Directions: The next three items are based on a survey on occurrence of vowels in a certain book irrespective of whether they are in upper or lower case. Vowel: A | E | I | O | U Percentage: 20 | 45 | 15 | 8 | 12 Among the three vowels which occur minimum number of times, what is the percentage of occurrence of the letter that occurs the maximum number of times among them?

Correct — A, 42 6/7 %. Two steps, and the second is where the question is won. First identify the three vowels that occur least often. Ranking the shared table — A 20, E 45, I 15, O 8, U 12 — the two commonest are E at 45 and A at 20, so the three least frequent are O at 8, U at 12 and I at 15. Second, the question asks for the percentage of occurrence of the most frequent letter among those three, which means its share within that group of three and not its share of the whole book. Among O, U and I the largest is I at 15, and the group totals 8 + 12 + 15 = 35. So the answer is 15/35 = 3/7 = 42.857…%, which is 42 6/7 %.

• (b)41 5/7 % — Equals 292/7. No pair of numbers from this table produces it; it is a near-miss built by walking down from the correct value in sevenths.
• (c)40 4/7 % — Equals 284/7, again unreachable from the table. Its purpose is to make the string of sevenths look like a graded scale so that no single value stands out.
• (d)39 2/7 % — Equals 275/7. The four options are deliberately close together, so the fraction has to be computed exactly rather than estimated.

This is a re-basing problem. A percentage always needs a stated base, and the question quietly changes it: the printed 15% for I is measured against every letter in the book, while the answer is measured against only the three least-frequent vowels. Changing the base from 100 to 35 is the whole computation — 15 out of 100 becomes 15 out of 35.

Recognising 3/7 saves the arithmetic. Since 15/35 cancels to 3/7, and 1/7 = 0.142857…, three sevenths is 0.428571…, or 42 6/7 %. It also gives an instant check against the options: only one of them is 42-and-something, so once the fraction is identified the answer follows without long division. The most common error is answering 15%, the figure printed in the table, which is not offered — a deliberate signal from the examiner that the base has changed. The second most common is picking the smallest of the three least-frequent vowels rather than the largest; the phrase 'occurs the maximum number of times among them' points to I, not to O. Notice too that O, U and I are chosen simply by ranking, so the earlier step never involves any arithmetic at all.

• The shared data set: A 20%, E 45%, I 15%, O 8%, U 12%.
• The three least frequent vowels are O (8), U (12) and I (15); their total is 35%.
• The most frequent among those three is I at 15%.
• 15/35 simplifies to 3/7, and 3/7 as a percentage is 300/7 = 42 6/7 %.
• One seventh is 0.142857… recurring, so sevenths of a hundred always land on a repeating decimal — the reason the options are printed as mixed fractions.

• Answering 15%, the figure printed in the table, instead of the share within the group of three.
• Taking the minimum of the three least-frequent vowels rather than the maximum.
• Estimating instead of computing when all four options lie within four percentage points of each other.

Asked as the middle item of a three-question data set, where the arithmetic is trivial and the whole difficulty lies in identifying the correct base.

• practice — not a real PYQ

  Using the same distribution (A 20, E 45, I 15, O 8, U 12), among the two vowels that occur most often, what percentage of their combined occurrences is contributed by A?

  • (a)20%
  • (b)25 5/13 %
  • (c)30 10/13 %
  • (d)44 4/9 %

  Answer(c) 30 10/13 % — the two commonest are E 45 and A 20, so A's share is 20/65 = 4/13 = 400/13 % = 30 10/13 %.

• practice — not a real PYQ

  In the same book, what percentage of all vowel occurrences is accounted for by O and U together?

  • (a)8%
  • (b)12%
  • (c)20%
  • (d)35%

  Answer(c) 20% — the five percentages already sum to 100, so O and U together are 8 + 12 = 20% of the whole.