Question

Amit’s present age is 1.5 times of Hari’s age. After 4 years, Amit’s age will be twice of Hari’s age 4 years ago. What is the sum of the present ages of Amit and Hari?

Answer 1

Hint: We assume the present age of Hari as a variable and using that variable and given relation between the ages write the present age of Amit. Find the age of Hari 4 years earlier and use the relation given to find the age of amit after 4 years. Equate the equations for present age of Amit to find the respective ages. Calculate the sum of present ages.

Complete step-by-step solution:
Let us assume the present age of Hari as ‘x’ years……………...… (1)
Since we are given Amit’s present age is 1.5 times of Hari’s age
\[ \Rightarrow \]Present age of Amit\[ = 1.5x\] years………………...… (2)
Now we can write the age of Hari 4 years ago \[ = x - 4\] years……………..… (3)
After 4 years from present age, Amit’s age will be twice of Hari’s age 4 years ago
From equation (3) we can form the equation
\[ \Rightarrow \]Age of Amit after 4 years\[ = 2(x - 4)\] years……………..… (4)
From equation (2) we can write the age of Amit after 4 years as\[ = 1.5x + 4\] years……….… (5)
Since age of amit will be same after 4 years, we can equate the equations (4) and (5)
\[ \Rightarrow 2(x - 4) = 1.5x + 4\]
Multiply the terms within the bracket in LHS
\[ \Rightarrow 2x - 8 = 1.5x + 4\]
Bring all the constant terms to one side of the equation
\[ \Rightarrow 2x - 1.5x = 4 + 8\]
\[ \Rightarrow 0.5x = 12\]
Divide both sides by 0.5
\[ \Rightarrow \dfrac{{0.5x}}{{0.5}} = \dfrac{{12}}{{0.5}}\]
Cancel same terms from numerator and denominator
\[ \Rightarrow x = 24\]................… (6)
So, present age of Hari is 24 years
Substitute the value of x in equation (2)
\[ \Rightarrow \]Present age of Amit\[ = 1.5 \times 24\] years
Convert decimal to fraction
\[ \Rightarrow \]Present age of Amit\[ = \left( {\dfrac{{15}}{{10}} \times 24} \right)\] years
Cancel same factors from numerator and denominator
\[ \Rightarrow \]Present age of Amit\[ = 36\] years
So, the sum of present ages of Hari and Amit \[ = 24 + 36\]
\[ \Rightarrow \] Sum of present ages of Hari and Amit \[ = 60\] years

\[\therefore \]Sum of their present ages is 60 years.

Note: Many students assume the wrong age as independent variable; they take present age of Amit as a variable and try to form an equation accordingly, keep in mind by looking at the statement of the question we have to find out whose age is an independent factor and whose age is dependent on that independent factor, then we assume the independent age factor as a variable.
Students are likely to make mistakes while forming the equation of age after 4 years as it is linked to age of Hari before 4 years, many students think that age of Hari as present age of Hari as present age is 4 years before than the age in future. Keep in mind the relation is between Amit’s age in future (4 years) and Hari’s age in past (4 years).
Also, while multiplying the decimal to a number you can take help of conversion of decimal to fraction by writing numerator as it is and denominator as \[{10^n}\] where n is number of digits after the decimal.