Real Numbers

14 Grade 10 Essential Mathematics questions on Real Numbers, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   3 marks

   One night the lowest temperatures recorded were −4 °C at Molo, 3 °C at Embu and −11 °C at a station on Mount Kenya.

   (a) Arrange the three temperatures in ascending order.

   (b) Calculate the difference between the highest and the lowest of the three temperatures.

   (c) Explain, referring to the number line, why −11 °C is lower than −4 °C even though 11 is greater than 4.

   (3 marks)

   Marking scheme

   (a) −11 °C, −4 °C, 3 °C (1 mark; all three in the correct order). (b) 3 − (−11) = 3 + 11 = 14 °C (1 mark). (c) On the number line −11 lies further to the left of zero than −4, and values decrease as you move to the left, so −11 < −4; the sign, not the size of the digits, fixes the position (1 mark).

2. Question 2

   1 mark

   At a weather station in Nyahururu the temperature at 6.00 a.m. was −6 °C. By midday it had risen by 11 °C. The midday temperature was:

   A. −17 °C

   B. −5 °C

   C. 5 °C

   D. 17 °C

   (1 mark)

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3. Question 3

   3 marks

   A fish trader at Dunga beach received 3$\frac{3}{4}$ crates of tilapia in the morning and sold 2$\frac{1}{3}$ crates by midday. Calculate the number of crates remaining, giving your answer as a mixed number.

   (3 marks)

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4. Question 4

   1 mark

   A shopkeeper writes four discount offers in different forms. Which one of the offers is the largest?

   A. 86% off

   B. $\frac{4}{5}off$

   C. $\frac{7}{8}off$

   D. 0.87 off

   (1 mark)

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5. Question 5

   1 mark

   Which one of the following fractions gives a terminating decimal?

   A. $\frac{1}{3}$

   B. $\frac{2}{7}$

   C. $\frac{5}{8}$

   D. $\frac{4}{11}$

   (1 mark)

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6. Question 6

   1 mark

   A lorry at a maize depot in Kitale carried 4.0672 tonnes of grain. Written correct to three significant figures, this mass is:

   A. 4.07 tonnes

   B. 4.067 tonnes

   C. 4.06 tonnes

   D. 4.00 tonnes

   (1 mark)

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7. Question 7

   1 mark

   Which one of the following numbers is irrational?

   A. $\sqrt{11}$

   B. $\sqrt{36}$

   C. $\frac{7}{3}$

   D. 0.125

   (1 mark)

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8. Question 8

   1 mark

   Between which two consecutive whole numbers does $\sqrt{30}$ lie?

   A. 5 and 6

   B. 6 and 7

   C. 3 and 4

   D. 4 and 5

   (1 mark)

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9. Question 9

   4 marks

   A school bursar buys three items costing Ksh 1 870, Ksh 3 240 and Ksh 985.

   (a) Estimate the total cost by rounding each amount to the nearest hundred shillings.

   (b) Calculate the exact total.

   (c) State the difference between the estimate and the exact total.

   (d) Suggest one reason why the bursar may prefer to use the estimate.

   (4 marks)

   Marking scheme

   (a) 1 900 + 3 200 + 1 000 = Ksh 6 100 (1 mark; all three roundings correct and added). (b) 1 870 + 3 240 + 985 = Ksh 6 095 (1 mark). (c) 6 100 − 6 095 = Ksh 5 (1 mark). (d) Any one reason (1 mark): it is quick to work out mentally; it checks whether the exact answer is reasonable; it helps him know in advance roughly how much money to carry.

10. Question 10

    2 marks

    Distinguish between a rational number and an irrational number, giving one example of each.

    (2 marks)

    Marking scheme

    Rational number: any number that can be written as a fraction a/b where a and b are integers and b ≠ 0; its decimal either terminates or recurs — e.g. $\frac{3}{4}$ (= 0.75) (1 mark for description with a correct example). Irrational number: a number that cannot be written as a fraction of two integers; its decimal neither terminates nor recurs — e.g. $\sqrt{5}$ or π (1 mark for description with a correct example). Accept any correct example.

11. Question 11

    1 mark

    Evaluate $20-12÷4\times 2+5$.

    A. 19

    B. 31

    C. 9

    D. 14

    (1 mark)

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12. Question 12

    6 marks

    A boda boda rider earned Ksh 1 450 in one day. He spent $\frac{2}{5}$ of the earnings on fuel and $\frac{1}{4}$ of the remainder on lunch, and saved what was left.

    (a) Calculate the amount spent on fuel.

    (b) Calculate the amount remaining after buying fuel.

    (c) Calculate the amount spent on lunch.

    (d) Calculate the amount saved.

    (e) Express the savings as a percentage of the day's earnings.

    (f) Find his total savings if he saves the same amount on each of 6 days.

    (6 marks)

    Marking scheme

    (a) $\frac{2}{5}\times 1450=Ksh$ 580 (1 mark). (b) 1 450 − 580 = Ksh 870 (1 mark). (c) $\frac{1}{4}\times 870=Ksh$ 217.50 (1 mark). (d) 870 − 217.50 = Ksh 652.50 (1 mark). (e) $(652.50÷1450)\times 100=45$% (1 mark). (f) $6\times 652.50=Ksh$ 3 915 (1 mark). Follow through on earlier errors where the method is correct.

13. Question 13

    4 marks

    A shopkeeper in Machakos weighs sugar to the nearest 0.1 kg and labels a packet 3.5 kg.

    (a) State the least possible mass of the packet.

    (b) State the greatest possible mass of the packet.

    (c) Explain why the total mass of 10 such packets may be up to 0.5 kg more than 35 kg.

    (4 marks)

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14. Question 14

    3 marks

    Express the recurring decimal 0.272727… as a fraction in its simplest form, showing your working.

    (3 marks)

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