Number
Bounds & Accuracy
Year 10 · Year 11
- ✓By the end of this lesson students will be able to round numbers to a specified number of decimal places or significant figures.
- ✓By the end of this lesson students will be able to determine the upper and lower bounds of a rounded number.
- ✓By the end of this lesson students will be able to calculate the upper and lower bounds for calculations involving rounded numbers.
- ✓By the end of this lesson students will be able to express the range of possible values for a measurement using error intervals.
Key concepts
Rounding to a given number of decimal places involves looking at the digit immediately to the right of the last required decimal place. If this digit is 5 or more, round up the last required digit. If it is less than 5, keep the last required digit as it is. All digits after the last required decimal place are removed.
Significant figures are digits in a number that carry meaning and contribute to its precision. To round to a given number of significant figures: 1. The first non-zero digit is the first significant figure. 2. Zeros between non-zero digits are significant. 3. Trailing zeros in a whole number are not significant unless indicated (e.g., by a decimal point). 4. Trailing zeros after a decimal point are significant. 5. Leading zeros (before the first non-zero digit) are not significant. Once the significant figures are identified, apply the rounding rule: if the digit immediately after the last required significant figure is 5 or more, round up; otherwise, keep it the same. Replace any digits to the right of the last significant figure with zeros if they are before the decimal point, or remove them if they are after the decimal point.
The upper bound (UB) and lower bound (LB) of a number represent the maximum and minimum possible values that the original unrounded number could have taken. To find the bounds, first identify the degree of accuracy (the unit to which the number has been rounded). The bounds are then found by adding and subtracting half of this unit of accuracy from the rounded number. For example, if a number is rounded to the nearest whole number (unit of 1), the bounds are ± 0.5. If rounded to 1 decimal place (unit of 0.1), the bounds are ± 0.05.
An error interval expresses the range of possible values for an unrounded number, x, given its rounded value. It is written using inequalities, where the lower bound (LB) is less than or equal to x, and x is strictly less than the upper bound (UB). This is because the lower bound itself would round up to the given value, but the upper bound would round to the next value. The standard notation is LB ≤ x < UB.
Key facts to remember
- 1When rounding, if the next digit is 5 or more, round up; otherwise, keep it the same.
- 2Significant figures start from the first non-zero digit.
- 3To find the upper and lower bounds of a rounded number, add and subtract half of the unit of accuracy from the rounded value.
- 4The error interval for a value 'x' rounded to a certain degree of accuracy is written as LB ≤ x < UB.
- 5For addition (A + B): LB = LB_A + LB_B, UB = UB_A + UB_B.
- 6For subtraction (A - B): LB = LB_A - UB_B, UB = UB_A - LB_B.
- 7For multiplication (A × B, where A, B > 0): LB = LB_A × LB_B, UB = UB_A × UB_B.
- 8For division (A ÷ B, where A, B > 0): LB = LB_A ÷ UB_B, UB = UB_A ÷ LB_B.
Worked examples
Example 1
Round the number 34.567 to: a) 2 decimal places b) 2 significant figures
Answer
a) 34.57 b) 35
Example 2
A length is measured as 150 cm to the nearest 10 cm. Find its error interval.
Answer
145 cm ≤ length < 155 cm
Remember the strict inequality (<) for the upper bound in error intervals.
Example 3
A rectangle has a length of 8.6 cm (to 1 d.p.) and a width of 3.5 cm (to 1 d.p.). Calculate the upper and lower bounds for its area.
Answer
Lower bound for area = 29.4975 cm² Upper bound for area = 30.7175 cm²
For multiplication, the lower bound of the product is the product of the lower bounds, and the upper bound of the product is the product of the upper bounds (assuming positive values).
Common mistakes
- ✗Incorrectly identifying the unit of accuracy (e.g., for 'nearest 10', using 1 instead of 10).
- ✗Using '≤' for both ends of an error interval (e.g., LB ≤ x ≤ UB instead of LB ≤ x < UB).
- ✗Rounding intermediate calculations before finding the final bounds, which can lead to inaccuracies.
- ✗Confusing which bounds to use for calculations, especially division and subtraction (e.g., for max A/B, using UB_A / UB_B instead of UB_A / LB_B).
- ✗Incorrectly counting significant figures, particularly with zeros (e.g., 0.005 is 1 s.f., not 3 s.f.).
Exam tips
- ★Always clearly state the unit of accuracy and half the unit of accuracy for each number involved in a bounds problem.
- ★For calculations involving bounds, always determine which combination of individual bounds will give the minimum and maximum possible result. Think logically: to get the largest answer, you generally want the largest numerator and smallest denominator (for division), or largest values added/multiplied.
- ★Write down the error interval for each rounded number before performing any calculations to avoid errors.
- ★Do not round your final answer until the very last step, and only if specified in the question. Work with the full precision of the bounds throughout.
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