OCR GCSE Maths · OCR 4

Approximation and Estimation

50 maths questions with worked answers with answers for OCR 4 Approximation and Estimation in OCR GCSE Maths (spec J560). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official OCR material.

All 50 questions and answers

Question 1 · Rounding · OCR 4.01a
Round 4.7362 to 2 decimal places.
Answer:
The third decimal digit is 6, so round up.
4.74.
Question 2 · Rounding · OCR 4.01a
Round 38 472 to 2 significant figures.
Answer:
The third significant digit is 4, so round down.
38 000.
Question 3 · Rounding · OCR 4.01a
Round 0.0506 to 1 significant figure.
Answer:
The first significant digit is 5; the next is 0.
0.05.
Question 4 · Rounding · OCR 4.01a
Round 0.08964 to 3 significant figures.
Answer:
The digits are 8, 9, 6 then 4.
0.0896.
Question 5 · Error intervals · OCR 4.01a
A length is 7.4 cm to 1 decimal place. Write its error interval.
Answer:
Half of 0.1 is 0.05.
7.35 ≤ l < 7.45.
Question 6 · Error intervals · OCR 4.01a
A mass is 600 g to the nearest 100 g. Write its error interval.
Answer:
Half of 100 is 50.
550 ≤ m < 650.
Question 7 · Truncation · OCR 4.01a
x = 5.8 truncated to 1 decimal place. Write the error interval for x.
Answer:
Truncation removes digits, so x could be up to but not including 5.9.
5.8 ≤ x < 5.9.
Question 8 · Error intervals · OCR 4.01a
n = 40 to 1 significant figure. Write its error interval.
Answer:
Half of 10 is 5.
35 ≤ n < 45.
Question 9 · Rounding · OCR 4.01a
Round 2.505 to the nearest whole number.
Answer:
The first decimal digit is 5, so round up.
3.
Question 10 · Rounding · OCR 4.01a
Give 7 ÷ 3 to 3 significant figures.
Answer:
7 ÷ 3 = 2.3333….
2.33.
Question 11 · Rounding · OCR 4.01a
Round 3 23 to 1 decimal place.
Answer:
323 = 3.666….
3.7.
Question 12 · Rounding · OCR 4.01a
Round 1259.6 to the nearest ten.
Answer:
The units digit is 9, so round up.
1260.
Question 13 · Rounding · OCR 4.01a
Round 48,763 to the nearest hundred.
Answer:
The tens digit is 6, so round the hundreds digit up.
48,763 rounds to 48,800.
Question 14 · Accuracy · OCR 4.01a
A length is 8 cm to the nearest centimetre. Give its error interval.
Answer:
Half the rounding unit is 0.5 cm.
8 − 0.5 ≤ length < 8 + 0.5.
7.5 cm ≤ length < 8.5 cm.
Question 15 · Accuracy · OCR 4.01a
A mass is 4.2 kg to the nearest 0.1 kg. Give its error interval.
Answer:
Half the rounding unit is 0.05 kg.
4.2 − 0.05 ≤ mass < 4.2 + 0.05.
4.15 kg ≤ mass < 4.25 kg.
Question 16 · Accuracy · OCR 4.01a
A time is 120 seconds to the nearest 10 seconds. Give its error interval.
Answer:
Half of 10 seconds is 5 seconds.
120 − 5 ≤ time < 120 + 5.
115 s ≤ time < 125 s.
Question 17 · Significant figures · OCR 4.01a
How many significant figures are in 0.03040?
Answer:
Leading zeros do not count; zeros between or after significant digits do.
The significant digits are 3, 0, 4, 0: four figures.
Question 18 · Context · OCR 4.01a
A coach seats 48 people. How many coaches are needed for 100 people?
Answer:
100 ÷ 48 is just over 2.
Two coaches hold only 96 people, so 3 coaches are needed.
Question 19 · Accuracy · OCR 4.01a
Why keep extra digits until the final step?
Answer:
Rounding an intermediate result changes the number used in later calculations.
Keep extra digits and round at the end to reduce accumulated rounding error.
Question 20 · Accuracy · OCR 4.01a
A number rounded to the nearest 10 is 70. Can its true value be 75?
Answer:
Values that round to 70 satisfy 65 ≤ value < 75.
75 rounds to 80, so it cannot be the true value.
Question 21 · Rounding · OCR 4.01a
Round 0.004 965 to 2 significant figures.
Answer:
The first significant figure is 4; the third is 6, so round up.
0.0050.
Question 22 · Error intervals · OCR 4.01a
Write the error interval for d = 3.5 to 1 decimal place using inequality notation.
Answer:
Half of 0.1 is 0.05.
3.45 ≤ d < 3.55.
Question 23 · Estimation · OCR 4.01b
Estimate 48.7 × 3.10.49.
Answer:
Round each number to 1 s.f.: 50 × 30.5.
1500.5 = 300.
Question 24 · Estimation · OCR 4.01b
Estimate 612 × 0.19.
Answer:
Round: 600 × 0.2.
120.
Question 25 · Estimation · OCR 4.01b
Estimate √(98.7).
Answer:
√100 = 10.
About 10 (slightly less).
Question 26 · Estimation · OCR 4.01b
Estimate the cost of 29 tickets at £11.95 each.
Answer:
Round: 30 × £12.
About £360.
Question 27 · Checking · OCR 4.01b
Is 3.8 × 21.2 = 805.6 reasonable? Explain.
Answer:
Estimate 4 × 20 = 80.
No; the answer should be about 80 (it is 80.56).
Question 28 · Estimation · OCR 4.01b
Estimate 7.9²2.1.
Answer:
Round: 8²2 = 642.
32.
Question 29 · Estimation · OCR 4.01b
Estimate the mean of 19, 21, 38 and 41.
Answer:
Round: 20, 20, 40 and 40, total 120.
120 ÷ 4 = 30.
Question 30 · Estimation · OCR 4.01b
Estimate 40119.8.
Answer:
Round: 40020.
20.
Question 31 · Estimation · OCR 4.01b
Estimate 19.8 × 4.9 by rounding each number to 1 significant figure.
Answer:
19.8 rounds to 20 and 4.9 rounds to 5.
Estimated product = 20 × 5 = 100.
Question 32 · Estimation · OCR 4.01b
Estimate (39.8 + 61.1) ÷ 5.1 using 1 significant figure for each number.
Answer:
Round to 1 significant figure: 39.8 ≈ 40, 61.1 ≈ 60 and 5.1 ≈ 5.
(40 + 60) ÷ 5 = 20.
Question 33 · Estimation · OCR 4.01b
Is 2,300 a reasonable answer to 4.1 × 5.9? Explain.
Answer:
Round to 1 significant figure: 4.1 ≈ 4 and 5.9 ≈ 6.
4 × 6 = 24, so 2,300 is far too large.
Question 34 · Estimation · OCR 4.01b
A bill is £19.80, £9.90 and £5.10. Estimate the total to the nearest pound for each.
Answer:
To the nearest pound: £19.80 ≈ £20, £9.90 ≈ £10 and £5.10 ≈ £5.
Estimated total = £20 + £10 + £5 = £35.
Question 35 · Accuracy · OCR 4.01b
Estimate the area of a 9.8 cm by 3.1 cm rectangle using 1 significant figure.
Answer:
9.8 cm ≈ 10 cm and 3.1 cm ≈ 3 cm.
Estimated area = 10 × 3 = 30 cm².
Question 36 · Estimation · OCR 4.01b
Estimate 62 × 0.489.7.
Answer:
Round: 60 × 0.510 = 3010.
3.
Question 37 · Estimation · OCR 4.01b
Use an estimate to decide whether 52 × 18 = 1236 could be correct.
Answer:
Estimate 50 × 20 = 1000.
1236 is too large (52 × 18 = 936), so it is wrong.
Question 38 · Bounds · OCR 4.01c · Higher only
A rectangle is 6 cm by 4 cm, each to the nearest cm. Find the upper bound of its area.
Answer:
Upper bounds: 6.5 cm and 4.5 cm.
6.5 × 4.5 = 29.25 cm².
Question 39 · Bounds · OCR 4.01c · Higher only
a = 12 to the nearest whole number and b = 3 to the nearest whole number. Find the upper bound of a ÷ b.
Answer:
Use the largest a and the smallest b: 12.5 ÷ 2.5.
5.
Question 40 · Bounds · OCR 4.01c · Higher only
A runner covers 100 m (to the nearest m) in 12.0 s (to 1 d.p.). Find the lower bound of the speed.
Answer:
Use the smallest distance and the largest time: 99.5 ÷ 12.05.
8.257… ≈ 8.26 m/s.
Question 41 · Bounds · OCR 4.01c · Higher only
x = 8.6 to 1 d.p. and y = 2.3 to 1 d.p. Find the lower bound of x − y.
Answer:
Use the smallest x and the largest y: 8.55 − 2.35.
6.2.
Question 42 · Bounds · OCR 4.01c · Higher only
Why should a final answer be given to a suitable degree of accuracy using bounds?
Answer:
Give the answer to the accuracy on which the upper and lower bounds agree.
For example, if bounds are 8.257 and 8.296, both round to 8.3 to 1 d.p.
Question 43 · Bounds · OCR 4.01c · Higher only
The side of a square is 5 cm to the nearest mm. Find the lower bound of its perimeter.
Answer:
Lower bound of side = 4.95 cm.
4 × 4.95 = 19.8 cm.
Question 44 · Bounds · OCR 4.01c · Higher only
A tank holds 200 litres to the nearest 10 litres. Buckets hold 5 litres to the nearest litre. What is the greatest number of full buckets the tank could fill?
Answer:
Use the largest tank and the smallest bucket: 205 ÷ 4.5 = 45.55….
45 full buckets.
Question 45 · Bounds · OCR 4.01c · Higher only
p = 4.2 to 1 d.p. and q = 0.6 to 1 d.p. Find the upper bound of p ÷ q.
Answer:
Use the largest p and the smallest q: 4.25 ÷ 0.55.
7.727… ≈ 7.73.
Question 46 · Accuracy · OCR 4.01c · Higher only
A length is 6.4 m to the nearest 0.1 m. State its lower bound.
Answer:
Half of 0.1 m is 0.05 m.
Lower bound = 6.4 − 0.05 = 6.35 m.
Question 47 · Accuracy · OCR 4.01c · Higher only
A mass is 9 kg to the nearest kilogram. State its upper bound.
Answer:
Half of 1 kg is 0.5 kg.
Upper bound = 9 + 0.5 = 9.5 kg; the true mass is below it.
Question 48 · Bounds · OCR 4.01c · Higher only
A rectangle has sides 5 cm and 8 cm, each rounded to the nearest cm. Find the lower bound for its area.
Answer:
Lower bounds of the positive sides are 4.5 cm and 7.5 cm.
Lower area bound = 4.5 × 7.5 = 33.75 cm².
Question 49 · Bounds · OCR 4.01c · Higher only
A distance is 120 m and a time is 20 s, both to the nearest unit. Find the upper bound for speed.
Answer:
Use the upper distance bound 120.5 m and lower time bound 19.5 s.
Upper speed bound = 120.519.5 = 24139 m/s ≈ 6.18 m/s.
Question 50 · Bounds · OCR 4.01c · Higher only
A length is 12.0 cm to the nearest 0.1 cm. State the interval for twice its length.
Answer:
The length satisfies 11.95 cm ≤ length < 12.05 cm.
Double both bounds: 23.9 cm ≤ twice the length < 24.1 cm.

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