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Upper and lower bounds in multiplicative calculations

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Lesson details

Learning outcome

I can calculate upper and lower bounds for calculations involving rounded numbers.

Key learning points

  1. The biggest possible product comes from multiplying the upper bounds together.
  2. The smallest possible product comes from multiplying the lower bounds.
  3. The biggest possible quotient comes from dividing an upper bound by a lower bound.
  4. The smallest possible quotient comes from dividing a lower bound by an upper bound.

Keywords

  • Upper bound - The upper bound for a rounded number is the smallest value that would round up to the next rounded value.

  • Lower bound - The lower bound for a rounded number is the smallest value that the number could have taken prior to being rounded.

  • Error interval - An error interval for a number x shows the range of possible values of x. It is written as an inequality a ≤ x < b

Common misconception

Assuming that when dividing the UB will be the result of the UB ÷ UB.

Get the pupils to investigate dividing numbers. Keep the divisor the same and change the dividend. Ask the question, "What happens as the dividend increases?".

Teacher tip

Pupils who struggle with the misconception can be reminded that they can try the different combinations and choose the appropriate one.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Prior knowledge starter quiz

Download starter quiz questions (PDF)

6 Questions

Q1.
The numbers either end of an interval are the lower and upper bounds.

Correct Answer: error

Q2.
The length of a leaf is measured as 6.7 cm correct to 1 decimal place. What are the lower and upper bounds?

6.2 and 7.2
6.75 and 6.85
6.65 and 6.74999999999
Correct answer: 6.65 and 6.75

Q3.
The length of a leaf is measured as 6.7 cm correct to 1 decimal place. The lower and upper bounds are 6.65 and 6.75. What is the error interval?

$$6.65\leq{x}>6.75$$
$$6.65<{x}\leq6.75$$
$$6.65\geq{x}<6.75$$
Correct answer: $$6.65\leq{x}<6.75$$
$$6.65\leq{x}\leq6.75$$

Q4.
The mass of a kitten is 1.3 kg correct to 2 significant figures. What is the error interval?

$$1.25\leq{x}\leq1.35$$
$$1.25\leq{x}<1.349999999999$$
$$1.25<{x}<1.35$$
Correct answer: $$1.25\leq{x}<1.35$$

Q5.
Here are the error intervals for $$x$$ and $$y$$: $$12.5\leq{x}<13.5$$ and $$3.5\leq{y}<4.5$$. What are the lower and upper bounds of $$x + y$$?

16 and 17
Correct answer: 16 and 18
17 and 17
17 and 18

Q6.
Here are the error intervals for $$x$$ and $$y$$: $$12.5\leq{x}<13.5$$ and $$3.5\leq{y}<4.5$$. What are the lower and upper bounds of $$x - y$$?

8 and 9
9 and 9
Correct answer: 8 and 10
9 and 10

Assessment exit quiz

Download exit quiz questions (PDF)

6 Questions

Q1.
The lower bound for a rounded number is the value the number could have taken prior to rounding.

Correct Answer: smallest, lowest, small, low, least

Q2.
$$x$$ = 4.5 correct to 1 decimal place. What is the error interval?

$$4.45\leq{x}\leq4.55$$
$$4.45<{x}\leq4.55$$
$$4.45\leq{x}>4.55$$
Correct answer: $$4.45\leq{x}<4.55$$
$$4.45\geq{x}<4.55$$

Q3.
$$y$$ = 26 correct to 2 significant figures. What is the error interval?

Correct answer: $$25.5\leq{y}<26.5$$
$$25.5 \geq{y}<26.5$$
$$25.5<{y}\leq26.5$$
$$25.5\leq{y}\leq26.5$$
$$25.5\leq{y}>26.5$$

Q4.
The error interval of $$x$$ is $$4.45\leq{x}<4.55$$. The error interval of $$y$$ is $$25.5\leq{y}<26.5$$. What are the lower and upper bounds of $$xy$$? (Give you answers to 3 decimal places)

Correct Answer: 113.475 and 120.575, 113.475 & 120.575, 120.575 and 113.475, 120.575 & 113.475, 113.475 120.575

Q5.
The error interval of $$x$$ is $$4.45\leq{x}<4.55$$. The error interval of $$y$$ is $$25.5\leq{y}<26.5$$. What are the lower and upper bounds of $$x\over{y}$$? (Give you answers to 3 s.f.)

Correct Answer: 0.168 and 0.178, 0.168 & 0.178, 0.178 and 0.168, 0.178 & 0.168, 0.178 0.168

Q6.
Which calculation will give the lower bound of $$x={{a+b}\over{c}}$$?

Correct answer: $$LB_x={{LB_a+LB_b}\over{UB_c}}$$
$$LB_x={{UB_a+LB_b}\over{LB_c}}$$
$$LB_x={{LB_a+UB_b}\over{LB_c}}$$
$$LB_x={{UB_a+UB_b}\over{LB_c}}$$
$$LB_x={{UB_a+UB_b}\over{UB_c}}$$

To help you plan your 10 maths lesson on: Upper and lower bounds in multiplicative calculations, download all teaching resources for free and adapt to suit your pupils' needs...