Approximation and estimation Study Guide

When we work with measurements, they are often rounded to a certain degree of accuracy. For example, a length measured as 10 cm to the nearest centimetre actually lies between 9.5 cm and 10.5 cm. These boundary values are called the lower bound (LB) and the upper bound (UB). The lower bound is the smallest possible value the original measurement could have been, and the upper bound is the largest possible value.

When we perform calculations involving these rounded measurements, we need to consider how the uncertainties accumulate. To find the upper bound or lower bound of a result, we must choose the bounds of the input values carefully. If we are calculating a sum or product, to find the upper bound of the result, we typically use the upper bounds of the input values. To find the lower bound of a sum or product, we typically use the lower bounds of the input values.

However, when performing division or subtraction, it's different. To find the upper bound of a division (e.g., A / B), we use the upper bound of the numerator (A) and the lower bound of the denominator (B). To find the lower bound, we use the lower bound of the numerator (A) and the upper bound of the denominator (B). Similarly for subtraction, to maximise the result (upper bound), we take the upper bound of the first value and subtract the lower bound of the second. To minimise the result (lower bound), we take the lower bound of the first value and subtract the upper bound of the second.

These principles are crucial for determining the possible range of a calculated value when the initial measurements are not exact, which is common in real-world applications involving ratios, proportions, and rates of change like speed or density.

This is Higher-tier content, so solutions should communicate multi-step reasoning precisely. Bounds describe the interval of possible original values after rounding. A value rounded to the nearest unit has half a unit subtracted for its lower bound and added for its upper bound. In division, the greatest possible quotient uses the greatest numerator and smallest positive denominator; the least uses the smallest numerator and greatest denominator.

If distance 120 km is rounded to the nearest 10 km, then 115 ≤ d < 125. If time 2.0 h is rounded to the nearest 0.1 h, then 1.95 ≤ t < 2.05. The upper bound for speed d/t is 125/1.95, while the lower bound is 115/2.05. Keep inequalities clear: lower bounds are included and upper bounds are excluded.

A strong solution states the relevant rule, substitutes or transforms values carefully, shows intermediate reasoning, and checks that the result is reasonable and in the required form. Common errors come from omitted terms, mishandled signs, premature rounding or using a rule that does not match the structure of the problem.