In a city, 80% population eat rice and 90% of the rice eaters are non-vegetarians. Then what percent of the population are vegetarian rice eaters?

Correct — B, (b) 8. The whole item turns on whose ninety per cent it is. The second figure in the stem is not ninety per cent of the city; it is ninety per cent of the rice eaters. So the vegetarians among the rice eaters are the remaining tenth of that smaller group, and a tenth of a group that is itself only four fifths of the city is not a tenth of the city.

Take the population as 100 people, which is the fastest way to handle a chain of percentages. Eighty of them eat rice. Of those eighty, ninety per cent are non-vegetarians, which is 72 people, so the remaining eight are vegetarian rice eaters. Eight people out of a hundred is 8 per cent, which is option (b).

In multiplier form the same calculation is one line: 0·80 × 0·10 = 0·08, that is 8 per cent. The order of the two factors carries the meaning — first narrow the population to the rice eaters, then take the vegetarian share within that narrower group — and the answer is the product because the second percentage is conditional on the first.

It is worth seeing what the four numbers in the problem actually describe, because the arithmetic is trivial once the groups are straight. Out of a hundred people: 72 are non-vegetarian rice eaters, 8 are vegetarian rice eaters, and 20 do not eat rice at all and are left entirely undescribed by the stem, since nothing is said about the vegetarian habits of that fifth of the city. Only the second group is asked for. A candidate who lays out those three boxes before doing any multiplication will never take a percentage of the wrong one.

Why the others are wrong

• (a)7·2 — The figure 7·2 is one further step down the same chain, taken on the wrong branch of it: it is ten per cent of 72, the non-vegetarian rice eaters, rather than ten per cent of 80, the rice eaters as a whole. A candidate who has correctly worked out that 72 people are non-vegetarian rice eaters and then applies the vegetarian tenth to that number has confused the group being divided with the group already separated out. The same value can be reached by multiplying all three percentages together, 0·8 × 0·9 × 0·1, which is the signature of a candidate multiplying every figure in sight without asking what each one is a percentage of. It is also the only option carrying a decimal point, and a decimal appearing among three whole numbers is usually a sign that one factor too many has been applied — the true answer to a problem built from tenths of tenths is rarely a fraction of a person.
• (c)9 — Nine per cent arises from treating the ninety as a share of the population and then taking a tenth of it: one tenth of ninety is nine. The step is wrong twice over. The ninety per cent is not a share of the population at all — it belongs to the rice eaters — and the tenth that is wanted is the tenth of the rice eaters who are vegetarians, not a tenth of the non-vegetarians. The value is also easy to reject by an outer bound: the vegetarian rice eaters are a subset of the rice eaters, who are eighty per cent of the city, and they are the smaller part of that subset, so the answer must be well under ten per cent and cannot be a tenth of a figure as large as ninety. Sanity bounds of this kind settle most percentage items faster than the arithmetic does.
• (d)10 — Ten per cent is the answer for a candidate who correctly finds that a tenth of the rice eaters are vegetarian and then reports that tenth as though it were a share of the whole city. It is the commonest error on chained percentages, and it consists of dropping the first factor: the ten per cent is of the rice eaters, so it must still be multiplied by the eighty per cent that defines them. The distinction is the difference between a conditional proportion and an absolute one, and it matters far beyond arithmetic — a figure such as 'ten per cent of those examined were found ineligible' means something quite different from 'ten per cent of the population were found ineligible', and the second cannot be inferred from the first without knowing how many were examined. Here that missing link is the eighty per cent, and using it brings ten down to eight.

A percentage is meaningless without its base, and this item is built on a chain of two percentages with different bases. The first, eighty per cent, has the population as its base. The second, ninety per cent, has the rice eaters as its base — it is a conditional proportion, a statement about a subgroup rather than about the whole. To convert a conditional proportion into a share of the whole population, multiply it by the share of the population that the subgroup represents. In symbols, the fraction of the population that is both P and Q equals the fraction that is P, multiplied by the fraction of those P that are also Q. Here the fraction that eats rice is 0·80, the fraction of rice eaters who are vegetarian is 1 − 0·90 = 0·10, and the product is 0·08. The complementary habit is just as useful: whenever a percentage is given for one part of a subgroup, write down the other part at once, because questions often ask for the complement rather than the stated share, as this one does. The most reliable practical technique for the whole family is to take the population as a hundred, or as whatever number makes every group come out whole, and to count people rather than manipulate decimals.

Key facts

• The 90 per cent in the stem has the rice eaters as its base, not the population, so it is a conditional proportion.
• Vegetarians among rice eaters are the complement of that figure within the same group: 100 − 90 = 10 per cent of rice eaters.
• The share of the population that is both is the product of the two: 0·80 × 0·10 = 0·08, that is 8 per cent.
• Taking the population as 100: 80 eat rice, of whom 72 are non-vegetarian and 8 are vegetarian rice eaters.
• Twenty people in every hundred do not eat rice, and the stem says nothing at all about their food habits.
• A conditional proportion cannot be reported as a share of the whole population without multiplying by the size of the subgroup.
• Sanity bound: the answer must be smaller than the 80 per cent who eat rice and smaller than a tenth of the population.

Common traps

• Reporting a percentage of a subgroup as though it were a percentage of the whole population.
• Applying the vegetarian share to the non-vegetarian group, which produces a fraction of a person and a fractional option.
• Treating the second percentage as though its base were the population, and taking a tenth of ninety.
• Multiplying every figure in the stem together without asking what each is a percentage of.
• Forgetting the fifth of the population who do not eat rice and about whom the stem says nothing at all.

• practice — not a real PYQ

  In a colony, 60 per cent of the residents subscribe to a newspaper, and 25 per cent of those subscribers take more than one newspaper. What percentage of the residents of the colony take more than one newspaper?

  • (a)10
  • (b)15
  • (c)20
  • (d)25

  Answer(b) 15

• practice — not a real PYQ

  In a town, 80 per cent of the people eat rice and 12 per cent of the people are vegetarians who eat rice. What percentage of the rice eaters are vegetarians?

  • (a)9·6
  • (b)12
  • (c)15
  • (d)20

  Answer(c) 15