Year 9 Maths The Thai Guy

Lesson 4.3 · 25 min

Lower and Upper Bounds

Every rounded number stands for a range of possible values. Finding the smallest and largest, and writing them as an inequality.

In this lesson you will learn to…

  • Find the lower and upper bounds of a number rounded to the nearest 10, 100, whole number or decimal place
  • Find bounds for numbers rounded to significant figures
  • Write the bounds as an inequality
  • Use bounds in a simple calculation, such as the largest possible perimeter

Why this matters

A football stadium holds 62 540 people. If that is rounded to 63 000 and 63 000 tickets are sold, nearly 500 people will have nowhere to sit. Rounded numbers hide a range of true values, and engineers, builders and event planners need to know the whole range — not just the rounded figure.

Which numbers round to 40?

To the nearest 10, all of these round to 40:

35,  36,  38.724,  39,  40.5,  41,  44,  44.935, \; 36, \; 38.724, \; 39, \; 40.5, \; 41, \; 44, \; 44.9

The smallest number that rounds to 40 is 35. There is no single largest — 44.944.9, 44.9944.99, 44.999…44.999\dots all round to 40. So we use 45 as the upper bound and write

35≤n<4535 \le n < 45

Harry’s method

  1. Find what the number was rounded to (nearest 100, nearest 0.1, …).
  2. Halve it.
  3. Subtract for the lower bound, add for the upper bound.

It works every time, as long as you are careful in step 1 — especially with significant figures.

Rounded valueRounded toHalf a unitBounds
340nearest 105335≤n<345335 \le n < 345
4500nearest 100504450≤n<45504450 \le n < 4550
0.71 d.p.0.050.65≤n<0.750.65 \le n < 0.75
3.782 d.p.0.0053.775≤n<3.7853.775 \le n < 3.785
40001 s.f.5003500≤n<45003500 \le n < 4500

Watch out for unusual units

Rounded to the nearest 50? Half of 50 is 25, so 350 has bounds 325≤n<375325 \le n < 375. Rounded to the nearest 0.5? Half of 0.5 is 0.25, so 7.5 has bounds 7.25≤n<7.757.25 \le n < 7.75.

A common error. Ruth says a number that is 4800 to the nearest 10 lies in 4750≤n<48504750 \le n < 4850. She has used half of 100, not half of 10. The correct bounds are 4795≤n<48054795 \le n < 4805.

Key rules and formulas

  • Half a unit either side

    Find the unit the number was rounded to, halve it, and go that far below and above. 1200 kg to the nearest 100 kg: half of 100 is 50, so the bounds are 1150 and 1250.

  • The inequality

    The lower bound itself rounds up to the number, so it is included (). The upper bound would round up to the next value, so it is not ().

  • Decimal places and significant figures

    To 2 d.p. the unit is 0.01, so go 0.005 either way: . To 2 s.f., 3400 means the nearest 100, so .

  • Largest and smallest results

    For the largest possible sum or product, use the upper bounds. For the smallest, use the lower bounds.
Try it

What rounds to 40?

30
n
30

Drag the marker along. Every value from 35 up to (but not including) 45 rounds to 40 to the nearest 10. 35 is the lower bound; 45 is the upper bound, even though 45 itself would round to 50.

Worked examples

Worked example 1

Three kinds of rounding

Write an inequality for a a car of mass kg, 1200 kg to the nearest 100 kg b a pencil of length cm, 14 cm to the nearest cm c a time s, 23.48 s to 2 d.p.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Bounds to significant figures

A number rounded to 2 significant figures is 520. Find its bounds.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Upper bound of a perimeter

A rectangle measures 9 cm by 4 cm, each correct to the nearest cm. Find the upper bound of its perimeter.

Try it yourself first, then reveal the solution one step at a time.

Practice: bounds

11 questions

Ten questions. Always ask first: what was this rounded to? Then halve it.
  1. Question 1 of 11Choose all that applyFoundation
    A number rounded to the nearest 10 is 240. Which of these could it be?

    Select every correct answer — there may be more than one.

  2. Question 2 of 11Complete the table
    Complete the bounds for each rounded number.
    Number, nRounded toLower boundUpper bound
    4500nearest 100
    7000nearest 1000
    83 200nearest 100
    690nearest 10
  3. Question 3 of 11Complete the table
    Now with decimal places and significant figures.
    Number, nRounded toLower boundUpper bound
    0.71 d.p.
    3.782 d.p.
    34002 s.f.
    40001 s.f.
  4. Question 4 of 11Explore and match
    A pencil is 14 cm long, correct to the nearest cm. Drag the marker to the lower bound of its length.

    Move the controls until you match the question, then press Check.

    13
    Length (cm)
    13
  5. Question 5 of 11Multiple choice
    Ruth says a number that rounds to 4800 to the nearest 10 lies in . What is the correct inequality?
  6. Question 6 of 11Fill the gaps
    The Burj Khalifa is 828 m tall, correct to the nearest metre. Complete the inequality for its height .

  7. Question 7 of 11Multiple choice
    A number rounded correct to the nearest 50 is 350. Which inequality shows its bounds?
  8. Question 8 of 11Choose all that apply
    The bounds for a number are . Which of these could be how it was rounded?

    Select every correct answer — there may be more than one.

  9. Question 9 of 11Type the answer
    A rectangle measures 9 cm by 4 cm, both correct to the nearest cm. Work out the upper bound of its perimeter.
    cm
  10. Question 10 of 11Type the answer
    A rugby pitch is 68 m wide, correct to the nearest metre. What is the minimum width it could be?
    m
  11. Question 11 of 11Type the answerChallenge
    A number rounded to 2 s.f. is 520. Jenna says it lies in . Give a number that is not in Jenna's interval but does round to 520 to 2 s.f.