

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

Significant figures

Two numbers, two significant figures

Round 45 382 and 0.004 518 each to 2 significant figures.

Which is closest to what you would do?

Keep the first two digits and drop the rest: 45 and 0.0045.Every digit counts, zeros too — so the first two figures of 0.004 518 are 0 and 0: 45 000 and 0.00.Two significant figures means two decimal places: 45 382.00 and 0.00.Start at the first non-zero digit, keep two figures, then fill any places before the decimal point with zeros: 45 000 and 0.0045.

How sure are you?

I'm sureI think soI'm guessing

Keep the first two digits and drop the rest: 45 and 0.0045.

This is a common misconception

For 0.004 518 that happens to work, because the zeros in front already hold the 4 and 5 in place. But 45 is about a thousand times smaller than 45 382. At 2 significant figures the neighbours of 45 382 are 45 000 and 46 000, and it is nearer 45 000. The zeros after 45 are place-holders: they keep the 4 in the ten-thousands column, so they cannot be dropped.

Every digit counts, zeros too — so the first two figures of 0.004 518 are 0 and 0: 45 000 and 0.00.

This is a common misconception

The zeros in front of the 4 only show where the 4 sits; they are not significant figures. And 0.00 is zero — nowhere near 0.004 518. Start counting at the first non-zero digit, the 4: the two figures are 4 and 5, so the answer is 0.0045.

Two significant figures means two decimal places: 45 382.00 and 0.00.

This is a common misconception

Decimal places count from the decimal point; significant figures count from the first non-zero digit. 45 382.00 has not been rounded at all — it still has five significant figures in front of the point — and 0.00 has lost the value entirely. Keep the 4 and the 5 in each number: 45 000 and 0.0045.

Start at the first non-zero digit, keep two figures, then fill any places before the decimal point with zeros: 45 000 and 0.0045.

This matches the evidence

Yes. In 45 382 the next digit after 4 and 5 is 3, so it rounds down to 45 000, and the three zeros keep it at about forty-five thousand. In 0.004 518 the next digit after 4 and 5 is 1, so it rounds down to 0.0045 — the leading zeros stay because they place the 4 in the thousandths column.

A rounded number must be about the same size as the original. Count significant figures from the first non-zero digit, round using the next digit, then fill the gap back to the decimal point with zeros.

Rounding in context

Up, down or nearest?

Pick a situation, then choose the rounding it needs.

Still to sort

Round up (0)

A part-group still needs a whole one.

Where the line is: Ask what happens to the leftover. If it still has to be carried, covered or served, one more whole one is needed — round up.

Round down (0)

Only complete items count.

Where the line is: If the leftover is too little to make or buy one more whole item, it is wasted — round down.

Round to the nearest (0)

Nothing forces a direction — choose a sensible accuracy.

Where the line is: When the answer is an amount of money or a measurement rather than a count of whole things, neither direction is forced: round to the nearest at the accuracy the context uses, such as the nearest penny.

6 of 6 still to sort.

Each calculation gives a decimal. The situation decides what to do with it.

Watch out: Several of these round the opposite way to 'nearest' — the context wins over the half-way rule.

Rounding part-way through

Find where this answer goes wrong

A square has an area of 50 cm². Work out its perimeter, giving your answer to 1 decimal place.

A student's answer — which line goes wrong?

Side length = √50
Side length = 7.1 cm
Perimeter = 4 × 7.1
Perimeter = 28.4 cm

A length L is shown as 6.4 cm

6.4 from rounding to 1 d.p.vs6.4 from truncating to 1 d.p.

Same display, different possible lengths — the operation that made it decides the interval.

Focus

The error interval

6.4 from rounding to 1 d.p.

6.35 ≤ L < 6.45

6.4 from truncating to 1 d.p.

6.4 ≤ L < 6.5

The insight

Two different intervals behind one display: always check whether a value was rounded or truncated before writing its interval.

What happens to the extra digits

6.4 from rounding to 1 d.p.

The length moves to whichever of 6.3, 6.4 or 6.5 is nearest — up or down.

6.4 from truncating to 1 d.p.

They are simply cut off, so this positive length never goes up.

Lower end

6.4 from rounding to 1 d.p.

6.35 — half-way to 6.3, and it rounds up to 6.4

6.4 from truncating to 1 d.p.

6.4 — cutting nothing off 6.4 leaves 6.4

Upper end

6.4 from rounding to 1 d.p.

6.45 — it rounds up to 6.5

6.4 from truncating to 1 d.p.

6.5 — it truncates to 6.5, while 6.499… still truncates to 6.4

Which end is included

6.4 from rounding to 1 d.p.

Lower end included (≤), upper end not (<)

6.4 from truncating to 1 d.p.

Lower end included (≤), upper end not (<)

A length of 6.47 cm

6.4 from rounding to 1 d.p.

Shown as 6.5 — it is past 6.45

6.4 from truncating to 1 d.p.

Shown as 6.4 — the 7 is cut off

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Pick the nearer neighbour — then read the same picture backwards for the error interval.

What you need to know

To round, find the two neighbouring values allowed at the stated accuracy and choose the nearer one; a value exactly half-way between them rounds up.
Decimal places count from the decimal point; significant figures count from the first non-zero digit, and place-holder zeros are kept so the rounded number stays the same size.
In context, round up when a part-group still needs a whole one, round down when only complete items count, and otherwise round to the nearest at a sensible accuracy.
Keep full accuracy through every step of a calculation and round only the final answer.
An error interval uses inequalities: for a rounded value the ends are the half-way points either side, with the lower end included (≤) and the upper end not (<); for a truncated positive value the interval starts at the shown value, with the same signs.

The big picture

Rounding replaces a number with the nearer of its two neighbours at a stated number of decimal places or significant figures. This lesson shows that choice on a zoomed-in number line, reads the same picture backwards to write the error interval of a rounded or truncated value, and practises choosing the right rounding for a context and rounding only at the end of a calculation.

Key points

1Rounding picks the nearest neighbour; truncating chops digits off.

2Significant figures start at the first non-zero digit — keep the place-holder zeros.

3The context decides the direction: up, down or nearest.

4Round once, at the end of a calculation.

52.6 rounded to 1 d.p. means 2.55 ≤ m < 2.65; 2.6 truncated to 1 d.p. means 2.6 ≤ m < 2.7.

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Worked example

Problem

Round 0.007 251 6 to (a) 2 significant figures, (b) 3 decimal places.

⚠ Watch out

Chopping digits off instead of rounding. It ignores which neighbour is nearer (6.47 becomes 6.4 instead of 6.5), and with significant figures it drops the place-holder zeros, so 45 382 to 2 significant figures becomes 45 instead of 45 000.

🧠

Memory hook

Zoom in until only two answers are left and pick the nearer one. Read the picture backwards for the error interval: lower end in, upper end out.

✓

Check yourself

Without looking back: round 38 516 to 2 significant figures, then write the error interval for a distance d given as 38 000 m to 2 significant figures.

Flashcards

(9)

What does rounding to a stated accuracy do to a number?

It replaces the number with the nearer of its two neighbouring values at that accuracy.

A value is exactly half-way between its two neighbours. Which way does it round?

Up — for example, 3.45 rounds to 3.5 at 1 decimal place.

Where do you start counting significant figures?

At the first non-zero digit. Zeros in front of it (as in 0.003) are place-holders, not significant figures.

Why is 61 480 to 2 significant figures 61 000 and not 61?

The place-holder zeros keep the number the same size; 61 is about a thousand times too small.

When should an answer in context be rounded up?

When a part-group still needs a whole one — like the boxes needed to pack every item.

When should an answer in context be rounded down?

When only complete items count — like how many full tickets you can afford.

When should you round in a calculation with several steps?

Only at the final answer. Keep full calculator values in every step before it.

In the error interval of a rounded value, which end is included?

The lower end (≤). The upper end is not included (<), because it rounds up to the next value.

x is truncated to 1 decimal place, giving 3.2. What is the error interval?

3.2 ≤ x < 3.3 — truncation only cuts digits off, so the interval starts at 3.2.

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.

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Which masses round to 2.6 kg?

Two numbers, two significant figures

Up, down or nearest?

Find where this answer goes wrong

More AQA GCSE Maths topics