GCSE · Maths · AQA · Spec 8300 · Foundation

Error intervals (truncation and rounding)

A ruler says 7.3 cm, but the pencil could be 7.27 or 7.32 cm. '7.3' labels a whole stretch of lengths — and chopping instead of rounding moves that stretch.

Start here

Which numbers could have become 7.3?

A length x cm was rounded to 1 decimal place, and the answer was 7.3. Drag the two ends so the bar covers every length that rounds to 7.3. Then decide: is each end included, or not?

Now chop it instead

Same 7.3, different story

This time nobody rounded. A length y cm was truncated to 1 decimal place: every digit after the first decimal place was simply chopped off. The result was 7.3. (So 7.31 would truncate to 7.3 — and so would 7.38.)

Which error interval for y is closest to what you think right now?
How sure are you?

The rounding underneath

Decimal places vs significant figures

Round 0.0046827 (a) to 3 decimal places and (b) to 2 significant figures. Then round 38 517 (c) to 2 significant figures.

  1. (a) Decimal places count from the decimal point. Three places: 0.004 | 6827 — the last digit you keep is the 4.
  2. missing step
Which line is step 2?

Rounding in real life

Which way should you round?

Pick an answer, then the rounding its situation needs.

Still to sort

Round up (0)

Rounding down would leave something short.

Where the line is: Ask: if I round down, is something left undone or someone left out? Then go up — even when the decimal is small.

Round down (0)

Only complete things count.

Where the line is: Ask: can the leftover part actually be used? If only complete things count, go down — even when the decimal is big.

Round to a sensible accuracy (0)

Round normally, to a level of detail that suits the units.

Where the line is: Nothing is left short and nothing has to be whole: round in the usual way, to an accuracy that suits the units — money to the nearest penny.

6 of 6 still to sort.

Every one of these calculations gives a decimal. The situation decides what to do with it.

Watch out: Don't just round the calculator's answer in the usual way. First ask what the number is counting — normal rounding can land on an answer the situation can't allow.

Round at the end

Where does this answer go wrong?

A recipe uses 250 g of flour for 6 people. How much flour is needed for 14 people? Give your answer to the nearest gram.

A student's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A rounded number is really a label for a whole stretch of possible values. Learn to say exactly which stretch.

What you need to know

  • A rounded or truncated number stands for a whole stretch of possible values. That stretch is its error interval.
  • For positive numbers like the ones here, write it as lower ≤ x < upper: the bottom end is included, the top end is not.
  • Rounded: the interval runs half a step either side of the value. Truncated: it starts at the value and runs one whole step up.
  • To round, find the last digit you keep and look at the digit next door. Significant figures are counted from the first non-zero digit.
  • Let the situation decide which way to round, and never round in the middle of a calculation.

The big picture

An error interval shows every value a rounded or truncated number could have started as. You'll set the two ends on a number line and write them with ≤ and <, see why a chopped number sits differently from a rounded one, sharpen your rounding to decimal places and significant figures, decide which way to round in real situations, and see why you only round at the very end of a calculation.

Key points

1The step is the place value you rounded to: 0.1 for 1 d.p., 10 for the nearest 10. A rounded value's error interval runs half a step either side: 7.3 to 1 d.p. gives 7.25 ≤ x < 7.35.
2Truncating a positive number chops digits off and never makes it bigger, so the truncated value is the bottom of its interval, which runs one whole step up: 7.3 truncated to 1 d.p. gives 7.3 ≤ x < 7.4.
3The top end is the first value that gives the next answer up, so it is written with <. There is no 'last number' just below it, so never use 7.34 or 7.349 as the top end.
4Decimal places are counted from the decimal point. Significant figures are counted from the first non-zero digit, and every digit after that counts, zeros included.
5When rounding big numbers to significant figures, keep place-holder zeros so the number stays the right size: 38 517 is 39 000 to 2 s.f.
6In context: round up when rounding down would leave something short, round down when only complete things count, and otherwise round to a sensible accuracy (money to the nearest penny).
7Keep full values through every step of a calculation and round only the final answer.

Worked example

Problem

A parcel's mass is 2400 g, correct to 2 significant figures. Write down the error interval for its mass, m grams.

⚠ Watch out

Treating a truncated value like a rounded one. A time truncated to 23 seconds lies in 23 ≤ t < 24, not 22.5 ≤ t < 23.5 — chopping never rounds up, so nothing below 23 could have become 23.

🧠

Memory hook

Rounded? The value sits in the middle. Chopped? It sits on the floor. Either way, the top end is the door to the next number — so the door is shut: <.

✓

Check yourself

Why do 9.2 rounded to 1 d.p. and 9.2 truncated to 1 d.p. have different error intervals? (Rounded: 9.15 ≤ x < 9.25. Truncated: 9.2 ≤ x < 9.3 — chopping never rounds up.)

Flashcards

(10)
What does an error interval show?
Every value a number could have had before it was rounded or truncated.
How do you find the error interval of a rounded value?
Work out the step it was rounded to, then go half a step down and half a step up. 6.4 to 1 d.p. gives 6.35 ≤ x < 6.45.
What does truncating mean?
Chopping off digits after a certain place, with no rounding. 8.69 truncated to 1 d.p. is 8.6.
Where does a truncated value sit in its error interval?
At the bottom (for positive numbers). 8.6 truncated to 1 d.p. gives 8.6 ≤ x < 8.7 — one whole step up.
6.4 is rounded to 1 d.p. Why isn't the top of its interval 6.44 or 6.449?
Numbers like 6.4499 still round to 6.4, so there's no 'last' one. Write the first number that doesn't work, 6.45, and leave it out with <.
Where do you start counting significant figures?
At the first non-zero digit. After that every digit counts, zeros included: 0.0508 has 3 significant figures.
Why does 52 719 to 2 s.f. need zeros?
The zeros are place-holders that keep the number the right size: 53 000, not 53.
When should you round UP in a real-life problem?
When rounding down would leave something short — people without seats, a wall not fully painted.
When should you round DOWN in a real-life problem?
When only complete things count — boxes that can be completely filled, tickets you can actually afford.
When should you round in a calculation with several steps?
Only at the end. Keep full calculator values for every step in between.

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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