# Upper and lower bounds

## In a nutshell

An upper and lower bound for a rounded measurement gives a range of values within which the true values of the measurement could lie. The range of numbers is called an error interval.

## Upper and lower bounds

When a measurement is given correct to a certain number of decimal places or significant figures, find the upper and lower bound by halving the number the unit is rounded to, and then adding or subtracting this number from the measurement to find the upper and lower bounds.

**Example 1**

*The mass of a man is $93kg$ to the nearest $kg$. Find the upper and lower bounds for the mass of the man.*

*The mass is rounded to the nearest $1kg$. Half of this is $0.5kg$. Add $0.5kg$ onto $93kg$ to find the upper bound and subtract $0.5kg$ from $93kg$ to find the lower bound.*

*The lower bound is $\underline{92.5 \ kg}$ and the upper bound is $\underline{93.5 \ kg}$.*

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## Upper and lower bounds within a calculation

You may have to find the value of a quantity to an appropriate degree of accuracy from a formula, where the quantities in the formula have been rounded. Find the upper and lower bounds of the quantities in the formula to work out all possible answers for the quantity being calculated. Then it is possible to give the answer to an appropriate degree of accuracy.

**Example 2**

*A plot of land is $2.5$m wide and $6.0$m long, correct to $2$ significant figures. Find the area of the plot of land to an appropriate degree of accuracy.*

*Find the upper and lower bounds for the width.*

*Upper bound of the width $= 2.55$m*

*Lower bound of the width $= 2.45$m*

*$2.45m\le width \lt 2.55m$*

*Find the upper and lower bounds for the length.*

*Upper bound of length $= 6.05$m*

*Lower bound of length $= 5.95$m*

*$5.95m \le length \lt 6.05m$*

*Multiply the lower bound for each measurement to find the minimum area and multiply the higher bound for each length to find the maximum area.*

*Minimum area $= 5.95 \times 2.45 = 14.5775$*

*Maximum area $= 6.05 \times 2.55 = 15.4275$*

*The answer lies between $14.5775m^{2}$ and $15.4275m^{2}$. Rounding both of these numbers to $2$ significant figures gives the same answer.*

*$\underline{Area = 15m^{2} \ (2 \ s.f.)}$*