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Limits Of Accuracy

Limits Of Accuracy

Bounds in calculations

Indirect Quesiton Extended 9 min read

On this page
  1. In this note
  2. Formula bank
  3. Biggest and smallest possible answers
  4. Differences and rearranged formulas
  5. Certain to fit; appropriate accuracy
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 1.10: bounds of calculations Extended
  • 1.10Find the bounds of a sum, difference, product and quotient of rounded values.
  • 1.10Use bounds with formulas: area, perimeter, speed, density, rearranged formulas.
  • 1.10Decide how many items are certain to fit; give an answer to an appropriate accuracy.

Before you start: upper and lower bounds.

Formula bank

Add / multiplyLearn itUB = UB + UB (or × ); LB = LB + LB (or ×)same direction for both.Use it when a sum or product (area, perimeter).
SubtractLearn itUB = UB\(_a\) − LB\(_b\); LB = LB\(_a\) − UB\(_b\)cross over: take away the least for the biggest answer.Use it when a difference.
DivideLearn itUB = UB\(_a\) ÷ LB\(_b\); LB = LB\(_a\) ÷ UB\(_b\)cross over: divide by the smallest for the biggest answer.Use it when speed, density, a rearranged formula.
Certain to fitLearn itworst case: LB capacity ÷ UB item, round DOWNUse it when "certain to carry".
Appropriate accuracyLearn itround LB and UB until they agreeUse it when "to an appropriate degree of accuracy".

Biggest and smallest possible answers

Each rounded input could be anywhere in its interval. For the biggest possible answer, pick for EACH input the bound that pushes the answer up. Subtraction and division cross over: taking away LESS leaves more, and dividing by a SMALLER number gives more (12 ÷ 2 = 6, but 12 ÷ 3 = 4).

General methodHow to find the bounds of any calculation

  1. Write the bounds of every rounded input (same units first).
  2. Decide: do you want the biggest (UB) or smallest (LB) answer?
  3. For each input pick the bound that pushes that way: + and × use the same bound; − and ÷ cross over for the second number.
  4. Calculate. Don't round the bound to the original accuracy.

Example A car travels 450 km (nearest 10 km) in 6.4 h (nearest 0.1 h). Lower bound of its average speed?

Slowest = shortest distance ÷ longest time.

  1. \(445\div6.45=68.99\ldots\)

Answer 68.99 km/h

Try it yourself

1
A rectangle is 8.4 cm by 5.2 cm, each to the nearest mm. Upper bound of its area?
Hint
8.45 × 5.25.
Show answer

Answer 44.3625

2
A square has side 5.1 cm (1 d.p.). Upper bound of its area?
Show answer

Answer 26.5225

3
Distance 16 km (nearest km), time 2.3 h (1 d.p.). Upper bound of the speed (3 d.p.)?
Hint
16.5 ÷ 2.25.
Show answer

Answer 7.333

Differences and rearranged formulas

Example A ribbon is 35 cm (nearest cm). A 128 mm piece (nearest mm) is cut off. Bounds of what is left, in mm?

Same units first, then cross over.

  1. 345 to 355 mm; 127.5 to 128.5 mm
  2. LB \(=345-128.5=216.5\)
  3. UB \(=355-127.5=227.5\)

Answer 216.5 mm and 227.5 mm

Example Area 96 m² (nearest m²), length 12 m (nearest m). Upper bound of the width?

width = area ÷ length: biggest area ÷ smallest length.

  1. \(96.5\div11.5=8.391\ldots\)

Answer 8.39 m

Try it yourself

1
\(P=\dfrac Wt\), \(W=600\) (nearest 10), \(t=48\) (nearest whole number). Upper bound of P (3 s.f.)?
Show answer

Answer 12.7

Certain to fit; appropriate accuracy

General methodHow to answer "how many is it certain to …"

  1. Use the worst case: the smallest capacity (LB) and the biggest item (UB).
  2. Divide, then round DOWN to a whole number.

Example A lift carries 1000 kg (nearest 10 kg). Boxes weigh 25 kg (nearest kg). How many boxes is it certain to carry?

Worst case: 995 ÷ 25.5, round down.

  1. \(995\div25.5=39.02\ldots\)

Answer 39 boxes

General methodHow to give an answer to an appropriate accuracy

  1. Work out the LB and the UB.
  2. Round both to 3 s.f., then 2 s.f., then 1 s.f. …
  3. The answer is the most accurate rounding where they AGREE.

Example \(x=\dfrac ab\), \(a=12.3\), \(b=1.7\) (both 1 d.p.). Give \(x\) to an appropriate accuracy.

Both bounds, then see where they agree.

  1. LB \(=12.25\div1.75=7\)
  2. UB \(=12.35\div1.65=7.48\ldots\)
  3. 2 s.f.: 7.0 and 7.5 disagree; 1 s.f.: both 7

Answer 7

Ways they test it

Type 1 Bound of a quotient

"Calculate the lower bound of the average speed." (3 marks)

Spot it speed, density, cost per item.

  1. Bounds of each input (B1).
  2. Correct combination, e.g. LB ÷ UB (M1).
  3. Answer (A1).

Type 2 Bound of a product or sum

"Find the upper bound of the area / perimeter."

Spot it area, perimeter, total.

  1. UB with UB (or LB with LB).

For example 2.4 cm by 1.7 cm (nearest mm), perimeter

  1. LB \(=2(2.35+1.65)=8\)
  2. UB \(=2(2.45+1.75)=8.4\)

Answer 8 cm and 8.4 cm

Type 3 Rearranged formula

"Area 96 m², length 12 m. Upper bound of the width."

Spot it a formula to rearrange first.

  1. Rearrange.
  2. Cross over for ÷.

Type 4 Certain to fit

"How many crates is the lift certain to carry?"

Spot it "certain", "guaranteed".

  1. LB capacity ÷ UB item.
  2. Round down.

Type 5 Appropriate accuracy

"Give x to an appropriate degree of accuracy. Give a reason." (4 marks)

Spot it "appropriate accuracy".

  1. Both bounds.
  2. Where they agree, with the reason.

Mistakes to avoid

  • ✗ UB of a ÷ b = UB ÷ UB✓ UB ÷ LBDivide by the smallest.
  • ✗ UB of a − b = UB − UB✓ UB − LBTake away the least.
  • ✗ Bounds found before changing units✓ same units first35 cm and 128 mm.
  • ✗ 39.02 boxes → 40✓ 39Certain to fit: round down.
  • ✗ Using the given values✓ use boundsGives 0 marks.

Summary

  • Bound every input first (same units).
  • + and ×: UB with UB, LB with LB.
  • − and ÷: cross over for the second number.
  • "Certain to fit": worst case, round down.
  • Appropriate accuracy: where both bounds agree.

If you see …

  • speed, density, ratea quotient: cross over.
  • "certain"worst case.

Practice

Practice: easy, medium, hard

Easy

1
9 tiles, each 4 cm to the nearest cm, in a row. Upper bound of the total length?
Show working
  1. \(9\times4.5\)

Answer 40.5

2
A card is 13 cm by 8 cm, each to the nearest cm. Upper bound of its area?
Show working
  1. \(13.5\times8.5\)

Answer 114.75

3
Same card: lower bound of its area?
Show working
  1. \(12.5\times7.5\)

Answer 93.75

Medium

1
Zain runs 400 m (nearest 10 m) in 62 s (nearest second). Upper bound of his speed (2 d.p.)?
Guide me

What is it asking? UB speed

What do we know? 405 m max, 61.5 s min

Plan UB ÷ LB

Show working
  1. \(405\div61.5=6.585\ldots\)

Answer 6.59

2
A car travels 450 km (nearest 10 km) in 6.4 h (nearest 0.1 h). Upper bound of the speed (2 d.p.)?
Guide me

What is it asking? UB speed

What do we know? 455 and 6.35

Plan UB distance ÷ LB time

Show working
  1. \(455\div6.35=71.653\ldots\)

Answer 71.65

3
Area 96 m² (nearest m²), length 12 m (nearest m). Upper bound of the width (3 s.f.)?
Guide me

What is it asking? UB width

What do we know? w = A ÷ l

Plan 96.5 ÷ 11.5

Show working
  1. \(8.391\ldots\)

Answer 8.39

Hard

1
A van carries 1000 kg (nearest 10 kg). Crates are 24 kg (nearest kg). Greatest number it is certain to carry?
Guide me

What is it asking? a whole number

What do we know? worst case

Plan 995 ÷ 24.5, round down

Show working
  1. \(995\div24.5=40.6\ldots\)

Answer 40

PartAnswer / workingMarksGuidance
995 and 24.5B1
\(995\div24.5\)M1
40A1
Total: 3 marks
2
A bag holds 85 g (nearest 5 g). Marbles are 7.5 g (1 d.p.). Largest number of marbles it could possibly hold?
Guide me

What is it asking? the most marbles possible

What do we know? best case this time

Plan UB bag ÷ LB marble, round down

Show working
  1. \(87.5\div7.45=11.7\ldots\)

Answer 11

PartAnswer / workingMarksGuidance
87.5 and 7.45B1
\(87.5\div7.45\)M1
11A1
Total: 3 marks
3
\(x=\dfrac ab\), \(a=12.3\), \(b=1.7\) (1 d.p.). Give x to an appropriate accuracy.
Guide me

What is it asking? x to a sensible accuracy

What do we know? LB 7, UB 7.48

Plan where do they agree?

Show working
  1. LB 7.0, UB 7.48
  2. agree to 1 s.f.: 7

Answer 7

PartAnswer / workingMarksGuidance
LB 7B1
UB 7.48…B1
7 with reasonB2
Total: 4 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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