Number Bases: Mathematics Tingkatan 4 questions

Section A: objective questions

15 marks

Choose the correct answer. Each question carries 1 mark.

Which of the following is not a number in base five?

1423₅
2503₅
3041₅
4312₅

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Base five uses only the digits 0, 1, 2, 3 and 4. 2503₅ contains the digit 5, which is not allowed. The other three use digits below 5 only.

For the teacher: Remembering, low difficulty, standard 2.1.1

Textbook p. 35

What is the place value of the underlined digit in 7231₈?

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From the right the place values are 8⁰, 8¹, 8², 8³. The 2 is the third digit from the right, so the place value is 8². The base is 8, so 2² is wrong.

For the teacher: Remembering, low difficulty, standard 2.1.1

Textbook p. 36

What is the value of the underlined digit in 5341₇?

147
21
1 029
3

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The 3 is in the 7² place: 3 × 49 = 147. 21 uses 7¹, 1 029 uses 7³ and 3 ignores the place value.

For the teacher: Understanding, low difficulty, standard 2.1.1

Textbook p. 37

Find the value of 110101₂ in base ten.

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32 + 16 + 0 + 4 + 0 + 1 = 53. Leaving out the 32 gives 21.

For the teacher: Applying, low difficulty, standard 2.1.1

Textbook p. 38

Find the value of 2043₅ in base ten.

1 059
2 043
273
258

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(2 × 125) + (0 × 25) + (4 × 5) + (3 × 1) = 250 + 20 + 3 = 273. 1 059 is the value of 2043₈; 2 043 has not been changed at all.

For the teacher: Applying, low difficulty, standard 2.1.1

Textbook p. 39

Convert 345₁₀ to a number in base five.

2340₅
0432₅
2430₅
1345₅

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345 ÷ 5 = 69 r 0; 69 ÷ 5 = 13 r 4; 13 ÷ 5 = 2 r 3; 2 ÷ 5 = 0 r 2. Remainders upwards: 2340₅. 0432₅ reads them downwards.

For the teacher: Applying, low difficulty, standard 2.1.2

Textbook p. 41

Convert 1011010₂ to a number in base eight.

231₈
132₈
552₈
26₈

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Group from the right: 1 | 011 | 010 gives 1, 3, 2. So 132₈. 231₈ reverses the digits.

For the teacher: Applying, low difficulty, standard 2.1.2

Textbook p. 43

Convert 6154₈ to a number in base two.

11001101100₂
110001101₂
100110001100₂
110001101100₂

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6 = 110, 1 = 001, 5 = 101, 4 = 100. Joined: 110 001 101 100. Dropping the zeros of 001 loses digits.

For the teacher: Applying, low difficulty, standard 2.1.2

Textbook p. 44

Convert 2134₅ to a number in base three.

100221₃
12212₃
2134₃
101220₃

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Step 1: (2 × 125) + (1 × 25) + (3 × 5) + 4 = 294₁₀. Step 2: divide by 3 repeatedly: remainders 0, 2, 2, 1, 0, 1, so 101220₃ (read upwards).

For the teacher: Applying, high difficulty, standard 2.1.2

Textbook p. 42

Calculate 1101₂ + 1011₂.

10000₂
2112₂
11000₂
11001₂

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13 + 11 = 24 = 11000₂. In a column, 1 + 1 = 10₂: write 0 and carry 1. 2112₂ is not possible because 2 is not a base-two digit.

For the teacher: Applying, low difficulty, standard 2.1.3

Textbook p. 46

Calculate 4023₅ − 1344₅.

3321₅
2124₅
2134₅
3312₅

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4023₅ = 513 and 1344₅ = 224. 513 − 224 = 289 = 2124₅. Borrow 5, not 10, in each column.

For the teacher: Applying, medium difficulty, standard 2.1.3

Textbook p. 47

Which of the following has the greatest value?

10110₂
32₅
24₈
21₉

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Change each to base ten: 10110₂ = 16 + 4 + 2 = 22, 32₅ = 17, 24₈ = 20, 21₉ = 19. The greatest is 22. A longer number is not always greater: 10110₂ has the most digits but also the smallest base.

For the teacher: Analysing, medium difficulty, standard 2.1.4

Textbook p. 49

Given that 110₂ + y₇ = 12₉, find the value of y.

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110₂ = 6 and 12₉ = 11. So y = 11 − 6 = 5. y must be less than 7, which rules out 11. 6 comes from using 12₉ = 12.

For the teacher: Analysing, high difficulty, standard 2.1.4

Textbook p. 50

Which number has the number value (2 × 3³) + (0 × 3²) + (1 × 3¹) + (2 × 3⁰)?

2102₃
2012₂
54₁₀
2012₃

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The digits are the multipliers of the place values from 3³ down to 3⁰: 2, 0, 1, 2. So the number is 2012₃. 54 is only the value of the first term.

For the teacher: Understanding, low difficulty, standard 2.1.1

Textbook p. 38

Hana says, "1011₂ and 1011₅ are equal because they have the same digits." Which statement about her claim is correct?

It is correct, because the digits are the same.
It is wrong, because the same digits have different place values in different bases.
It is wrong, because 1011₂ is not a number.
It is correct, because both numbers have four digits.

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1011₂ = 8 + 0 + 2 + 1 = 11, but 1011₅ = 125 + 0 + 5 + 1 = 131. The place values depend on the base, so the values are different.

For the teacher: Evaluating, medium difficulty, standard 2.1.1

Textbook p. 39

Section B: structured questions

29 marks

Answer all questions. Show your working.

Answer the questions about the number 4302₅.

[1 mark] (a)State the place value of the digit 3.

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5² (or 25) [1]1

Tick each point you got.

[1 mark] (b)State the digit value of the digit 4.

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4 × 5³ = 500 [1]1

Tick each point you got.

[2 marks] (c)Find the value of the number in base ten.

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(4 × 125) + (3 × 25) + (0 × 5) + 2 [1]1
577 [1]1

Tick each point you got.

For the teacher: Remembering, low difficulty, standard 2.1.1

Textbook p. 36

Number bases.

[2 marks] (a)Convert 275₁₀ to a number in base six.

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275 ÷ 6 = 45 r 5; 45 ÷ 6 = 7 r 3; 7 ÷ 6 = 1 r 1; 1 ÷ 6 = 0 r 1 [1]1
1135₆ [1]1

Tick each point you got.

[3 marks] (b)Convert 342₅ to a number in base two.

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(3 × 25) + (4 × 5) + 2 = 97 [1]1
97 divided by 2 repeatedly, remainders read upwards [1]1
1100001₂ [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 2.1.2

Textbook p. 42

Calculate each of the following. Give the answer in the base of the question.

[3 marks] (a)4326₇ + 1654₇

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Column sums with carrying in base 7: 6 + 4 = 13₇ (write 3, carry 1) and so on [1]1
Correct digits in the answer [1]1
6313₇ [1]1

Tick each point you got.

[3 marks] (b)3012₄ − 1233₄

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Borrowing 4 (not 10) in the columns, or 198 − 111 [1]1
87 in base four [1]1
1113₄ [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 2.1.3

Textbook p. 47

Compare the numbers 10111₂, 32₅ and 25₈.

[3 marks] (a)Write each number in base ten.

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10111₂ = 23 [1]1
32₅ = 17 [1]1
25₈ = 21 [1]1

Tick each point you got.

[1 mark] (b)Arrange the three numbers in ascending order.

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32₅, 25₈, 10111₂ [1]1

Tick each point you got.

[1 mark] (c)10111₂ is greater than 25₈.

TrueFalse

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True

10111₂ = 23 and 25₈ = 21, and 23 is greater than 21.

For the teacher: Analysing, medium difficulty, standard 2.1.4

Textbook p. 50

A club has 120 pupils. 35% of them are girls and the rest are boys.

[2 marks] (a)Find the number of girls and write it in base five.

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35 ÷ 100 × 120 = 42 [1]1
132₅ [1]1

Tick each point you got.

[2 marks] (b)Find the number of boys and write it in base eight.

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120 − 42 = 78 [1]1
116₈ [1]1

Tick each point you got.

[2 marks] (c)Calculate the difference between the number of boys and girls and write it in base two.

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78 − 42 = 36 [1]1
100100₂ [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 2.1.4

Textbook p. 49

Create your own number.

[1 mark] (a)Write a 3-digit number in base five whose value in base ten is greater than 60 and less than 70.

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Any valid answer, for example 221₅ (61) to 234₅ (69) [1]1

Tick each point you got.

[2 marks] (b)Show the working that proves its value in base ten.

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Place values 25, 5, 1 used with the digits [1]1
Sum of digit values is between 61 and 69, e.g. (2 × 25) + (2 × 5) + 1 = 61 [1]1

Tick each point you got.

For the teacher: Creating, high difficulty, standard 2.1.1

Textbook p. 38

Section A: objective questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Section B: structured questions

Question 16

Question 17

Question 18

Question 19

Question 20

Question 21