Year 10 Extended · Unit 1 Number sets, surds, indices and bounds · Lesson 6 of 6

What is the biggest and smallest the truth could be?

A rounded number stands for every number that would round to it, so it is really an interval. Calculating with rounded numbers gives an interval too, and you find its ends by choosing, for each number, the end that pushes the result furthest in the direction you want.

Number and AlgebraGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 8: round 0.04567 to 2 significant figures.
    Answer0.046. Keep 4 and 5; the next digit is 6, so round up.
  2. Which whole numbers round to 20 to the nearest 10?
    Answer15 to 24. Halfway, 15, rounds up; 25 rounds to 30.
  3. Work out 12.5 × 6.5.
    Answer81.25. 12.5 × 6 = 75 and 12.5 × 0.5 = 6.25.
Words used in this lesson
upper bound
the value just above every number that rounds to the stated one; it is not reached. For example: the 0.125 for 0.12 to 2 d.p.
lower bound
the smallest value that rounds to the stated number. For example: the 7.35 for 7.4 to 1 d.p.
accuracy
the place a value was rounded to. For example: to the nearest 10 cm.
interval
all the values between a lower and an upper bound. For example: from 2.5 up to 7.5.
limit of accuracy
another name for a bound of a rounded value. For example: 8.75 and 8.85 for 8.8.
maximize
make a result as large as possible. For example: use the largest first number.

Is the truck under the limit?

A weighbridge in Durban shows a grain truck as 24 300 kg, to the nearest 100 kg. The bridge it must cross has a limit of 24 320 kg. The scale never lies about its reading, but its reading is rounded, and the truck's true mass is somewhere inside an interval.

Predict first: can the driver be certain the truck is under the limit? And is the interval for “6 to the nearest whole number” 5 to 7, or narrower? Use the figures, then check.

45
1
44.5lower bound
45.5upper bound

45 to the nearest 1: it lies from 44.5 up to, but not including, 45.5.

Show me the steps
  1. 45 to the nearest 1: half a unit is 0.5, so the interval runs from 44.5 up to 45.5. 44.5 rounds up to 45, so it is included; 45.5 rounds up to 46, so it is excluded.
  2. Set the unit to 10: 45 means 450, and the interval is from 445 up to 455. Half a unit is now 5.
  3. Set the unit to 0.1: 45 means 4.5, and the interval is from 4.45 up to 4.55. Half a unit is 0.05.
  4. Whatever the unit, the width of the interval is exactly one unit, and the rounded value is in the middle.
12
7
74.75lower bound
93.75upper bound

12 × 7 = 84 from the rounded numbers, but the true result is between 74.75 and 93.75.

Show me the steps
  1. 12 × 7 from the rounded numbers is 84. The lower bound uses both lower bounds, 11.5 × 6.5 = 74.75, and the upper bound uses both upper bounds, 12.5 × 7.5 = 93.75.
  2. Press a + b: the bounds are 11.5 + 6.5 = 18 and 12.5 + 7.5 = 20. For a sum, the half units add: the result is 19 ± 1.
  3. Press a − b: the largest result is 12.5 − 6.5 = 6 and the smallest is 11.5 − 7.5 = 4. The errors still add, and the lower bound of b is used to make the result biggest.
  4. Press a ÷ b: the largest result is 12.5 ÷ 6.5 = 1.923… (the smallest divisor) and the smallest is 11.5 ÷ 7.5 = 1.533… (the largest divisor).

A value to a stated accuracy lies half a unit either side. The lower bound is included and the upper bound is not: 7.55 ≤ x < 7.65 for 7.6 to the nearest tenth, because 7.65 would round up to 7.7.

To make a result as large as possible, use the upper bound in a sum, an upper bound times an upper bound, the upper bound of the first minus the lower bound of the second, and the upper bound over the lower bound in a quotient. For the smallest result, use the opposite bounds.

Then decide the rounding the context needs: a count of items that must fit rounds down.

Two common mistakes. Taking a full unit instead of half: 7.6 to the nearest tenth is not 7.5 to 7.7. And using the same kind of bound for everything: in a subtraction or a division, the number being taken away or divided by needs its opposite bound to make the result as large as possible.

Worked examples

1. A town has 6 800 people, to 2 significant figures. Give the bounds.

  1. The second significant figure is in the hundreds, so the accuracy is the nearest 100. Half a unit is 50.
  2. 6 800 − 50 = 6 750 and 6 800 + 50 = 6 850: 6 750 ≤ n < 6 850.

2. A court is 28 m by 15 m, each to the nearest metre. Find the bounds of its area.

  1. Length: 27.5 ≤ l < 28.5. Width: 14.5 ≤ w < 15.5.
  2. Lower: 27.5 × 14.5 = 398.75. Upper: 28.5 × 15.5 = 441.75.
  3. The area is between 398.75 m² and 441.75 m².

3. Two lengths are 52.4 cm and 19.6 cm, each to 1 decimal place. Find the smallest possible difference.

  1. 52.4 is at least 52.35, and 19.6 is below 19.65.
  2. The difference is smallest with the lowest first length and the highest second: 52.35 − 19.65 = 32.7 cm.

Your go. The first step is given. A runner covers 100 m (to the nearest metre) in 12.3 s (to the nearest 0.1 s). Find the greatest possible speed, to 3 significant figures.

  1. Speed = distance ÷ time, so use the upper bound of the distance, 100.5 m, and the lower bound of the time.
  2. Find the lower bound of the time, then divide.
Check the answer

The lower bound of the time is 12.25 s, and 100.5 ÷ 12.25 = 8.204…, which is 8.20 m/s to 3 significant figures.

With a GDC (optional)

A GDC can work out the extreme values for you, but you must choose which bounds to use. Enter the product as one line with brackets, for example (28.5)(15.5) for an upper bound, and write down the lower-bound line separately. Do not round in the middle: store each bound with its full digits, and round only the final answer.

Write each bound on the page before you type it, so the choice of bounds is visible. The bounds themselves are the thinking: everything here can be done by hand.

Check before you go on

1. A parcel's mass is 85 kg, to the nearest 5 kg. Which gives its bounds?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The span of a bridge is 2350 m, to the nearest 10 m. What is the upper bound of its length, in metres?

3. A time is 0.0640 s, correct to 3 significant figures. What is its lower bound, in seconds?

Level 2 · Building

4. A rectangular plot is 9.2 cm by 6.5 cm, each correct to 1 decimal place. What is the upper bound of its area, in cm²?

5. A cyclist rides 240 m (to the nearest 10 m) in 32 s (to the nearest second). What is the greatest possible speed, in m/s, to 2 decimal places?

6. a = 14.2 and b = 8.6, each to 1 decimal place. What is the upper bound of a − b?

Level 3 · Stretch

7. A cargo plane in Johannesburg carries pallets whose mass is 850 kg each, to the nearest 10 kg. Its load limit is 102 500 kg. What is the largest number of pallets that is certain to be within the limit?

8. p = 6.4 and q = 2.5, each to 1 decimal place. What is the lower bound of (p + 3) ÷ (q − 1), to 2 decimal places?

Something to think about

A number is 4.0 to 1 decimal place, and another is 4 to the nearest whole number. Are they the same statement? Write the interval for each. Then find the interval for their sum, and say which of the two numbers is responsible for most of the uncertainty.

One way to think about it

4.0 to 1 d.p. lies in 3.95 ≤ x < 4.05, a width of 0.1. 4 to the nearest whole number lies in 3.5 ≤ x < 4.5, a width of 1. So 4.0 and 4 are different statements: the 4.0 claims ten times the accuracy. The sum lies in 7.45 ≤ s < 8.55, a width of 1.1. The whole number is responsible for almost all of it. A result is only as accurate as the least accurate number in it.

Exit ticket

9. x = 400, correct to 1 significant figure. What is the upper bound of x?