Rounding to decimal places and significant figures, estimating by rounding, and finding upper and lower bounds of rounded values at Higher tier.

What this dot point is asking

AQA wants you to round numbers to a given number of decimal places or significant figures, estimate the answer to a calculation by rounding, and at Higher tier find the upper and lower bounds of a rounded measurement and use them in calculations. Estimation checks that an answer is sensible; bounds quantify the uncertainty in a measurement, and both are tested on each paper.

Rounding to decimal places and significant figures

To round to a number of decimal places, look at the digit just after the last place you keep: if it is $5$ or more round up, otherwise leave the last digit unchanged. So $3.847$ to 2 decimal places is $3.85$, and $0.1923$ to 2 decimal places is $0.19$.

Significant figures count from the first non-zero digit. In $0.004\,82$ the first significant figure is $4$; to 2 significant figures it is $0.0048$. In $58\,300$ the first significant figure is $5$; to 2 significant figures it is $58\,000$. Watch the place-holding zeros: rounding $4\,962$ to 2 significant figures gives $5\,000$, not $50$.

Estimating calculations

Estimating means rounding every number to 1 significant figure, then doing the simpler sum. It gives a quick sense-check and is an exam skill in its own right.

Upper and lower bounds at Higher tier

To find the bounds, halve the rounding unit and add or subtract. For $8.6$ measured to 1 decimal place, the unit is $0.1$, half of it is $0.05$, so the bounds are $8.55$ and $8.65$.

When combining bounds, think about what makes the result largest or smallest. For a sum or product, the maximum uses both upper bounds. For a difference $a - b$, the maximum uses the upper bound of $a$ with the lower bound of $b$. For a quotient $\tfrac{a}{b}$, the maximum uses the upper bound of $a$ over the lower bound of $b$.

A worked bounds calculation

Suppose a car travels a distance of $150\,\text{m}$ measured to the nearest $10\,\text{m}$ in a time of $12\,\text{s}$ measured to the nearest second, and you want the upper bound for its speed. The distance bounds are $145\,\text{m}$ to $155\,\text{m}$ (half of $10$ is $5$), and the time bounds are $11.5\,\text{s}$ to $12.5\,\text{s}$. Speed is distance over time, so the maximum speed uses the largest distance over the smallest time: $\dfrac{155}{11.5} \approx 13.48\,\text{m/s}$. Choosing the wrong pairing (the smaller time for a maximum is correct, but using the smaller distance is not) is the usual error, so always reason about what makes the answer biggest.

Deciding a sensible degree of accuracy

A related skill is giving an answer to a suitable accuracy. The upper and lower bounds of a calculated quantity tell you how many figures are actually reliable: if the upper and lower bounds of a length agree to $14.3$ but differ at the next digit, then $14.3\,\text{cm}$ is the most you can safely claim. AQA sometimes asks you to find both bounds of a result and then state the value to an appropriate degree of accuracy, which means quoting the figures on which the two bounds agree. This connects rounding back to the real precision of a measurement.