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

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

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

Using upper and lower bounds practically

Maths

Upper and lower bounds

Upper and lower bounds in additive calculations

Upper and lower bounds in multiplicative calculations

Advanced problem solving with rounding, estimation and bounds

In the prior unit, pupils learnt how to round to a given number of decimal places or significant figures and explored the effect of using rounded numbers for calculations. In this unit, pupils will investigate the bounds and perform calculations to determine a best/worst case scenario.

This unit explores the impact of upper and lower bounds on additive and multiplicative calculations.

Rounding, estimation and bounds

The degree of {{  }} determines the upper and lower bounds of a number or measurement. 

Considering the maximum number of boxes to safely fit in a lift.  Which of the following should be used? (Where $$l$$ represents the capacity of the lift and $$b$$ represents the mass of a box?

A sack contains 16 kg of flour correct to the nearest kilogram.  What are the lower and upper bounds of the mass of the flour?

A sack contains 16 kg of flour correct to the nearest kilogram.  This is put into smaller bags containing 500 g correct to 10g. What are the lower and upper bounds of the mass of the bags?

A sack contains 16 kg of flour correct to the nearest kilogram. This is put into smaller bags containing 500 g correct to 10g. Which calculation is correct to find the minimum number of bags produced?

A sack contains 16 kg of flour correct to the nearest kilogram.  This is put into smaller bags containing 500 g correct to 10g.  What is the minimum number of bags produced?

The {{ }} bound for a rounded number is the smallest value that would round up to the next rounded value. 

$$x$$ = 5.2 correct to 1 decimal place.  What is the error interval?

$$y$$ = 45 correct to 2 significant figures.    What is the error interval?

The error interval of $$x$$ is $$5.15\leq{x}<5.25$$.  The error interval of $$y$$ is $$44.5\leq{y}<45.5$$.  What are the lower and upper bounds of $$xy$$?  (Give you answers to 3 decimal places)

The error interval of $$x$$ is $$5.15\leq{x}<5.25$$.  The error interval of $$y$$ is $$44.5\leq{y}<45.5$$.  What are the lower and upper bounds of $$xover{y}$$?  (Give you answers to 3 s.f.)

Which calculation will give the upper bound of $$x={{a^2}\over{b+c}}$$

Starter Quiz

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