Measures, estimation and bounds

Maths · Number. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Measurements are never exact, so this topic is about units and about how accurate a number is.

When you convert area or volume units, the conversion factor must be squared or cubed. When you estimate, round every number to 1 significant figure and then calculate.

A rounded value hides a range of possible true values, called the error interval. Its ends are the lower bound and the upper bound. For a rounded number they are half a unit below and half a unit above. For a truncated number (digits cut off, not rounded) the true value can only be the same or bigger. In a calculation, choose the bounds that make the answer as big, or as small, as possible.

Rules to remember

1 m = 100 cm, so 1 m² = 100² = 10 000 cm² and 1 m³ = 100³ = 1 000 000 cm³.
Time is not decimal: 1 hour = 60 minutes, so 0.1 hour = 6 minutes and 0.25 hour = 15 minutes.
Significant figures are counted from the first digit that is not zero: 0.004 72 to 2 significant figures is 0.0047.
Rounding: the bounds are half a unit each side. 7.4 to 1 decimal place gives 7.35 ≤ x < 7.45.
Truncating: 6.8 truncated to 1 decimal place gives 6.8 ≤ x < 6.9.
Largest value of a + b or a × b: use both upper bounds. Smallest: use both lower bounds.
Largest value of a − b or a ÷ b: upper bound of a with lower bound of b. Smallest: lower bound of a with upper bound of b.

Common mistake

Using both upper bounds to find the largest value of a division or a subtraction. To make a ÷ b or a − b as large as possible, make a as large as possible and b as small as possible.

Worked example

x = 9.6 correct to 1 decimal place and y = 4 correct to the nearest whole number. Find the upper bound of x − y.

Bounds of x: 9.55 and 9.65. Bounds of y: 3.5 and 4.5.
For the largest difference, take the largest x and the smallest y.
Upper bound of x − y = 9.65 − 3.5 = 6.15.

Answer: 6.15

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