Sheet № 07 · Higher only · AQA · Edexcel · OCR
Bounds and Error Intervals –
Bounds and error intervals are a Higher tier topic that tests your understanding of accuracy and measurement. When a number has been rounded, the true value could lie anywhere within a range — and exam questions ask you to find that range, then use it in calculations. This topic links closely to rounding, significant figures, and real-wor
§Key definitions
Truncation
means cutting off digits without rounding. If 7.86 is truncated to 1 decimal place, it becomes 7.8 (not 7.9). The error interval for a truncated value is different:
Question:
A field is 120 m long, measured to the nearest 10 m. Write down the error interval for the length.
Answer:
115 ≤ length < 125
Q1:
A mass is given as 250 g, rounded to the nearest 10 g. Write the error interval.
Q2:
A number is truncated to 2 decimal places, giving 4.37. Write the error interval.
§Formulas to memorise
For a value x rounded to a given degree of accuracy: lower bound ≤ x < upper bound
For a value x truncated to 1 d.p.: truncated value ≤ x < truncated value + 0.1
Maximum value of A + B = upper bound of A + upper bound of B
Minimum value of A + B = lower bound of A + lower bound of B
Maximum value of A − B = upper bound of A − lower bound of B
Minimum value of A − B = lower bound of A − upper bound of B
Maximum value of A × B = upper bound of A × upper bound of B
Minimum value of A × B = lower bound of A × lower bound of B
Maximum value of A ÷ B = upper bound of A ÷ lower bound of B
Minimum value of A ÷ B = lower bound of A ÷ upper bound of B
Worked example
A field is 120 m long, measured to the nearest 10 m. Write down the error interval for the length.
Working:
⚠ Common mistakes
- ✗Using a strict inequality at the lower bound. For rounding, the lower bound IS included (≤), and the upper bound is NOT included (<). Do not write < on both sides.
- ✗Confusing rounding and truncation error intervals. For rounding, the error is ± half the unit. For truncation, the value can only be equal to or larger than the truncated value, up to one full unit above.
- ✗Using the wrong combination of bounds for division. To find the maximum of A ÷ B, you need the upper bound of A divided by the lower bound of B, not upper ÷ upper.
- ✗Forgetting to state the suitable degree of accuracy. If the question asks for it, you must include a justification — show that both bounds round to the same value at that degree of accuracy.
- ✗Rounding bounds themselves. Bounds should be exact — do not round 8.35 to 8.4; that defeats the purpose.
✦ Exam tips
- →Draw a table of upper and lower bounds before you start the calculation. This keeps your working organised and reduces errors.
- →Remember the subtraction and division rules by thinking about extremes. The biggest difference comes from a big number minus a small number. The biggest quotient comes from a big number divided by a small number.
- →On calculator papers, keep full precision in your working and only round at the very end.
Bonus
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