Number - Measures and accuracy: apply and interpret limits of accuracy, including upper and lower bounds (Higher tier)

The idea

This is Higher tier only. It adds calculating with bounds, not only describing the range a rounded measurement stands for. Write the lower and upper bound of every measurement first, then choose the pair that drives the answer to the extreme you want. Each of these gives the largest answer: for a sum, add the two upper bounds; for a difference, take the upper bound of the first quantity and the lower bound of the second; for a product of positive quantities, multiply the upper bounds; for a division of positive quantities, divide the upper bound of the top by the lower bound of the bottom. With a distance of 100 m measured to the nearest metre and a time of 12.4 s to one decimal place, the fastest possible speed is 100.5 ÷ 12.35, about 8.14 m/s, and the slowest is 99.5 ÷ 12.45, about 7.99 m/s. Both bounds round to 8 m/s to one significant figure, so that is the accuracy worth quoting.

What earns the marks

• The upper and lower bound of each measurement written down before any calculation.
• Choosing the combination that gives the bound asked for, such as the largest numerator with the smallest denominator for an upper bound of a positive quotient.
• The calculation carried out on the unrounded bounds, even if the final rounding is wrong.
• A suitable degree of accuracy, quoting only the figures on which the upper and lower bounds agree, together with a reason.

Common mistakes and how to put them right

• Dividing the upper bound by the upper bound when the largest possible quotient of positive quantities needs the smallest divisor.

• Taking 12.45 as the lower bound of a time recorded as 12.4 s to one decimal place, when 12.35 is the lower and 12.45 the upper.

• Rounding each bound before the calculation, which pushes the final answer outside the true range.

• Quoting the answer to several decimal places when the two bounds stop agreeing much earlier than that.

Exam tips

• Label the four bounds clearly, upper and lower for each measurement, so the pair you pick is visible to the examiner.
• For a suitable degree of accuracy, round both bounds to the same accuracy and quote the most accurate value at which the two agree.
• Keep the bounds in the calculator memory rather than retyping rounded versions of them.