Number: Standard Form and Rounding Worksheet with Answers

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KS3 · Number

KS3.M-N3 Number: Standard Form and Rounding

AQA AQA KS3 · Calculator allowed · about 80 minutes

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

Round the number 3847 to:

(a)the nearest 10(1)

(b)the nearest 100(1)

(c)the nearest 1000(1)

2

The number 5.6789 is given.

(a)Round 5.6789 to 1 decimal place.(1)

(b)Round 5.6789 to 2 decimal places.(1)

3

The number 0.04563 is given. Round it to:

(a)1 significant figure(1)

(b)2 significant figures(1)

(c)3 significant figures(1)

4

By rounding each number to 1 significant figure, estimate the value of (398 x 5.2) / 19.8. You must show your rounded values.

(2)

5

Write these numbers in standard form:

(a)480000(1)

(b)0.00072(1)

(c)6030000(1)

6

Write these standard form numbers as ordinary numbers:

(a)3.4 x 10⁴(1)

(b)9.05 x 10^-3(1)

(c)1.2 x 10⁷(1)

7

Standard form must have a single digit from 1 to 9 before the decimal point.

(a)Which of the following correctly represents 4500 in standard form?(1)

A) 45 x 10²
B) 4.5 x 10³
C) 0.45 x 10⁴
D) 4.5 x 10²

(b)Rewrite 45 x 10² correctly in standard form.(1)

8

Calculate the following, giving each answer in standard form.

(a)(2.5 x 10³) x (3 x 10⁴)(2)

(b)(6 x 10⁵) x (4 x 10^-2)(2)

9

Calculate the following, giving each answer in standard form.

(a)(8 x 10⁷) / (2 x 10³)(2)

(b)(9 x 10^-2) / (3 x 10⁵)(2)

10

In 2024 the population of the UK was approximately 68500000.

(a)Write this population in standard form.(2)

(b)The population of a small town is 3.24 x 10⁴. Round this population to the nearest thousand, giving your answer as an ordinary number.(2)

11

Priya buys three items costing 4.65 pounds, 12.20 pounds and 7.85 pounds.

(a)Work out the total cost of the three items.(1)

(b)Round the total cost to the nearest pound.(1)

(c)Using the exact total from part a, work out how much change Priya receives from a 30 pounds note.(1)

12

A school trip needs tickets for 187 students at 8.95 pounds each, plus a coach costing 396 pounds. By rounding each value to 1 significant figure, estimate the total cost of the trip. You must show your rounded values.

(3)

13

Work out (3.2 x 10⁵) + (4.5 x 10⁴). Give your answer in standard form.

(3)

14

Work out (7.8 x 10⁶) - (2.5 x 10⁵). Give your answer in standard form.

(3)

15

Write these numbers in order, starting with the smallest: 3.6 x 10⁴, 9.1 x 10³, 4.2 x 10^-2, 3.6 x 10^-3

(3)

16

Consider the number 6.478.

(a)Round 6.478 to 2 decimal places.(1)

(b)The number 6.478 is truncated (not rounded) to 2 decimal places. State the truncated value and explain why it is different from your answer to part a.(2)

17

A bacterium has a width of 0.0000025 metres.

(a)Write this width in standard form.(2)

(b)4000 of these bacteria are placed end to end. Calculate the total length in metres, giving your answer in standard form.(2)

18

Light travels at a speed of 3 x 10⁸ metres per second.

(a)Calculate how far light travels in 60 seconds. Give your answer in standard form.(2)

(b)A spacecraft is 9 x 10¹⁰ metres from Earth. Work out how many seconds it would take light to travel this distance. Give your answer as an ordinary number.(3)

19

A number, when rounded to the nearest 100, is 700.

(a)Write down the smallest number it could have been.(2)

(b)Assuming the original number is a whole number, write down the largest number it could have been.(2)

20

In 2023, a company's annual revenue was 4.6 x 10⁹ pounds and its costs were 3.85 x 10⁹ pounds.

(a)Calculate the company's profit. Give your answer in standard form.(2)

(b)The company has 15000 employees. Work out the profit per employee, giving your answer as an ordinary number.(2)

(c)Round the total revenue, 4.6 x 10⁹ pounds, to 1 significant figure and explain what this rounded value tells you about the size of the company.(2)

21

The diameter of a hydrogen atom is about 1 x 10^-10 metres. The diameter of the Milky Way galaxy is about 9.5 x 10²⁰ metres.

(a)Work out how many times bigger the diameter of the galaxy is than the diameter of the atom. Give your answer in standard form.(3)

(b)Round your answer to part a to 1 significant figure and comment on whether this level of precision is reasonable, given that the original diameters were only given to 2 significant figures.(2)

22

Amara estimates the value of (612 x 0.48) / 0.099 by rounding each number to 1 significant figure. Find Amara's estimate. You must show your rounded values.

(3)

Mark scheme · KS3.M-N3 Number: Standard Form and Rounding

Question 1

(a) B1 3850 cao
(a) Answer: 3850
(b) B1 3800 cao
(b) Answer: 3800
(c) B1 4000 cao
(c) Answer: 4000

Question 2

(a) B1 5.7 cao
(a) Answer: 5.7
(b) B1 5.68 cao
(b) Answer: 5.68

Question 3

(a) B1 0.05 cao
(a) Answer: 0.05
(b) B1 0.046 cao
(b) Answer: 0.046
(c) B1 0.0456 cao
(c) Answer: 0.0456

Question 4

M1 correctly rounds all three values to 1 sf: 400, 5, 20 (oe)
A1 100 cao, from (400 x 5) / 20
Answer: 100

Question 5

(a) B1 4.8 x 10⁵ cao
(a) Answer: 4.8 x 10⁵
(b) B1 7.2 x 10^-4 cao
(b) Answer: 7.2 x 10^-4
(c) B1 6.03 x 10⁶ cao
(c) Answer: 6.03 x 10⁶

Question 6

(a) B1 34000 cao
(a) Answer: 34000
(b) B1 0.00905 cao
(b) Answer: 0.00905
(c) B1 12000000 cao
(c) Answer: 12000000

Question 7

(a) B1 selects B
(a) Answer: B (4.5 x 10³)
(b) B1 4.5 x 10³ cao
(b) Answer: 4.5 x 10³

Question 8

(a) M1 multiplies coefficients and adds indices: 2.5 x 3 = 7.5, 10³⁺⁴
(a) A1 7.5 x 10⁷ cao
(a) Answer: 7.5 x 10⁷
(b) M1 multiplies coefficients and adds indices: 6 x 4 = 24, 10^5+(-2)
(b) A1 2.4 x 10⁴ cao (from 24 x 10³ converted to standard form)
(b) Answer: 2.4 x 10⁴

Question 9

(a) M1 divides coefficients and subtracts indices: 8/2 = 4, 10^7-3
(a) A1 4 x 10⁴ cao
(a) Answer: 4 x 10⁴
(b) M1 divides coefficients and subtracts indices: 9/3 = 3, 10^-2-5
(b) A1 3 x 10^-7 cao
(b) Answer: 3 x 10^-7

Question 10

(a) M1 coefficient 6.85 identified
(a) A1 6.85 x 10⁷ cao
(a) Answer: 6.85 x 10⁷
(b) M1 converts to ordinary number 32400 (oe)
(b) A1 32000 cao
(b) Answer: 32000

Question 11

(a) B1 24.70 pounds cao
(a) Answer: 24.70 pounds
(b) B1 25 pounds cao ft from part a
(b) Answer: 25 pounds
(c) B1 5.30 pounds cao ft from part a
(c) Answer: 5.30 pounds

Question 12

M1 rounds all three values correctly to 1 sf: 200, 9, 400 (oe)
M1 correct method: (200 x 9) + 400
A1 2200 pounds cao
Answer: 2200 pounds

Question 13

M1 converts both numbers to the same power of 10 or to ordinary numbers: 320000 and 45000 (oe)
M1 adds correctly: 320000 + 45000 = 365000 (oe)
A1 3.65 x 10⁵ cao
Answer: 3.65 x 10⁵

Question 14

M1 converts both numbers to the same power of 10 or to ordinary numbers: 7800000 and 250000 (oe)
M1 subtracts correctly: 7800000 - 250000 = 7550000 (oe)
A1 7.55 x 10⁶ cao
Answer: 7.55 x 10⁶

Question 15

M1 converts all four values to ordinary numbers or a comparable form (oe)
M1 correctly identifies the smallest and largest values
A1 fully correct order: 3.6 x 10^-3, 4.2 x 10^-2, 9.1 x 10³, 3.6 x 10⁴ cao
Answer: 3.6 x 10^-3, 4.2 x 10^-2, 9.1 x 10³, 3.6 x 10⁴

Question 16

(a) B1 6.48 cao
(a) Answer: 6.48
(b) B1 6.47 stated cao
(b) B1 correct explanation, e.g. truncation simply removes the extra digits without checking them, whereas rounding checks the next digit and rounds up if it is 5 or more (oe)
(b) Answer: 6.47; truncation cuts off the remaining digits without checking them, rounding checks the next digit

Question 17

(a) M1 coefficient 2.5 identified
(a) A1 2.5 x 10^-6 cao
(a) Answer: 2.5 x 10^-6
(b) M1 correct method: 4000 x 2.5 x 10^-6, e.g. (4 x 10³) x (2.5 x 10^-6)
(b) A1 1 x 10^-2 cao
(b) Answer: 1 x 10^-2 metres

Question 18

(a) M1 correct method: 3 x 10⁸ x 60 (oe)
(a) A1 1.8 x 10¹⁰ cao
(a) Answer: 1.8 x 10¹⁰ metres
(b) M1 correct method: time = distance / speed = (9 x 10¹⁰) / (3 x 10⁸)
(b) M1 correct standard form division: 3 x 10² (oe)
(b) A1 300 seconds cao
(b) Answer: 300 seconds

Question 19

(a) M1 identifies the halfway point between 600 and 700 as the lower boundary
(a) A1 650 cao
(a) Answer: 650
(b) M1 identifies the upper boundary as 750 (the midpoint between 700 and 800)
(b) A1 749 cao
(b) Answer: 749

Question 20

(a) M1 correct subtraction: 4.6 x 10⁹ - 3.85 x 10⁹ = 0.75 x 10⁹ (oe)
(a) A1 7.5 x 10⁸ cao
(a) Answer: 7.5 x 10⁸ pounds
(b) M1 correct division: (7.5 x 10⁸) / 15000 (ft from part a)
(b) A1 50000 pounds cao
(b) Answer: 50000 pounds
(c) B1 5 x 10⁹ cao
(c) B1 sensible explanation, e.g. the rounded value shows the revenue is approximately 5 billion pounds, giving a quick sense of the company's scale (oe)
(c) Answer: 5 x 10⁹ pounds; shows revenue is about 5 billion pounds

Question 21

(a) M1 correct division set up: (9.5 x 10²⁰) / (1 x 10^-10)
(a) M1 correct index subtraction: 20 - (-10) = 30 (oe)
(a) A1 9.5 x 10³⁰ cao
(a) Answer: 9.5 x 10³⁰
(b) B1 1 x 10³¹ cao ft from part a
(b) B1 sensible comment, e.g. since the original data only had 2 significant figures of accuracy, giving the answer to more figures would suggest a false level of precision, so rounding to 1 sf is reasonable (oe)
(b) Answer: 1 x 10³¹; reasonable, since the original data was only accurate to 2 sf

Question 22

M1 rounds all three values correctly to 1 sf: 600, 0.5, 0.1 (oe)
M1 correct order of operations: (600 x 0.5) / 0.1
A1 3000 cao
Answer: 3000

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