Question

The sum of the ages of a man and his wife is 6 times the sum of the ages of their children. Two years ago the sum of their ages was 10 times the sum of the ages of their children at that time. After six years the sum of their ages will be 3 times the sum of the ages of their children. How many children do they have?
(a) 2 children
(b) 3 children
(c) 6 children
(d) 7 children

Answer 1

Hint: Assume that the present sum of the ages of the man and his wife is ‘x’ and the present sum of the ages of their children is ‘y’. Assume that they have ‘n’ children. Form three linear equations in three variables according to the given information and solve them to find the value of ‘n’.

Complete step-by-step answer:
Let us assume that the present sum of the ages of the man and his wife is ‘x’ and the present sum of the ages of their children is ‘y’. Let them have ‘n’ children.
Now, it is given that the sum of the ages of a man and his wife is 6 times the sum of the ages of their children. Therefore,
x = 6y………………..(i)
Second condition is that, two years ago the sum of their ages was 10 times the sum of the ages of their children at that time. So we have to subtract 2 years from each individual. Therefore,
Sum of the ages of man and wife, two years ago = x – 4, and
Sum of the ages of all the children, two years ago = y – 2n.
So, according to the question, we have,
x – 4 = 10(y – 2n)………………(ii)
Third condition is that, after six years the sum of their ages will be 3 times the sum of the ages of their children. So we have to add 6 years to each individual. Therefore,
Sum of the ages of man and wife, after 6 years = x + 12, and
Sum of the ages of all the children, after 6 years = y + 6n.
So, according to the question, we have,
x + 12 = 3(y + 6n)………………(iii)
Now, substituting the value of ‘x’ from equation (i) in equation (ii), we get,
6y – 4 = 10y – 20n
4y = 20n – 4
y = 5n – 1……………..(iv)
Similarly, substituting the value of ‘x’ from equation (i) in equation (iii), we get,
6y + 12 = 3y + 18n
3y = 18n – 12
y = 6n – 4……………..(v)
From equations (iv) and (v), we have,
5n – 1 = 6n – 4
n = 3
Hence, option (b) is the correct answer.

Note: One may note that, in the second condition we have subtracted ‘4’ from ‘x’ while calculating the sum of the age of man and his wife. You may get confused about why we haven’t subtracted only ‘2’. This is because ‘x’ is the sum of the ages of two persons, so, when we are considering 2 years back then we have to subtract 2 years from the age of both man and his wife. So a total of 4 years gets subtracted from ‘x’. Since there are ‘n’ children, so when we will subtract 2 years from each then a total of ‘2n’ years will get subtracted from ‘y’. Similar is the case with third condition also. This time we have to add 6 years to the ages of each individual.