Number System Questions for CAT XAT & MBA

30 Base System Questions with Detailed Solutions

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📘 Base System – 30 Practice Questions with Step-by-Step Solutions

These questions are divided into Levels of Difficulty (LOD) from 1 to 3 with increasing complexity. Click "Show/Hide Solution" to view each explanation.

1. Convert (1011)₂ to base 10

(a) 11
(b) 13
(c) 8
(d) 16
(e) None of these

\( (1011)_{2} = 1 \times 2^0 + 1 \times 2^1 + 0 \times 2^2 + 1 \times 2^3 = 11 \)

2. Convert (123)₄ to base 10

(a) 27
(b) 29
(c) 24
(d) 32
(e) None of these

\( (123)_{4} = 3 \times 4^0 + 2 \times 4^1 + 1 \times 4^2 = 27 \)

3. Convert (321)₅ to base 10

(a) 86
(b) 88
(c) 83
(d) 91
(e) None of these

\( (321)_{5} = 1 \times 5^0 + 2 \times 5^1 + 3 \times 5^2 = 86 \)

4. Convert (231)₆ to base 10

(a) 91
(b) 93
(c) 88
(d) 96
(e) None of these

\( (231)_{6} = 1 \times 6^0 + 3 \times 6^1 + 2 \times 6^2 = 91 \)

5. Convert (3210)₃ to base 10

(a) 102
(b) 104
(c) 99
(d) 107
(e) None of these

\( (3210)_{3} = 0 \times 3^0 + 1 \times 3^1 + 2 \times 3^2 + 3 \times 3^3 = 102 \)

6. Convert (321)₇ to base 10

(a) 162
(b) 164
(c) 159
(d) 167
(e) None of these

\( (321)_{7} = 1 \times 7^0 + 2 \times 7^1 + 3 \times 7^2 = 162 \)

7. Convert (123)₈ to base 10

(a) 83
(b) 85
(c) 80
(d) 88
(e) None of these

\( (123)_{8} = 3 \times 8^0 + 2 \times 8^1 + 1 \times 8^2 = 83 \)

8. Convert (201)₅ to base 10

(a) 56
(b) 58
(c) 53
(d) 61
(e) None of these

\( (201)_{5} = 1 \times 5^0 + 0 \times 5^1 + 2 \times 5^2 = 56 \)

9. Convert (1023)₄ to base 10

(a) 75
(b) 77
(c) 72
(d) 80
(e) None of these

\( (1023)_{4} = 3 \times 4^0 + 2 \times 4^1 + 0 \times 4^2 + 1 \times 4^3 = 75 \)

10. Convert (431)₆ to base 10

(a) 169
(b) 171
(c) 166
(d) 174
(e) None of these

\( (431)_{6} = 1 \times 6^0 + 3 \times 6^1 + 4 \times 6^2 = 169 \)

11. Convert 121 to base 4

(a) 1321
(b) 1231
(c) 1431
(d) 1341
(e) None of these

Divide 121 repeatedly by 4:
121 ÷ 4 = 30 R1
30 ÷ 4 = 7 R2
7 ÷ 4 = 1 R3
1 ÷ 4 = 0 R1
\( \boxed{(1321)_{4}} \)

12. Convert 145 to base 6

(a) 401
(b) 411
(c) 321
(d) 421
(e) None of these

Divide 145 repeatedly by 6:
145 ÷ 6 = 24 R1
24 ÷ 6 = 4 R0
4 ÷ 6 = 0 R4
\( \boxed{(401)_{6}} \)

13. Convert 85 to base 3

(a) 10011
(b) 11001
(c) 11100
(d) 10110
(e) None of these

Divide 85 repeatedly by 3:
85 ÷ 3 = 28 R1
28 ÷ 3 = 9 R1
9 ÷ 3 = 3 R0
3 ÷ 3 = 1 R0
1 ÷ 3 = 0 R1
\( \boxed{(10011)_{3}} \)

14. Convert 100 to base 5

(a) 400
(b) 440
(c) 344
(d) 204
(e) None of these

Divide 100 repeatedly by 5:
100 ÷ 5 = 20 R0
20 ÷ 5 = 4 R0
4 ÷ 5 = 0 R4
\( \boxed{(400)_{5}} \)

15. Convert 73 to base 8

(a) 111
(b) 101
(c) 110
(d) 121
(e) None of these

Divide 73 repeatedly by 8:
73 ÷ 8 = 9 R1
9 ÷ 8 = 1 R1
1 ÷ 8 = 0 R1
\( \boxed{(111)_{8}} \)

16. Convert 199 to base 7

(a) 403
(b) 340
(c) 430
(d) 404
(e) None of these

Divide 199 repeatedly by 7:
199 ÷ 7 = 28 R3
28 ÷ 7 = 4 R0
4 ÷ 7 = 0 R4
\( \boxed{(403)_{7}} \)

17. Convert 300 to base 9

(a) 363
(b) 336
(c) 633
(d) 343
(e) None of these

Divide 300 repeatedly by 9:
300 ÷ 9 = 33 R3
33 ÷ 9 = 3 R6
3 ÷ 9 = 0 R3
\( \boxed{(363)_{9}} \)

18. Convert 154 to base 6

(a) 414
(b) 341
(c) 431
(d) 443
(e) None of these

Divide 154 repeatedly by 6:
154 ÷ 6 = 25 R4
25 ÷ 6 = 4 R1
4 ÷ 6 = 0 R4
\( \boxed{(414)_{6}} \)

19. Convert 127 to base 2

(a) 1111111
(b) 1010101
(c) 1110111
(d) 1101111
(e) None of these

Divide 127 repeatedly by 2:
127 ÷ 2 = 63 R1
63 ÷ 2 = 31 R1
31 ÷ 2 = 15 R1
15 ÷ 2 = 7 R1
7 ÷ 2 = 3 R1
3 ÷ 2 = 1 R1
1 ÷ 2 = 0 R1
\( \boxed{(1111111)_{2}} \)

20. Convert 99 to base 5

(a) 344
(b) 443
(c) 434
(d) 343
(e) None of these

Divide 99 repeatedly by 5:
99 ÷ 5 = 19 R4
19 ÷ 5 = 3 R4
3 ÷ 5 = 0 R3
\( \boxed{(344)_{5}} \)

21. If (101.11)₂ = (N)₁₀, find the value of N.

(a) 5.75
(b) 5.95
(c) 5.55
(d) 6.05
(e) None of these

\( (101.11)_{2} = 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^{-2} + 1 \times 2^{-3} = \boxed{5.75} \)

22. If (123.12)₅ = (N)₁₀, find the value of N.

(a) 38.28
(b) 38.48
(c) 38.08
(d) 38.58
(e) None of these

\( (123.12)_{5} = 1 \times 5^4 + 2 \times 5^3 + 3 \times 5^2 + 1 \times 5^{-2} + 2 \times 5^{-3} = \boxed{38.28} \)

23. If (210.21)₃ = (N)₁₀, find the value of N.

(a) 21.78
(b) 21.98
(c) 21.58
(d) 22.08
(e) None of these

\( (210.21)_{3} = 2 \times 3^4 + 1 \times 3^3 + 0 \times 3^2 + 2 \times 3^{-2} + 1 \times 3^{-3} = \boxed{21.7778} \)

24. If (321.1)₄ = (N)₁₀, find the value of N.

(a) 57.25
(b) 57.45
(c) 57.05
(d) 57.55
(e) None of these

\( (321.1)_{4} = 3 \times 4^3 + 2 \times 4^2 + 1 \times 4^1 + 1 \times 4^{-1} = \boxed{57.25} \)

25. If (432.32)₆ = (N)₁₀, find the value of N.

(a) 164.56
(b) 164.76
(c) 164.36
(d) 164.86
(e) None of these

\( (432.32)_{6} = 4 \times 6^4 + 3 \times 6^3 + 2 \times 6^2 + 3 \times 6^{-2} + 2 \times 6^{-3} = \boxed{164.5555} \)

26. If (301.13)₅ = (N)₁₀, find the value of N.

(a) 81.56
(b) 81.76
(c) 81.36
(d) 81.86
(e) None of these

\( (301.13)_{5} = 3 \times 5^4 + 0 \times 5^3 + 1 \times 5^2 + 1 \times 5^{-2} + 3 \times 5^{-3} = \boxed{81.56} \)

27. If (1001.1)₂ = (N)₁₀, find the value of N.

(a) 9.5
(b) 9.7
(c) 9.3
(d) 9.8
(e) None of these

\( (1001.1)_{2} = 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^{-1} = \boxed{9.5} \)

28. If (124.21)₇ = (N)₁₀, find the value of N.

(a) 67.44
(b) 67.64
(c) 67.24
(d) 67.74
(e) None of these

\( (124.21)_{7} = 1 \times 7^4 + 2 \times 7^3 + 4 \times 7^2 + 2 \times 7^{-2} + 1 \times 7^{-3} = \boxed{67.4375} \)

29. If (231.1)₃ = (N)₁₀, find the value of N.

(a) 26.33
(b) 26.53
(c) 26.13
(d) 26.63
(e) None of these

\( (231.1)_{3} = 2 \times 3^3 + 3 \times 3^2 + 1 \times 3^1 + 1 \times 3^{-1} = \boxed{26.3333} \)

30. If (110.3)₄ = (N)₁₀, find the value of N.

(a) 20.75
(b) 20.95
(c) 20.55
(d) 21.05
(e) None of these

\( (110.3)_{4} = 1 \times 4^3 + 1 \times 4^2 + 0 \times 4^1 + 3 \times 4^{-1} = \boxed{20.75} \)

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Base System – 30 Practice Questions with Step-by-Step Solutions

Base System – 30 Practice Questions with Step-by-Step Solutions