EPFO EO/AO 2023 General Ability Test · Question 83 of 120

Correct — D, (d) 0·8125. The item prints its decimal point as a raised middle dot throughout, which is this booklet's convention and not a separate symbol; read (·1101) with a subscript two as the binary fraction point one one zero one. The positional method. To the right of the binary point the place values are successive negative powers of two: one half, one quarter, one eighth, one sixteenth, and so on. The digits are 1, 1, 0, 1, so the value is one half plus one quarter plus nothing plus one sixteenth. In decimal that is 0·5 plus 0·25 plus 0 plus 0·0625, which sums to 0·8125. That is option (d) exactly. The shorter method, and the one worth carrying into the hall. Ignore the point, read the digit string as an ordinary binary integer, and divide by two raised to the number of digits after the point. Here 1101 in binary is eight plus four plus zero plus one, which is 13, and there are four digits after the point, so the value is 13 divided by 16. Sixteen goes into 13 as 0·8125. One division replaces four additions of fractions, and it is far harder to slip a place with it. The elimination method, which settles this particular item without arithmetic at all. Every binary fraction of four places is a whole-number multiple of one sixteenth, and one sixteenth is 0·0625. Multiplying 0·0625 by any whole number gives a result whose fourth decimal place is 5 when the multiplier is odd and 0 when it is even — the fourth place can never be 2, 3 or 4. So options (a), (b) and (c) are not merely wrong for this string, they cannot be the exact decimal value of any four-place binary fraction whatever. Only option (d) is even a candidate. A candidate who has understood what a binary fraction is can answer this in a second, which is exactly what the paper is testing. A useful check on the size of the answer: the largest four-place binary fraction is ·1111, which is 15 divided by 16, or 0·9375, and every binary fraction is strictly less than 1. An answer of 0·8125 sits sensibly inside that range.

• (a)0·8122 — Wrong, and impossible. A fraction with four binary places is a multiple of one sixteenth, that is of 0·0625, and no multiple of 0·0625 ends in the digit 2 at the fourth decimal place — the fourth place of such a number is always 5 or 0. This option is one of three near-misses manufactured by changing the last digit of the true answer, and its only function is to catch a candidate who half-remembers the answer as 'zero point eight one two something' without having done the arithmetic.
• (b)0·8123 — Wrong for the same structural reason: the fourth decimal place of a four-place binary fraction can only be 5 or 0, never 3. It is also worth seeing what would have to be true for a value near this to arise — 0·8123 is not thirteen sixteenths, nor any other simple binary fraction, so no misreading of the digit string 1101 produces it. It is pure decoration on the answer key, present so that four options exist.
• (c)0·8124 — Wrong on the same test, since the fourth place would have to be 4. This is the option nearest to the true value, differing from it by one ten-thousandth, and it is the one that punishes a candidate who computes correctly and then transcribes carelessly. Note that no plausible slip in the positional method lands here either: dropping the one-sixteenth term gives 0·75, adding a one-eighth term that is not there gives 0·9375, and misreading the string as 1011 gives 0·6875. Every genuine error produces a value far from this option, so a result of 0·8124 can only come from copying, not from computing.

A positional number system extends to the right of the point exactly as it does to the left, with the base raised to negative powers. In decimal the places after the point are tenths, hundredths and thousandths; in binary they are halves, quarters, eighths and sixteenths. So converting a binary fraction to decimal is only a weighted sum, and it can always be replaced by one division: read the digits after the point as an integer and divide by two raised to the count of those digits. The reverse conversion, decimal fraction to binary, is done by repeated multiplication by two, writing down and then discarding the integer part each time and reading the collected digits downwards. Two consequences of the base matter in practice. First, a binary fraction of n places is always an exact terminating decimal, because 2 divides 10; second, the converse fails, and a decimal as ordinary as 0·1 has no terminating binary form at all, repeating for ever. That asymmetry is the reason a computer holding money as a binary floating-point number can produce a total that is a fraction of a paisa away from the expected figure, and it is why accounting systems either scale amounts to whole paise or use a decimal type.

Number-system conversion is a fixture of the EO/AO computers block, and this item takes the harder of the two directions by putting the digits after the point rather than before it. Nothing here needs a calculator; the paper is checking whether the candidate knows that the place values continue as negative powers of the base. The habit rewarded is doing a size check before choosing — every binary fraction lies between 0 and 1, and a four-place one is a multiple of 0·0625 — because that check alone disposes of three of the four printed options. The middle dot used for the decimal point is a printing convention of this booklet and appears in the stem and in all four options.

• The places after a binary point carry the weights one half, one quarter, one eighth, one sixteenth and so on.
• (·1101) in binary equals one half plus one quarter plus one sixteenth, which is 0·8125.
• A quicker route: read the digits as the integer 1101, which is 13, and divide by 2 to the power 4, which is 16.
• Any binary fraction of four places is a multiple of 0·0625, so its fourth decimal place must be 5 or 0.
• The largest four-place binary fraction is ·1111, equal to fifteen sixteenths or 0·9375; every binary fraction is less than 1.
• A decimal fraction is converted to binary by repeated multiplication by two, recording the integer part at each step.
• Every terminating binary fraction is an exact terminating decimal, but the converse is false — 0·1 in decimal is a repeating binary fraction.
• That non-terminating conversion is the source of rounding error in binary floating-point arithmetic on money.

• Applying the integer place values 1, 2, 4, 8 to digits that lie after the point, where the weights are halves, quarters, eighths and sixteenths.
• Reading the digits right to left after the point; the first digit after the point is the largest weight, not the smallest.
• Forgetting the zero in the third place and treating the string as 111.
• Computing 13 divided by 16 correctly and then picking the adjacent option — three of the four differ only in the last digit.

Number-system items in EO/AO papers come as a direct conversion, as arithmetic performed in a non-decimal base, or as a question about how many bits are needed to represent a given range. Learn the powers of two up to 1024 and their reciprocals as decimals — 0·5, 0·25, 0·125, 0·0625, 0·03125 — and both directions of conversion become mental arithmetic.

No directly related past PYQ was found.

• practice — not a real PYQ

  The decimal equivalent of the binary fraction (·011) is :

  • (a)0·375
  • (b)0·425
  • (c)0·625
  • (d)0·750

  Answer(a) 0·375

• practice — not a real PYQ

  Which one of the following decimal fractions has a terminating binary representation ?

  • (a)0·1
  • (b)0·2
  • (c)0·3
  • (d)0·75

  Answer(d) 0·75