NumberHigher focus★ Signature Higher
Bounds &
Error Intervals.
Every rounded measurement has a range it could have been. The lower bound and upper bound. To find max/min of calculations: think about whether you want the biggest possible or smallest possible inputs.
I · Key definitions
Lower bound
Smallest possible actual value. Half a unit below.
35 to nearest 1: LB = 34.5
Upper bound
Largest possible actual value. Half a unit above.
35 to nearest 1: UB = 35.5
Error interval
The range the value lies in. Write as LB ≤ x < UB.
34.5 ≤ x < 35.5
Precision
The unit the value is rounded to.
'nearest 10' → precision = 10
Max product
Use biggest values for each input.
Max area = UB × UB
Max quotient
Want biggest? Use UB on top, LB on bottom.
Max speed = UB-dist / LB-time
II · Finding bounds — the formulas
Bounds from rounded value
LB = value − ½ × precision
UB = value + ½ × precision
250 to nearest 10: LB=245, UB=255
Error interval notation
LB ≤ x < UB
(LB inclusive, UB exclusive)
8.3 to 1 d.p. → 8.25 ≤ x < 8.35
Add / multiply · max & min
Max: use UB of both
Min: use LB of both
Max(a + b) = UB(a) + UB(b)
Subtract / divide · max & min
Max(a − b) = UB(a) − LB(b)
Min(a − b) = LB(a) − UB(b)
Max(a ÷ b) = UB(a) ÷ LB(b)
Min(a ÷ b) = LB(a) ÷ UB(b)
opposites for − and ÷
III · Quick reference · bounds by rounding
Rounded to
Value
LB
UB
Nearest 10
250
245
255
Nearest 1
35
34.5
35.5
1 d.p.
8.3
8.25
8.35
2 d.p.
4.72
4.715
4.725
2 s.f.
5.3
5.25
5.35
Rule: bound = value ± ½ × precision. Precision is the rounding unit.
IV · Worked examples
State an error interval
2 marks
A length L = 4.7 cm, rounded to 1 d.p. Write the error interval for L.
i.Precision = 0.1 (1 d.p.). Half of that = 0.05
ii.LB = 4.7 − 0.05 = 4.65
iii.UB = 4.7 + 0.05 = 4.75
4.65 ≤ L < 4.75
A rectangle has sides length 8 cm and width 5 cm, both to the nearest cm. Find the max and min possible area.
i.Length: LB=7.5, UB=8.5. Width: LB=4.5, UB=5.5
ii.Max area = UB×UB = 8.5 × 5.5 = 46.75 cm²
iii.Min area = LB×LB = 7.5 × 4.5 = 33.75 cm²
Max: 46.75 cm² · Min: 33.75 cm²
Max speed (max quotient)
4 marks
Dist = 120 m (nearest 10 m). Time = 15 s (nearest s). Find max speed.
i.Dist: LB=115, UB=125. Time: LB=14.5, UB=15.5
ii.Max speed: want biggest dist / smallest time
iii.= UB(d) / LB(t) = 125 / 14.5 = 8.62 m/s
Max speed = 8.62 m/s (3 s.f.)
V · Common mistakes & examiner tips
Common mistakes
Using just ± half the value instead of half the precision.
Max quotient wrong: max doesn't mean UB/UB. It's UB/LB.
Forgetting the < at upper end. UB is exclusive: x < UB, not x ≤ UB.
Precision confusion. 'Nearest 10' precision is 10, not 1.
Calculating with the rounded value then stating bounds at the end.
Examiner tips
Always list LB and UB first before using them.
For max of a−b or a÷b: UB on top, LB on bottom.
For min of a−b or a÷b: LB on top, UB on bottom.
Write the interval in the form LB ≤ x < UB.
Sanity check: max > min always. If not, you've slipped.