The average age of a husband and his wife was 23 years when they were married 5 years ago. The average age of the husband, the wife and their child is 20 years now. How old is the child now ?

Correct — D, (d) 4 years. The step that decides this question is the one that looks trivial: when a group of people moves forward in time, the group's total age rises by the number of people multiplied by the number of years, not by the number of years alone. Two people, five years, ten years added to the total. Everything else is conversion between an average and a total. Five years ago there were two people with an average age of 23, so their ages then totalled 2 x 23 = 46 years. Five years later each of them is five years older, so the couple's combined age now is 46 + 10 = 56 years — an average of 28 each, which is the check to make in passing. Today the household is three people with an average of 20, so their ages total 3 x 20 = 60 years. The child's age is what is left over: 60 - 56 = 4 years. A sanity check finishes it. The couple married five years ago, so a child of the marriage must be under five, and four years satisfies that. It is worth noticing what the option list does not give you here. All four choices are under five years, so the marriage date eliminates none of them, and the arithmetic has to be done. The two commonest slips on this question are both worth naming because of where they land. Add five to the couple's total instead of ten and the child comes out at 60 - 51 = 9 years; read the average of 23 as their age now rather than five years ago and the child comes out at 60 - 46 = 14 years. Neither 9 nor 14 is on the paper. That absence is a gift: a candidate who arrives at a number the option list does not contain has been told, free of charge, to go back and look at the time shift again.

Why the others are wrong

• (a)9 months — Nine months. Every wrong option on this item can be tested the same way, by asking what the couple's combined age would have to be for it to be true. The three of them total 60 years, so a child of nine months would require the husband and wife together to be 59 years and 3 months old today, an average of a shade under 30 each. The stem fixes their combined age at 56 — from 46 five years ago plus ten years of ageing between two people — so this option is out by three and a quarter years. Notice also that this is the only option expressed in months while the other three are in years. It has to be converted before it can be compared with them, and a candidate who is scanning quickly can mistake it for the smallest sensible answer rather than checking what it implies.
• (b)1 year — One year. On the same test, a child of one year would require the husband and wife to total 59 years between them today, three more than the 56 the stem actually gives. Nothing in the question produces 59: the couple's total is pinned by two figures only, the average of 23 five years ago and the five years that have passed, and both are stated. This option is worth a moment because it is the sort of answer that feels right to a candidate reasoning loosely — a couple married five years, so a small child — and feelings about plausibility are exactly what an averages question punishes. The stem contains enough information to fix the answer exactly, so there is no place in it for an estimate.
• (c)3 years — Three years. A child of three would require the couple to total 57 years now, one year more than the 56 that the data give, so this is the near miss of the set and the option a candidate lands on after being one year out somewhere in the chain. That is a useful thing to see, because a single year of error in a problem like this almost always comes from the time shift rather than from the averages: it is the sort of slip made by counting the years for one spouse and forgetting the other, or by treating the marriage as four years ago. The defence is the habit of writing the totals down rather than carrying them — 46 then, 56 now, 60 for three people — because a written total can be checked in a second, while a half-remembered one cannot.

Two ideas carry every average-age question of this family. The first is that an average is only a total in disguise: with n people and an average of A, the total is nA, and the total is the quantity that can actually be added, subtracted and compared. Nothing useful can be done with the averages directly, so the first move in any such problem is to convert every average into a total and the last move is to convert back. The second idea is how a total behaves in time. If a group of n people is followed for t years, every one of them ages t years and the group's total rises by n x t. This is where almost all errors in the topic come from, because the t in the question is stated once and the n has to be supplied by the reader. A five-year gap adds 10 years to a couple, 15 to a family of three, and 5 to a single person, and the same stem can involve more than one group of different sizes — as here, where the couple is a group of two five years ago and the household is a group of three now. Keeping the two totals separate, each labelled with its size and its date, is what makes the arithmetic safe. The third component, when a question involves a birth, is that a new member joins with an age of zero and grows from there, which is why a child born since the marriage must be younger than the marriage.

Key facts

• An average is a total in disguise: n people with an average age of A have a combined age of nA, and the total is the only form in which ages from different sentences can safely be added or subtracted.
• Over t years, a group of n people gains n x t years in total. A five-year gap adds ten years to a couple and fifteen to a family of three, and forgetting the multiplier is the single commonest error in the topic.
• The working here: two people averaging 23 five years ago totalled 46 years; five years on their combined age is 56; three people averaging 20 now total 60; the child is 60 - 56 = 4 years old.
• The couple's combined age of 56 means an average of 28 each today, which is a useful intermediate check because it is a figure a reader can sanity-test against the original average of 23 five years earlier.
• The answer is consistent with the marriage: the couple married five years ago and the child is four, so the child is younger than the marriage — a check every problem of this shape should end with.
• Two slips produce numbers that are not on the paper at all — adding five instead of ten gives 9 years, and reading 23 as the present average gives 14 years — so an answer missing from the option list is itself a signal to recheck the time shift.

Common traps

• Adding the elapsed years once instead of once per person. Five years shifts a couple's combined age by ten, and this single multiplication is where most wrong answers in the topic are born.
• Reading an average stated for a past moment as though it applied now. Here 23 was the average at the time of marriage, five years ago, not today, and treating it as current gives a child of fourteen.
• Working with averages instead of totals. Averages cannot be added across groups of different sizes, so a step that adds or subtracts two averages directly is almost always wrong.
• Comparing options across mixed units. The first choice on this item is in months and the other three in years, so the four have to be put on a common footing before any of them can be judged plausible.

• practice — not a real PYQ

  The average age of a group of four friends was 20 years, three years ago. What will be the average age of the same four friends two years from now ?

  • (a)22 years
  • (b)23 years
  • (c)25 years
  • (d)27 years

  Answer(c) 25 years

• practice — not a real PYQ

  The average age of a family of five members is 24 years. If the youngest member is 8 years old, what was the average age of the family at the time of the birth of the youngest member ?

  • (a)16 years
  • (b)18 years
  • (c)20 years
  • (d)22 years

  Answer(c) 20 years