Calculating with Bounds in Compound Measures
Combine bounds in a calculation by picking the values that give the greatest and least results, including for speed, density and other compound measures.
What a learner can do afterwards
- Pick which bounds give the greatest result of a division
- Find the range of possible speeds from a bounded distance and time
- Give a final answer to an accuracy the bounds actually support
1 · Read
Speed divides distance by time, and rounded inputs make the answer a range for you. For the greatest speed, pair the longest distance with the shortest time: upper over lower. The smallest speed flips the pair: lower over upper. With 100 m (95 to 105) and 8 s (7.5 to 8.5), the greatest speed is 105 divided by 7.5 = 14.
Distance 250 m to the nearest 10 m spans 245 to 255 m, and time 12 s to the nearest second spans 11.5 to 12.5 s. The greatest speed is 255 divided by 11.5: longest distance, shortest time. State which bounds you used every time.
Products maximise differently: for a maximum, take upper times upper. Length 6 cm (5.5 to 6.5) by width 4 cm (3.5 to 4.5) gives at most 6.5 times 4.5 = 29.25. Density is mass over volume, so maximum density needs the heaviest mass in the smallest volume: 405 divided by 45 = 9 g per cm cubed. Pressure works the same way: maximum pressure is upper force over lower area.
Quote only the accuracy the bounds allow. Inputs to 2 figures earn about 2 figures back, never a five-decimal sprawl.
Max divides upper by lower, max multiplies upper by upper, then round to what the inputs support.
More to read
2.3 Time, Velocity, and Speed
Speed divides distance by time, and with rounded inputs the answer is a range. The greatest speed pairs the longest distance with the shortest time: upper over lower. The smallest flips the pair.
For 100 m in 8 s, bounds give at most 105 over 7.5 = 14 m/s. Writing 14 rather than a long decimal respects what the bounds support. State which bounds you used every time.
11.2 Density
Density is mass over volume, another division of two measurements. Maximum density needs the heaviest mass in the smallest volume. With 400 g and 50 cm cubed (each to the nearest 10), that is 405 over 45 = 9 g per cm cubed.
Units carry through the division, so keep them attached. Grams over cm cubed gives density units directly. Dropping units is how right numbers earn wrong marks.
C | Units and Conversion Factors
Compound measures multiply or divide base units, so bounds must follow the operation. For a maximum of a product like area, take upper times upper: 6.5 times 4.5 = 29.25. For a maximum of a division, take upper over lower.
Final answers deserve only the accuracy the bounds allow. If inputs are to 2 figures, quote the result to 2 figures, never 5.
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2 · Watch
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Where it sits
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Where this leads
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17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.