Approximation and estimation Revision Notes

Finding Upper and Lower Bounds of Individual Measurements

If a measurement is given to the nearest whole number, its true value is halfway between the half numbers. For example, '10 to the nearest whole number' means the value is between 9.5 and 10.5. The lower bound (LB) is 9.5 and the upper bound (UB) is 10.5. If it's to one decimal place, e.g., '10.0 cm', then it's between 9.95 cm and 10.05 cm. The rule is to take half of the degree of accuracy, and then subtract it for the lower bound and add it for the upper bound. For 'nearest 10', divide 10 by 2 = 5, so LB is measurement - 5, UB is measurement + 5. Remember, the lower bound is inclusive, but the upper bound is often considered exclusive in practice, though for calculation purposes we use the exact boundary value.

Calculations Involving Addition and Subtraction

When you need to find the bounds of a sum or difference, apply the following:

• Upper Bound of Sum (A + B): Use the upper bound of A (UB_A) + the upper bound of B (UB_B). For example, if A is 10 (9.5 to 10.5) and B is 5 (4.5 to 5.5), UB(A+B) = 10.5 + 5.5 = 16.
• Lower Bound of Sum (A + B): Use the lower bound of A (LB_A) + the lower bound of B (LB_B). LB(A+B) = 9.5 + 4.5 = 14.
• Upper Bound of Difference (A - B): Use the upper bound of A (UB_A) - the lower bound of B (LB_B). This maximises the result. UB(A-B) = 10.5 - 4.5 = 6.
• Lower Bound of Difference (A - B): Use the lower bound of A (LB_A) - the upper bound of B (UB_B). This minimises the result. LB(A-B) = 9.5 - 5.5 = 4.

Calculations Involving Multiplication and Division

For products and quotients, careful selection of bounds is critical:

• Upper Bound of Product (A × B): Use UB_A × UB_B. For example, if A is 10 (9.5 to 10.5) and B is 5 (4.5 to 5.5), UB(A×B) = 10.5 × 5.5 = 57.75.
• Lower Bound of Product (A × B): Use LB_A × LB_B. LB(A×B) = 9.5 × 4.5 = 42.75.
• Upper Bound of Quotient (A / B): Use UB_A / LB_B. This maximises the result by using the largest possible numerator and smallest possible denominator. UB(A/B) = 10.5 / 4.5 = 2.333... .
• Lower Bound of Quotient (A / B): Use LB_A / UB_B. This minimises the result by using the smallest possible numerator and largest possible denominator. LB(A/B) = 9.5 / 5.5 = 1.727... .

Applying Bounds to Rates of Change

Rates of change often involve division, such as speed (distance/time) or density (mass/volume). Use the rules for division:

• To find the maximum possible speed, use the upper bound of the distance and the lower bound of the time. (Speed_max = UB_Distance / LB_Time).
• To find the minimum possible speed, use the lower bound of the distance and the upper bound of the time. (Speed_min = LB_Distance / UB_Time).

Similar logic applies to other rates like density or pressure. Always consider what combination of bounds will give you the maximum or minimum possible outcome for the specific calculation.

Core method

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. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

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. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.