Calculations with bounds
Objective. Find bounds for sums, products and quotients.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Bounds calculations use the upper and lower bounds of rounded quantities to find the range of a result from a sum, product or quotient. To get the maximum possible sum, add the two upper bounds; for the minimum, add the two lower bounds. For a quotient such as distance ÷ time, the maximum value comes from the largest numerator divided by the smallest denominator, and the minimum from the smallest numerator divided by the largest denominator. For example, if x has bounds 4.5 ≤ x < 5.5 and y has bounds 1.5 ≤ y < 2.5, the maximum of x ÷ y is 5.5 ÷ 1.5. Mixing up which bound goes where is the main danger in this topic.
Worked example
Speed is calculated from distance 120 m (to nearest 10 m) and time 15 s (to nearest 1 s). Find the maximum possible speed using bounds.
- 1Distance bounds: 115 ≤ d < 125.
- 2Time bounds: 14.5 ≤ t < 15.5.
- 3Maximum speed uses largest distance and smallest time: 125 ÷ 14.5.
- 4125 ÷ 14.5 = 8.62... m/s (3 s.f.).
Answer: 8.62 m/s (3 s.f.)
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Practice
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Start practisingCommon mistake. Pupils often divide upper by upper to find a maximum quotient, giving too small a result. For division, always pair the largest numerator with the smallest denominator to maximise the answer.
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