Goal and prerequisites
Find the tight interval consistent with rounded measurements and use it in a derived quantity. Review number structure before starting.
Method and its conditions
Half the rounding unit lies on either side of the recorded value. This module uses round-half-up for positive measurements: use a closed lower endpoint and an open upper endpoint. For a positive quotient whose denominator has a strictly positive lower bound, increase its numerator and decrease its denominator to maximize it; reverse these choices to minimize it.
Worked example: show the decisions
A length recorded as 12 cm to the nearest centimetre satisfies 11.5≤L<12.5. A time recorded as 4 s to the nearest second satisfies 3.5≤t<4.5. The greatest possible limiting speed is 12.5/3.5=25/7 cm/s. It is an upper bound, not an attained maximum under these intervals.
Contrast: diagnose an incorrect approach
Using both upper endpoints in L/t does not maximize the quotient. A larger denominator makes a positive quotient smaller. For a difference A−B, its upper bound uses the upper bound of A and lower bound of B. Keep the rounding unit separate from a percentage error.
Transfer: try a changed situation
Distance is 20 m to the nearest metre and time is 5 s to the nearest second. Find the lower bound of speed and explain the denominator choice.
Transfer solution and interpretation
The lower bound is 19.5/5.5=39/11 m/s. The smallest distance and largest possible limiting time give the smallest limiting quotient. Because the upper time endpoint is excluded, this bound is not attained.
A length is recorded as 3 cm to the nearest centimetre. What is its upper bound, in cm? Use round-half-up for these positive measurements.
- Add half the one-centimetre rounding unit; the upper endpoint is a limiting bound.
- Using the stated quantities: 3+1/2 = 7/2.
- Check the domain, units and whether the answer is an exact value, bound or estimate.
Compare the other answer choices
Treat the rounded measurement as exact.
Add a whole rounding unit instead of half a unit.
Return the lower bound instead of the upper bound.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
0580 syllabus for exams in 2025–2027