Question

If the two digits of the ages of Mr. Manoj are reversed then the new ages obtained is the age of his wife. \[\dfrac{1}{11}\]of the sum of their ages is equal to the difference between their ages. If Mr. Manoj is older than his wife, then finds the difference between their ages.
(a) 10 years
(b) 8 years
(c) 7 years
(d) 9 years

Answer 1

Hint: In this question, let us first assume the ten's digit of manoj's age as x and one's digit as y. Now, let us write his age in terms of place value and his wife's age by changing the places of x and y in his age. Now, find the sum of their ages and difference between their ages and substitute in the given relation to simplify further. Then get the x and y values and find the difference of their ages.

Complete step-by-step solution -
Now, from the given condition that age of Manoj is a two digit
Let us assume the ten's place as x and one's place as y
Now, age of Manoj can be written as
\[\Rightarrow 10x+y\]
Here, given that on reversing the digits in age of Manoj we get age of his wife
Now, the age of wife can be written as
\[\Rightarrow 10y+x\]
Now, let us find the sum of their ages
\[\Rightarrow 10x+y+10y+x\]
Now, this can be further written in the simplified form as
\[\Rightarrow 11\left( x+y \right)\]
Let us now find the difference between their ages
\[\Rightarrow 10x+y-10y-x\]
Now, on further simplification we get,
\[\Rightarrow 9\left( x-y \right)\]
Here, given that \[\dfrac{1}{11}\]of the sum of their ages is equal to the difference between their ages
Now, on substituting the respective values we get,
\[\Rightarrow \dfrac{1}{11}\left( 11\left( x+y \right) \right)=9\left( x-y \right)\]
Now, on cancelling out the common terms we can further write it as
\[\Rightarrow x+y=9\left( x-y \right)\]
Now, on rearranging the terms on both the sides we get,
\[\Rightarrow 8x=10y\]
Now, let us bring variables to one side and constants to the other side
\[\Rightarrow \dfrac{x}{y}=\dfrac{10}{8}\]
Now, this can be further written in the simplified form as
\[\therefore \dfrac{x}{y}=\dfrac{5}{4}\]
Now, for the age to be a double digit we get,
\[\therefore x=5,y=4\]
Now, let us calculate the difference between their ages
\[\Rightarrow 9\left( x-y \right)\]
Now, on substituting the respective values we get,
\[\Rightarrow 9\left( 5-4 \right)\]
\[\Rightarrow 9\]
Thus, the difference between their ages is 9 years
Hence, the correct option is (d).

Note: Instead of assuming the digits of the age first we can also solve by assuming the ages as some variables and get the relation between them then solve further for the digits. Both the methods give the same result. It is important to note that we need to write the ages as their sum of place values like \[10x+y\] while solving instead of writing them as \[xy\]. Because writing the other does not give the result.