Question

Ann is $ 12 $ years old. Her mother is $ 3 $ times as old as Ann. How many years ago was her mother $ 4 $ times as old as Ann?

Answer 1

Hint: First of all assume “M” variable for the present age of Mother and “A” the age of Ann and then accordingly convert other given word statements in the form of mathematical expression and accordingly simplify the equations for the resultant value.

Complete step-by-step answer:
Given that: Ann is $ 12 $ years old
 $ A = 12 $
Ann’s mother is $ 3 $ times as old as Ann
 $ \Rightarrow 3A = M $ …. (A)
Let “y” represent the few number of years
years ago was her mother $ 4 $ times as old as Ann
 $ (A - Y)4 = (M - Y) $
Place the value of the equation (A) in the above equation –
 $ (A - Y)4 = (3A - Y) $
Simplify the above equation, multiplying the term outside the bracket with the term inside the bracket.
 $ 4A - 4Y = 3A - Y $
Move term with “A” on the right hand side from the left side. When you move any term from one side to another, the sign of the term also changes. Positive term changes to negative term and vice-versa.
 $ 4A - 3A = - Y + 4Y $
Simplify the above expression –
 $ A = 3Y $
The above expression can be re-written as –
 $ 3Y = A $
Term multiplicative on one side, if moved to the opposite side then it goes to the denominator.
 $ Y = \dfrac{A}{3} $
Place the value from the equation (A)
 $ Y = \dfrac{{12}}{3} $
Common factors from the numerator and the denominator cancels each other.
 $ Y = 4 $ years.
This is the required solution.
So, the correct answer is “ $ Y = 4 $ years”.

Note: In these types of problems first assume the unknown quantity, here it is mother’s age. Then use the assumed variable to form an equation and compare it with the assigned value in question and solve. It will give us the required value. Find the mathematical expression from the word statement and find correlation between the unknowns.