Year 9 Maths The Thai Guy

Lesson 4.4 · 35 min

Bounds in Calculations

Which bounds to use for the largest and smallest possible answer — adding, subtracting, multiplying and dividing measurements — and giving an answer to a suitable accuracy.

In this lesson you will learn to…

  • Find the upper and lower bounds of a sum, difference, product and quotient
  • Explain why the upper bound of uses the lower bound of
  • Use bounds in formulas such as speed, density and area
  • Decide an appropriate degree of accuracy by comparing the bounds of an answer

Which bound goes where?

For every calculation, ask: what makes the answer as big as possible? Then do the opposite for the smallest possible answer.

To make it as big as possibleUse
a+ba + bUBa+UBb\text{UB}_a + \text{UB}_b
a−ba - bUBa−LBb\text{UB}_a - \text{LB}_b
a×ba \times bUBa×UBb\text{UB}_a \times \text{UB}_b
a÷ba \div bUBa÷LBb\text{UB}_a \div \text{LB}_b

Subtracting

The biggest difference comes from the biggest starting value and the smallest amount taken away. Watch the units, too — convert before you find the bounds.

How accurate is my answer?

Measurements that are only rounded give an answer that is only known within a range. Find the upper and lower bounds of the answer, then round both to fewer and fewer significant figures until they agree.

Bounds in formulas

In a formula, work out what each measurement does to the answer. In a fraction, a bigger numerator increases it; a bigger denominator decreases it.

Key rules and formulas

  • Adding and multiplying

    Upper with upper, lower with lower (for positive measurements).

  • Subtracting

    To make a difference as big as possible, take away as little as possible.

  • Dividing

    This covers speed, density, pressure and any "per" measure.

  • A suitable degree of accuracy

    Round the upper and lower bounds of the answer to fewer and fewer significant figures until they agree. That common value is the answer you can be sure of.

Worked examples

Worked example 1

Area of a rectangle

A rectangle is cm by cm, both to 1 d.p. Find the upper bound of its area.

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

Worked example 2

Speed

A runner covers m, to the nearest metre, in s, to the nearest 0.1 s. Find the lower bound of her average speed.

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

Worked example 3

Finding a suitable accuracy

and , both to 2 d.p. Find to a suitable degree of accuracy.

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

Practice: bounds in calculations

7 questions

Ask what makes the answer as big — or as small — as possible.
  1. Question 1 of 7Type the answer
    and , both to 1 d.p. Find the upper bound of .
  2. Question 2 of 7Type the answer
    and , both to 1 d.p. Find the upper bound of .
  3. Question 3 of 7Type the answer
    A square has side cm, to the nearest cm. Find the lower bound of its area.
    cm²
  4. Question 4 of 7Type the answer
    A cyclist travels km (to the nearest km) in hours (to the nearest 0.1 h). Find the upper bound of her average speed, to 3 s.f.
    km/h
  5. Question 5 of 7Type the answer
    A metal block has mass g (to the nearest 10 g) and volume (to the nearest ). Find the lower bound of its density, to 3 s.f.
    g/cm³
  6. Question 6 of 7Multiple choice
    and , both to the nearest whole number. Which gives the lower bound of ?
  7. Question 7 of 7Step by stepChallenge
    A field in two parts.
    A rectangular field is m by m, both to the nearest 5 m. A square of side m, to the nearest metre, is fenced off for a garden.
    1. Upper bound of the area of the whole field.
      m²