Real Numbers

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

1. Question 1

   1 mark

   The length of a tea plot in Kericho is 46.5 m and its width is 12.4 m. The perimeter, correct to 3 significant figures, is:

   A. 118 m

   B. 117 m

   C. 120 m

   D. 117.8 m

   (1 mark)

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2. Question 2

   1 mark

   A water tank in Machakos holds 0.375 of its capacity. Expressed as a fraction in its simplest form, 0.375 is:

   A. $\frac{37}{50}$

   B. $\frac{3}{80}$

   C. $\frac{3}{8}$

   D. $\frac{375}{100}$

   (1 mark)

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3. Question 3

   1 mark

   A maize farmer harvests 8 400 kg of maize. Written in standard form, this mass is:

   A. $84\times 10^2$ kg

   B. $0.84\times 10^4$ kg

   C. $8.4\times 10^3$ kg

   D. $8.4\times 10^4$ kg

   (1 mark)

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4. Question 4

   4 marks

   A rectangular kitchen garden measures 8.4 m by 5.6 m, each measurement correct to 1 decimal place. State the lower and upper limits of the length, and hence, using the greatest possible width, calculate the maximum possible area of the garden.

   (4 marks)

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5. Question 5

   4 marks

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

   (4 marks)

   Marking scheme

   1 mark: a rational number can be written as a fraction a/b where a and b are integers and b ≠ 0. 1 mark: an irrational number cannot be written in that form (its decimal expansion neither terminates nor recurs). 1 mark: correct example of a rational number, e.g. $\frac{3}{4}$, -5, 0.25. 1 mark: correct example of an irrational number, e.g. $\sqrt{5}$, π. (Award the example marks only where the earlier definition is at least partly correct.)

6. Question 6

   4 marks

   Three bells at a school ring at intervals of 12 minutes, 18 minutes and 30 minutes. They all ring together at 8.00 a.m. Determine the next time they will all ring together.

   (4 marks)

   Marking scheme

   1 mark: recognition that the LCM of the three intervals is required. 1 mark: prime factorisation, $12=2^2\times 3$, $18=2\times 3^2$, $30=2\times 3\times 5$. 1 mark: $LCM=2^2\times 3^2\times 5=180$ minutes = 3 hours. 1 mark: next time = 11.00 a.m. (Award the final mark only for a stated clock time, not for 180 minutes alone.)

7. Question 7

   1 mark

   Evaluate: -$12+20÷(-4)\times 3$

   A. -6

   B. -3

   C. 6

   D. -27

   (1 mark)

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8. Question 8

   3 marks

   Express the recurring decimal 0.454545… as a fraction in its simplest form, showing all your working.

   (3 marks)

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9. Question 9

   4 marks

   Without using a calculator, evaluate $(\frac{3}{4}+\frac{1}{6})÷\frac{11}{12}$, showing all your working.

   (4 marks)

   Marking scheme

   1 mark: common denominator of 12 used, i.e. $\frac{3}{4}=\frac{9}{12}$ and $\frac{1}{6}=\frac{2}{12}$. 1 mark: $\frac{9}{12}+\frac{2}{12}=\frac{11}{12}$. 1 mark: division changed to multiplication by the reciprocal, $\frac{11}{12}\times \frac{12}{11}$. 1 mark: = 1. (No marks for the answer 1 alone without the fraction working, since the question requires working.)

10. Question 10

    3 marks

    Explain why the sum of a rational number and an irrational number is always irrational.

    (3 marks)

    Marking scheme

    1 mark: let the rational number be r and the irrational number be s, and suppose the sum r + s were rational, say t. 1 mark: then s = t - r, and the difference of two rational numbers is itself rational (closure of the rationals under subtraction). 1 mark: this would make s rational, which contradicts s being irrational; therefore r + s must be irrational. (Accept an equivalent argument in words provided the contradiction is clearly stated.)

11. Question 11

    2 marks

    The distance from Nairobi to Mombasa by road is about 485 km. Express this distance in metres, giving your answer in standard form.

    (2 marks)

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12. Question 12

    1 mark

    Simplify $\frac{6}{\sqrt{3}}$ by rationalising the denominator.

    A. $\frac{\sqrt{3}}{2}$

    B. $3\sqrt{2}$

    C. $6\sqrt{3}$

    D. $2\sqrt{3}$

    (1 mark)

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13. Question 13

    1 mark

    A trader at a market in Kisii records the following numbers on her stock sheet: -7, 0, $\frac{2}{5}$, $\sqrt{9}$ and $\sqrt{7}$. Which one of these is NOT a rational number?

    A. 0

    B. -7

    C. $\sqrt{7}$

    D. $\sqrt{9}$

    (1 mark)

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14. Question 14

    1 mark

    A bodaboda rider measures a trip as 14.8 km, correct to 1 decimal place. The smallest distance the trip could actually have been is:

    A. 14.7 km

    B. 14.79 km

    C. 14.85 km

    D. 14.75 km

    (1 mark)

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