General Base Conversion MCQs: 10 Solved GATE Questions with Explanations

Base conversion errors usually come from notation, not large arithmetic. A digit can invalidate a radix, fractional places use negative powers, and unknown-base equations require algebra plus digit constraints. Solve each item on paper before reading its explanation.

General base conversion rules: expansion, division and digit validity

For an integer numeral (d_k...d_1d_0)_b, expand it as d_k b^k + ... + d_1 b + d_0. For a fraction, continue with negative powers. Thus, (0.1101)_2 = 1/2 + 1/4 + 0/8 + 1/16. To convert a decimal integer to base b, divide repeatedly by b and read the remainders from last to first. For an unknown radix, expand every numeral algebraically, then require the base to be an integer greater than every digit used.

A single value can be converted from one radix to another through decimal. (132)_4 = 1 x 4^2 + 3 x 4 + 2 = 30. Dividing 30 by 5 gives remainders 0, 1, 1, which read upward as (110)_5. The reverse check is (110)_5 = 25 + 5 = 30. For fractions, use (0.1101)_2 = 0.8125 as a checkpoint. For broader practice in valid-digit checks, decimal conversion, and power-of-two grouping, use Number System Basics and Decimal Conversion MCQs. That foundation handles routine conversions. Binary fractions require negative powers, fixed-point exactness requires resolution checks, and unknown-radix equations require algebra plus digit constraints.

Direct base conversion and equivalent representations: Questions 1 to 3

Q1. Match one value across four radices

Source: GATE 2024, MSQ, from the practice page.

“Which of the following is/are EQUAL to 224 in radix-5 (i.e., base-5) notation?”

• A. 64 in radix-10

• B. 100 in radix-8

• C. 50 in radix-16

• D. 121 in radix-7

Answer: A, B and D. First convert (224)_5 to decimal: 2 x 25 + 2 x 5 + 4 = 64. Now check every option in decimal. (64)_10 = 64, (100)_8 = 1 x 8^2 = 64, (50)_16 = 5 x 16 = 80, and (121)_7 = 49 + 14 + 1 = 64. Option C is the only mismatch. For an MSQ, finding A does not finish the work; test every candidate.

Q2. Convert radix 4 to radix 5 through decimal

Source: GATE 2023, NAT, from the practice page.

“A particular number is written as 132 in radix-4 representation. The same number in radix-5 representation is ________.”

Answer: 110. Expand (132)_4 as 1 x 16 + 3 x 4 + 2 = 30. Repeated division gives 30 = 6 x 5 + 0, 6 = 1 x 5 + 1, and 1 = 0 x 5 + 1. Reading the remainders upward gives (110)_5. Before locking a NAT response, reverse the conversion: 1 x 25 + 1 x 5 + 0 = 30.

Q3. Convert a ternary number to hexadecimal

Source: GATE 2021, MCQ, from the practice page.

“Let the representation of a number in base 3 be 210. What is the hexadecimal representation of the number?”

• A. 15

• B. 21

• C. D2

• D. 528

Answer: A. 15. Convert (210)_3 to decimal: 2 x 3^2 + 1 x 3 + 0 = 21. Then split 21 by powers of 16: 21 = 1 x 16 + 5, so the hexadecimal representation is (15)_16. Option B is the notation trap. It is the decimal value, not that value converted to hexadecimal.

Binary fractions and fixed-point resolution: Questions 4 to 5

Q4. Recover decimal digits from a binary fraction

Source: GATE 2021, NAT, from the practice page.

“If $x$ and $y$ are two decimal digits and $(0.1101)_2 = (0.8xy5)_{10}$, the decimal value of $x+y$ is _____ .”

Answer: 3. Evaluate the binary fraction through place values: 1/2 + 1/4 + 0/8 + 1/16 = 0.8125. Match that result with the digit pattern 0.8xy5. This gives x = 1 and y = 2, so x + y = 3. Treating 1101 as a decimal integer and shifting its point would ignore binary place values and lead to a distractor.

Q5. Decide which decimals fit an unsigned fixed-point format

Source: GATE 2018, MCQ, from the practice page.

“Consider the unsigned 8-bit fixed point binary number representation below, b7 b6 b5 b4 b3 . b2 b1 b0 where the position of the binary point is between b3 and b2. Assume b7 is the most significant bit. Some of the decimal numbers listed below cannot be represented exactly in the above representation: (i) 31.500 (ii) 0.875 (iii) 12.100 (iv) 3.001 Which one of the following statements is true?”

• A. None of (i), (ii), (iii), (iv) can be exactly represented

• B. Only (ii) cannot be exactly represented

• C. Only (iii) and (iv) cannot be exactly represented

• D. Only (i) and (ii) cannot be exactly represented

Answer: C. Only (iii) and (iv) cannot be exactly represented. Three fractional bits give a resolution of 2^-3 = 0.125, so an exact fractional part must be a multiple of 0.125. The five integer bits cover 0 through 31. Therefore, 31.500 = 11111.100_2 and 0.875 = 0.111_2 fit. The fractional parts in 12.100 and 3.001 do not. As a quick check, 0.875 / 0.125 = 7, an integer, while 0.100 / 0.125 = 0.8, not an integer.

Unknown radix from equations and roots: Questions 6 to 8

Q6. Use Vieta's formulas when coefficients are in base b

Source: GATE 2017, NAT. Practise related questions on the general base conversion topic page.

“Consider the quadratic equation $x^2-13x+36=0$ with coefficients in a base $b$. The solutions of this equation in the same base $b$ are $x$ = 5 and $x$ = 6. Then $b$ = _____”

Answer: 8. Interpret the coefficient numerals in base b: (13)_b = b + 3 and (36)_b = 3b + 6. The roots sum to 5 + 6 = 11, so Vieta's formula gives b + 3 = 11, hence b = 8. Verify with the product: 5 x 6 = 30, while 3 x 8 + 6 = 30. The base must also exceed the largest digit 6, and base 8 does.

Q7. Count all valid pairs in a mixed-base equality

Source: GATE 2015, NAT, from the practice page.

“Consider the equation $(43)_x$ = $(y3)_8$ where 𝑥 and 𝑦 are unknown. The number of possible solutions is ______________”

Answer: 5. Expand both sides: (43)_x = 4x + 3 and (y3)_8 = 8y + 3, so x = 2y. Digit 4 requires x >= 5, while the leading octal digit y can be 1 through 7. Testing those integers leaves (x,y) = (6,3), (8,4), (10,5), (12,6), (14,7), exactly five pairs. Reject y = 1 and y = 2, because they produce bases x = 2 and x = 4, where digit 4 is invalid.

Q8. Solve a radix equation containing a fractional numeral

Source: GATE 2014, NAT, from the practice page.

“The base (or radix) of the number system such that the following equation holds is____________. $\frac {312} {20} = 13.1$”

Answer: 5. In base b, (312)_b = 3b^2 + b + 2, (20)_b = 2b, and (13.1)_b = b + 3 + 1/b. Substitution gives (3b^2 + b + 2)/(2b) = b + 3 + 1/b. Multiplying by 2b and simplifying gives 3b^2 + b + 2 = 2b^2 + 6b + 2, then b(b - 5) = 0. Reject zero and require the base to exceed digit 3, leaving b = 5. Check in decimal: (312)_5 = 82, (20)_5 = 10, and (13.1)_5 = 8.2, so 82/10 = 8.2.

Unknown digits, factor pairs and option testing: Questions 9 to 10

Q9. Count factor pairs that obey digit and base constraints

Source: GATE 2014, NAT, from the practice page.

“Consider the equation $(123)_5 = (x8)_y$ with $x$ and $y$ as unknown. The number of possible solutions is _____ .”

Answer: 3. Convert (123)_5 to decimal: 25 + 10 + 3 = 38. Since (x8)_y = xy + 8, the equation requires xy = 30. Digit 8 forces y > 8, and the leading digit must satisfy 1 <= x < y. The valid factor pairs are (x,y) = (1,30), (2,15), (3,10). Other orientations make the proposed base at most 8 or make x an invalid digit in base y.

Q10. Test candidate bases after expanding both numerals

Source: GATE 2004, MCQ, from the practice page.

“If $(73)_x$ (in base-x number system) is equal to $(54)_y$ (in base-y number system), the possible values of x and y are”

• A. 8, 16

• B. 10, 12

• C. 9, 13

• D. 8, 11

Answer: D. 8, 11. Expand to 7x + 3 = 5y + 4, or 7x - 5y = 1, with x > 7 and y > 5. Test the choices. A gives 59 and 84, B gives 73 and 64, C gives 66 and 69, and D gives 59 and 59. Only D satisfies the equality, and both bases permit their displayed digits. The subscript names the base; it does not mean multiplication by a variable.

General base conversion score map and the next practice step

For Q1 to Q3, practise expansion and repeated division. For Q4 to Q5, revisit negative powers and fixed-point resolution. For Q6 to Q8, expand each base-dependent numeral before solving. For Q9 to Q10, enforce digit validity after the algebra.

Use this five-line self-check:

1. Write the base under every numeral.

2. Expand all place values.

3. Solve the resulting equation in decimal.

4. Apply the largest-digit constraint.

5. Convert back to verify.

As a final check, (243)_5 = 2 x 25 + 4 x 5 + 3 = 73. Repeated division by 8 gives 73 = 9 x 8 + 1, 9 = 1 x 8 + 1, and 1 = 0 x 8 + 1. Reading the remainders upward gives (111)_8; the reverse expansion 64 + 8 + 1 = 73 verifies it.

The earlier Number System MCQs: 12 Solved (GATE, with answers) uses Q3 and Q10 inside a broad survey of conversions, base arithmetic, and signed representation. Q3 here establishes conversion between unrelated bases through decimal; Q10 follows three unknown-radix problems to train algebra and digit constraints. For broader study, browse the GATE CS Exam Preparation Courses & Test Series. If errors persist, use GATE Guidance by Sanchit Sir for the complete Digital Electronics concept sequence.