Estimate the value of:
8.1 × 38.20.49
8.1 × 38.20.49
Worked Solution & Answer
Round each number to 1 significant figure first, then calculate.
8 × 400.5 = 640
8 × 400.5 = 640
Answer: 8 × 400.5 = 640
Estimation — Notes & Worked Examples Show / Hide
Explanation:
To estimate the answer to a calculation, round every number in the question to 1 significant figure (1 s.f.) first (for example, 48.7 becomes 50, and 0.48 becomes 0.5). Always write down the rounded calculation clearly to guarantee the method mark.
Remember that dividing by 0.5 is the same as multiplying by 2, and dividing by 0.2 is the same as multiplying by 5. An estimate is an overestimate if the numbers were rounded up (or the denominator was rounded down), and an underestimate if rounded down.
Example 1: Work out an estimate for 58.4 × 21.30.48
Solution:
Round each value to 1 s.f.:
58.4 → 60, 21.3 → 20, 0.48 → 0.5
Estimate = 60 × 200.5 = 12000.5
1200 ÷ 0.5 = 1200 × 2 = 2400
Answer: 2400
Example 2: A ticket costs £19.25. Estimate the cost of 392 tickets. Is your answer an overestimate or underestimate?
Solution:
Round to 1 s.f.: 19.25 → 20 and 392 → 400
Estimate = 20 × 400 = £8000
Both numbers were rounded up (20 > 19.25 and 400 > 392).
Answer: £8000 (Overestimate)
Estimate the value of:
√(48) × 31.2
√(48) × 31.2
Worked Solution & Answer
√(n) is the positive number that multiplies by itself to give n.
e.g. √(9) = 3 because 3 × 3 = 9
√(49) × 30 = 7 × 30 = 210
e.g. √(9) = 3 because 3 × 3 = 9
√(49) × 30 = 7 × 30 = 210
Answer: √(49) × 30 = 7 × 30 = 210
Work out
By rounding each number to one significant figure, work out an estimate for
31.4 × 48.7
Show your working clearly.
By rounding each number to one significant figure, work out an estimate for
31.4 × 48.7
Show your working clearly.
Worked Solution & Answer
(a) Work out an estimate for 92 × 1.63. (2) (b) Given 2.96 × 3.2 = 9.472, find 29.6 × 32. (1)
(a) Work out an estimate for the value of 92 × 1.63
You must show all your working. [2 marks]
You must show all your working. [2 marks]
(b) find the value of 29.6 × 32 [1 mark]
Worked Solution & Answer
(a)
90 × 2 = 180 M1 A1
Answer (a): 180
(b)
9.472 × 10 × 10 = 947.2 B1
Answer (b): 947.2
Sophie drives a distance of 513 kilometres on a motorway in France.
She pays 0.81 euros for every 10 kilometres she drives.
She pays 0.81 euros for every 10 kilometres she drives.
(a) Work out an estimate for the total amount that Sophie pays. [3 marks]
euros
(b) Is your answer to part (a) an underestimate or an overestimate?
Give a reason for your answer. [1 mark]
Give a reason for your answer. [1 mark]
Worked Solution & Answer
(a)
500 ÷ 10 × 0.8 = 50 × 0.8 = 40 euros M2 A1
Answer (a): 40 euros
(b)
500 < 513 and 0.8 < 0.81 so underestimate C1
Answer (b): Underestimate, because both 513 and 0.81 were rounded down (to 500 and 0.8)
A cone has a volume of 98 cm3.
The radius of the cone is 5.13 cm.
The radius of the cone is 5.13 cm.
(a) Work out an estimate for the height of the cone. [3 marks]
cm
(b) Will your estimate be more than John's answer or less than John's answer?
Give reasons for your answer. [1 mark]
Give reasons for your answer. [1 mark]
Worked Solution & Answer
(a)
Estimate V ≈ 100, r ≈ 5, π ≈ 3. 100 = (1/3) × 3 × 5² × h M1
⇒ 100 = 25h M1
⇒ h = 4 cm A1
⇒ 100 = 25h M1
⇒ h = 4 cm A1
Answer (a): 4 (accept 3.5 to 4.5) cm
(b)
Numerator increased and denominator decreased, so the estimate is larger C1
Answer (b): More than John's answer, because the numerator (98 -> 100) was rounded up and the denominator (π -> 3 and 5.13 -> 5) was rounded down.
Balena has a garden in the shape of a circle of radius 10 m.
He is going to cover the garden with grass seed to make a lawn.
Grass seed is sold in boxes.
Each box of grass seed will cover 46 m2 of garden.
He is going to cover the garden with grass seed to make a lawn.
Grass seed is sold in boxes.
Each box of grass seed will cover 46 m2 of garden.
Balena wants to cover all the garden with grass seed.
(a) Work out an estimate for the number of boxes of grass seed Balena needs.
You must show your working. [4 marks]
You must show your working. [4 marks]
(b) Is your estimate for part (a) an underestimate or an overestimate?
Give a reason for your answer. [1 mark]
Give a reason for your answer. [1 mark]
Worked Solution & Answer
(a)
Area = π × 10² ≈ 3 × 100 = 300 m² M3
Boxes ≈ 300 ÷ 50 = 6 (or 314 ÷ 45 ≈ 7) A1
Boxes ≈ 300 ÷ 50 = 6 (or 314 ÷ 45 ≈ 7) A1
Answer (a): 6 to 8
(b)
Underestimate because 3 < π and 50 > 46 C1
Answer (b): Underestimate, because 3 was used for π (less than 3.14...) and 50 was used for 46 (greater than 46), giving a smaller value
(a) Estimate (√401 + 1.9³) / cos 58.7° by rounding to 1 s.f. [3 marks]
13 (b) Identify two errors Sam has made in his cosine rule working. [2 marks]
13 (b) Identify two errors Sam has made in his cosine rule working. [2 marks]
(a) By rounding each number to one significant figure,
[INDENT] estimate the value of √(401) + 1.93\cos 58.7°
You **must** show your working. [3 marks]
[INDENT] estimate the value of √(401) + 1.93\cos 58.7°
You **must** show your working. [3 marks]
Answer
(b) Here is a triangle. [2 marks]
Worked Solution & Answer
(a)
(√400 + 2³) / cos 60° = (20 + 8) / 0.5 = 28 / 0.5 = 56 M2 A1
Answer (a): Answer 56
(b)
Error 1: Cosine rule uses –2bc cos A, not +2bc cos A M1
Error 2: 3² + 7² = 9 + 49 = 58 ≠ 10² (and 100 + 42 cos x ≠ 142 cos x) A1
Error 2: 3² + 7² = 9 + 49 = 58 ≠ 10² (and 100 + 42 cos x ≠ 142 cos x) A1
Answer (b): Error 1: It should be – 2 × 3 × 7 × cos x (minus sign in cosine rule). Error 2: 3² + 7² = 58, not 10² (or 100 + 42 cos x ≠ 142 cos x).
Berenika wants to buy 35 T-shirts.
Each T-shirt costs £5.80
Berenika does the calculation 40 × 6 = 240 to estimate the cost of 35 T-shirts.
Each T-shirt costs £5.80
Berenika does the calculation 40 × 6 = 240 to estimate the cost of 35 T-shirts.
(a) Explain how Berenika’s calculation shows the actual cost will be less than £240 [1 mark]
(b) Work out the actual cost of buying 35 T-shirts using the special offer. [4 marks]
£
Worked Solution & Answer
(a)
She rounded both numbers up (35 to 40 and 5.80 to 6), so her estimate is an overestimate
Answer (a): She rounded both numbers up (35 to 40 and 5.80 to 6), so her estimate is an overestimate
(b)
35 × £5.80 = £203 M2
10% of £203 = £20.30 M1
£203 – £20.30 = £182.70 A1
10% of £203 = £20.30 M1
£203 – £20.30 = £182.70 A1
Answer (b): £ 182.70
A total of 700 tickets were on sale for a football match.
452 of the tickets were sold.
452 of the tickets were sold.
(a) How many tickets were **not** sold? [2 marks]
(b) Work out an estimate for the total amount of money paid for these tickets.
You must show all your working. [3 marks]
You must show all your working. [3 marks]
£
(c) Is your answer to part (b) an underestimate or an overestimate?
Give a reason for your answer. [1 mark]
Give a reason for your answer. [1 mark]
Worked Solution & Answer
(a)
700 – 452 = 248 M1 A1
Answer (a): 248
(b)
300 × £10 + 400 × £20 = £3000 + £8000 = £11 000 M2 A1
Answer (b): £ 11 000
(c)
Overestimate, because all four numbers were rounded up
Answer (c): Overestimate, because all four numbers were rounded up