GATE 2018 IN Q.18: solution and answer

GATE 2018 · INQ.18 · 1 mark · MCQ

The representation of the decimal number $(27.625)_{10}$ in base-2 number system is

(A)
$11011.110$
(B)
$11101.101$
(C)
$11011.101$
(D)
$10111.110$

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Answer and solution

ANSWER

$(27.625)_{10} = (11011.101)_{2}$ , so the answer is option (C).

Official answer: CSource: Official answer key, GATE 2018 IN (IIT Guwahati)

Convert the integer part and the fractional part separately: $(27.625)_{10} = 27 + 0.625$ .
Integer part: divide repeatedly by 2 and read the remainders from bottom to top. $27 \to 13 r 1$ , $13 \to 6 r 1$ , $6 \to 3 r 0$ , $3 \to 1 r 1$ , $1 \to 0 r 1$ , so $(27)_{10} = (11011)_{2}$ .
Check with place values: $16 + 8 + 0 + 2 + 1 = 27$ .
Fractional part: multiply repeatedly by 2 and read the integer carries from top to bottom. $0.625 \times 2 = 1.25 \to 1$ , $0.25 \times 2 = 0.5 \to 0$ , $0.5 \times 2 = 1.0 \to 1$ . The fraction is now zero, so the conversion ends.
So $(0.625)_{10} = (0.101)_{2}$ . Check: $\frac{1}{2} + 0 + \frac{1}{8} = 0.625$ .
Put the two parts together: $(27.625)_{10} = (11011.101)_{2}$ , which is option (C).

KEY IDEA

Convert the integer part and the fractional part separately. For the integer part, divide repeatedly by 2 and read the remainders from the last one up. For the fractional part, multiply repeatedly by 2 and read the carries from the first one down. $0.625 = \frac{1}{2} + \frac{1}{8}$ terminates in binary, so the result is exact.

WATCH OUT FOR

The usual slip is reading the fractional carries in the wrong order. Reading them bottom to top gives $.101$ again only because it is a palindrome, but on other fractions it gives a wrong answer. The distractors come from these slips: (A) $11011.110 = 27.75$ gets the fraction wrong, (B) $11101.101 = 29.625$ swaps bits in the integer part, and (D) $10111.110 = 23.75$ reverses the integer remainders (reading them top to bottom) and also gets the fraction wrong. A quick place-value check of the chosen option catches all three.

From the GATE 2018 IN paper. Every question of this paper · Download the full paper · All 5 Number systems questions, with notes

GATE 2018 IN Q.18: solution and answer

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