Accuracy, Bounds& Error Intervals.

How do you find upper and lower bounds?

Whenever a number is rounded, the true value could be slightly higher or lower than what is stated. The lower bound is the smallest value that would round to the given number, and the upper bound is the smallest value that would round up to the next level.

The method is straightforward: identify the degree of accuracy (e.g. nearest 10, 1 d.p., 2 s.f.) and then apply the half-precision rule:

• Lower bound = stated value − half the degree of accuracy
• Upper bound = stated value + half the degree of accuracy

For example, a mass of 74 kg rounded to the nearest integer has a degree of accuracy of 1 kg. Half of 1 is 0.5, so the bounds are 73.5 ≤ m < 74.5. For a length of 3.60 m given to 2 decimal places, the degree of accuracy is 0.01 m, giving bounds 3.595 ≤ l < 3.605.

When the value is rounded to significant figures, identify the place value of the last significant figure. For instance, 4800 to 2 s.f. has its last s.f. in the hundreds column (degree of accuracy = 100), so bounds are 4750 ≤ x < 4850.

What are error intervals and how do you write them?

An error interval is a concise way of expressing the range of possible values using inequality notation. For a rounded value, the error interval takes the form:

LB ≤ x < UB

Note the strict inequality (<) on the upper bound side. The lower bound is included (because it would round to the stated value) but the upper bound is excluded (because it would round up to the next value).

Truncation vs rounding

When a number is truncated rather than rounded, digits are simply removed without rounding up. For a value truncated to 7.3 (1 d.p.), the error interval is 7.3 ≤ x < 7.4. The lower bound equals the stated value itself (not half a unit below), because truncation always rounds down. Recognising whether a question says "rounded" or "truncated" is essential to writing the correct error interval.

How do you calculate with bounds?

IGCSE Higher questions frequently ask you to find the maximum or minimum of a calculation involving rounded values. The key rules are:

• To maximise a sum: UB + UB
• To minimise a sum: LB + LB
• To maximise a difference: UB − LB
• To minimise a difference: LB − UB
• To maximise a product: UB × UB
• To minimise a product: LB × LB
• To maximise a quotient: UB &div; LB
• To minimise a quotient: LB &div; UB

A useful way to remember this for fractions or speed/density formulas: to get the biggest answer, make the top as large as possible and the bottom as small as possible. To get the smallest answer, do the opposite.

Worked Examples

Common Mistakes With Bounds

Exam Tips