In an office, 40% of the employees are men and the rest women. Half of the employees are tall and half short. If 10% of the employees are men and short, and 40 employees are women and tall, the number of tall men employees is

Correct — A, (a) 60.

THE STEP THAT DECIDES THIS QUESTION is to draw a TWO-BY-TWO TABLE and to work in percentages of the whole workforce until the very last line. The employees are split two ways at once — by sex and by height — so there are four groups, and the stem gives both sets of totals plus one of the four groups. That is exactly enough to fill the table.

Let every entry be a percentage of the total number of employees, N.

Men are 40 per cent, so women are 60 per cent. Half are tall and half are short, so each is 50 per cent. Men who are short are given as 10 per cent.

Now the table completes itself by subtraction:

TALL SHORT TOTAL MEN 30 10 40 WOMEN 20 40 60 TOTAL 50 50 100

Men who are tall = 40 − 10 = 30 per cent. Women who are tall = 50 − 30 = 20 per cent. Women who are short = 60 − 20 = 40 per cent, and the short column checks out at 10 + 40 = 50.

NOW ANCHOR THE PERCENTAGES TO A COUNT. Every figure so far is a proportion; one absolute number is needed to fix the size of the office, and the stem supplies it: 40 employees are women and tall. From the table, women and tall is 20 per cent of the workforce, so

0·20 N = 40 → N = 200 employees.

TALL MEN are 30 per cent of that:

0·30 × 200 = 60

and the answer is option (a).

A CHECK BY DIVISIBILITY, which disposes of the option list almost without arithmetic and is worth learning as a technique. Whatever the total turns out to be, tall men must be exactly 30 per cent of it, and short men must be exactly 10 per cent, so N has to be a multiple of ten and the tall-men figure has to be three tenths of it. Test the options against that:

60 tall men needs N = 200. Then women and tall are 20 per cent of 200 = 40, which is precisely what the stem says. Consistent. 50 tall men needs N = 166·67, which is not a whole number of employees at all. Impossible. 40 tall men needs N = 133·33. Impossible for the same reason. 30 tall men needs N = 100, and then women and tall would be 20 per cent of 100 = 20, not the 40 the stem gives. Contradicted.

Only one option survives, and the elimination uses nothing beyond the two structural facts that men are 40 per cent and short men 10 per cent.

A FINAL SANITY READING OF THE TABLE. Men are more likely to be tall than women in this office — three quarters of the men are tall against a third of the women — which is why the tall column is dominated by men even though men are the smaller group. Reading the completed table back in words like that is a good habit: a table that cannot be described sensibly usually contains an arithmetic error.

The stem ends on the words "employees is" with no punctuation, and the options are bare numerals in descending order.

Why the others are wrong

• (b)50 — Fifty is the one option that corresponds to no cell of the table at all. The four cells are 30, 10, 20 and 40 per cent of the workforce, and fifty employees out of two hundred would be 25 per cent, a share that appears nowhere in the arrangement. The divisibility test rejects it immediately: tall men are exactly three tenths of the workforce, so a figure of fifty would require a total of 166 and two thirds employees, which is not a whole number, and short men would then be sixteen and two thirds. Its function on the list is structural rather than diagnostic — the options descend evenly from 60 to 30 in steps of ten, and an evenly spaced ladder gives nothing away about which rung is right. It is worth noticing how often quantitative option lists on this paper are built that way. The defence is always the same: complete the working and then verify the chosen figure against the stem's own numbers, rather than looking for an option that stands out.
• (c)40 — Forty is the number printed in the stem itself — the count of employees who are women and tall — and it is the strongest trap on the list. Questions of this kind supply exactly one absolute figure, and that figure has a job to do: it converts the percentages into counts. Its own value is almost never the answer, because if it were, the question would be asking for something it had already stated. Here 40 is the count of a DIFFERENT cell from the one asked about, and confusing the two amounts to reading the table's rows and columns the wrong way round. There is also a wrong route that lands on 40 by accident: treating the two classifications as independent and multiplying 40 per cent by 50 per cent to get 20 per cent of 200. That reasoning assumes men and women are equally likely to be tall, which the stem contradicts by telling us that only a quarter of the men are short.
• (d)30 — Thirty is the PERCENTAGE of the workforce that consists of tall men, mistaken for the number of them. The confusion is easy to fall into, because the whole of the working is done in percentages and only the last line converts to a count; a candidate who completes the table correctly and stops one step early has the right structure and the wrong units. The stem asks for "the number of tall men employees", not their share. The figure can also be tested directly against the stem: thirty tall men would mean a workforce of a hundred, and in a workforce of a hundred the women who are tall would number twenty, whereas the stem says there are forty of them. So the office must be twice that size, and every count in it doubles — sixty tall men, twenty short men, forty tall women and eighty short women, adding to two hundred. Always convert before answering, and always test the converted figure against the one absolute number the stem provided.

A TWO-WAY CLASSIFICATION IS A TABLE, AND A TABLE IS SOLVED BY ITS MARGINS. When a population is divided simultaneously by two attributes — here sex and height — every member falls into exactly one of four groups, and those four groups can be laid out in a two-by-two grid whose row totals and column totals are the two separate classifications.

The arithmetic fact that makes such problems easy is this: a two-by-two table has four interior cells, but only ONE of them is a free choice once both sets of margins are known. Give the row totals, the column totals and any single interior cell, and the other three follow by subtraction. That is why this stem gives exactly what it gives — 40 per cent men, half tall, and one interior cell at 10 per cent — and no more. Recognising that the information is exactly sufficient is itself reassuring: nothing has been left out and nothing needs to be assumed.

WORK IN PERCENTAGES, ANCHOR AT THE END. The great convenience of these problems is that the structure is entirely proportional. Every relationship among the cells holds whatever the size of the population, so the table can be completed in percentages without knowing N at all. The single absolute figure in the stem is then used once, at the end, to fix N — and it should be used once only. Candidates who try to carry an unknown N through every subtraction make the working heavier and the errors likelier.

THE INDEPENDENCE TRAP deserves its own warning, because it is the deep error in this area. It is tempting to compute the men-and-tall cell as 40 per cent of 50 per cent, that is 20 per cent, on the reasoning that height and sex are unrelated. That is the assumption of STATISTICAL INDEPENDENCE, and a question of this type almost always contradicts it deliberately. Here the men-and-short cell is given as 10 per cent, whereas independence would make it 20 per cent, so in this office men are markedly more likely to be tall than women are. The whole point of supplying an interior cell is to tell the candidate how the two attributes are related; if they were independent, the cell would not need to be given.

READING THE COMPLETED TABLE IN WORDS is the final discipline. Three quarters of the men here are tall and only a third of the women are; women make up three fifths of the workforce but only two fifths of the tall employees. A table that can be narrated like that is internally coherent. This same structure — margins, one interior cell, the rest by subtraction — underlies set problems with two overlapping groups, survey questions about who reads which newspaper, and the inclusion-exclusion principle in its simplest form.

Key facts

• The workforce is split two ways at once — 40 per cent men against 60 per cent women, and 50 per cent tall against 50 per cent short — which makes a two-by-two table with four groups.
• In a two-by-two table, both sets of margins plus any ONE interior cell determine the other three cells by subtraction; nothing further need be assumed.
• Men who are short are 10 per cent, so men who are tall are 40 − 10 = 30 per cent of the workforce.
• Women who are tall are then 50 − 30 = 20 per cent, and women who are short are 60 − 20 = 40 per cent, with the short column checking at 10 + 40 = 50.
• The single absolute figure, 40 women who are tall, equals 20 per cent of the workforce, so the office has 200 employees.
• Tall men are 30 per cent of 200, that is 60 — and 30 is the percentage while 60 is the count, a distinction the option list exploits.
• A divisibility check settles the list on its own: tall men must be three tenths of the total, so 50 would need 166⅔ employees and 40 would need 133⅓, neither of which is possible.
• Thirty tall men would mean a workforce of 100, in which the tall women would number 20 rather than the 40 the stem states.
• Height and sex are NOT independent here: independence would put men and short at 20 per cent, whereas the stem gives 10, so men in this office are more likely to be tall.

Common traps

• Handing back the number printed in the stem. Forty is the count of tall WOMEN, which is the other cell of the same column.
• Answering with the percentage instead of the count. Tall men are 30 per cent of the workforce but 60 in number.
• Assuming the two classifications are independent and computing 40 per cent of 50 per cent. The stem's interior cell exists precisely to contradict that.
• Carrying the unknown total through the whole calculation instead of completing the table in percentages and anchoring once at the end.
• Failing to check the completed table against both margins, which is a free verification that catches almost every subtraction error.

• practice — not a real PYQ

  In a class, 60 per cent of the students are girls and half of all the students wear spectacles. If 25 per cent of the students are girls who wear spectacles, what percentage of the students are boys who wear spectacles ?

  • (a)15 per cent
  • (b)20 per cent
  • (c)25 per cent
  • (d)35 per cent

  Answer(c) 25 per cent — those wearing spectacles make up 50 per cent of the class in all, and 25 per cent of the class are girls wearing them, so the boys wearing them make up the remaining 25 per cent. The 60 per cent figure is not needed for this particular cell.

• practice — not a real PYQ

  In an office of 200 employees, 40 per cent are men and half of all the employees are tall. If 10 per cent of the employees are men who are short, how many of the employees are women who are short ?

  • (a)40
  • (b)60
  • (c)80
  • (d)100

  Answer(c) 80 — the table gives men and tall at 30 per cent, women and tall at 20 per cent and women and short at 60 − 20 = 40 per cent of the workforce, which for 200 employees is 80. The short column then checks at 20 short men plus 80 short women, that is 100, or half the office.