Question

The present age of a woman is $ 3 $ years more than three times the age of her daughter. Three years hence, the woman's age will be $ 10 $ years more than twice the age of her daughter. Find their present ages.

Answer 1

Hint: First we will form the equations in two variables based on the given conditions and then we will use a simple and basic mathematical principle to solve linear equations in two variables by elimination method.

Complete step-by-step answer:
Let the present age of a woman is \[x\]
And the present date of her daughter is $ y $ .
Given, the present age of woman is $ 3 $ years more than $ 3 $ times the age of daughter which means
 $ x = 3 + 3y $
 $ \Rightarrow x - 3y = 3 $ …..(i)
Also, it is given that after $ 3 $ years, the woman’s age will be $ 10 $ years more than twice the age of her daughter that means:
Firstly after $ 3 $ years, woman’s age $ = x + 3 $
Daughter’s age $ = y + 3 $
According to question, $ (x + 3) = 10 + 2(y + 3) $
 $ \Rightarrow x + 3 = 10 + 2y + 6 $
 $ \Rightarrow x - 2y = 10 + 6 - 3 $
 $ \Rightarrow x - 2y = 13 $ ……(ii)
Now, on solving linear equation (i) and (ii) i.e. subtracting (ii) from (i)
 $ (x - 3y) - (x - 2y) = 3 - 13 $
 $ \Rightarrow x - 3y - x + 2y = - 10 $
 $ \Rightarrow - y = - 10 $
 $ \Rightarrow y = 10 $
Substitute the value of $ y = 10 $ in equation (i), we get
 $ x - 3(10) = 3 $
 $ \Rightarrow x = 3 + 30 $
 $ \Rightarrow x = 33 $
So, the present age of women is $ 33 $ years and daughter’s age is $ 10 $ years.

Note: In these types of questions, always assume the present age of a given entity/person as some variable and then solve the question according to the given data. We can also solve the equations simultaneously.