Question

The ages of Pramod and Rohit are $ 16 $ years and $ 18 $ years respectively. In what ratio must they invest money at $ 5\% $ p.a. compounded yearly so that both get the same sum on attaining the age of $ 25 $ years.

Answer 1

Hint: The formula for calculating amount is used here. The time period for which Pramod invests money is calculated by subtracting 16 from 25. Similarly, the time period for which Rohit invests money is calculated by subtracting 18 from 25. An equation is formed by taking both amounts as equal and the ratio of Pramod’s principal to Rohit’s principal is calculated.
Amount is given by,
\[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}\]
Where, P is principal, r is rate of interest and t is time.

Complete step-by-step answer:
Time for which Pramod invests money = \[25 - 16 = 9{\text{ years}}\]
Time for which Rohit invests money = \[25 - 18 = 7{\text{ years}}\]
Pramod’s amount is calculated as:
\[
\Rightarrow A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t} \\
\Rightarrow A = {P_{{\text{Pramod}}}}{\left( {1 + \dfrac{5}{{100}}} \right)^9} \\
 \]
Rohit’s amount is calculated as:
\[
\Rightarrow A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t} \\
\Rightarrow A = {P_{{\text{Rohit}}}}{\left( {1 + \dfrac{5}{{100}}} \right)^7} \\
 \]
As Pramod and Rohit are to attain the same sum, therefore,
\[{P_{{\text{Pramod}}}}{\left( {1 + \dfrac{5}{{100}}} \right)^9} = {P_{{\text{Rohit}}}}{\left( {1 + \dfrac{5}{{100}}} \right)^7}\]
Now, we need to find the ratio of principals of Pramod and Rohit.
\[
  \dfrac{{{P_{{\text{Pramod}}}}}}{{{P_{{\text{Rohit}}}}}} = \dfrac{{{{\left( {1 + \dfrac{5}{{100}}} \right)}^7}}}{{{{\left( {1 + \dfrac{5}{{100}}} \right)}^9}}} \\
   = \dfrac{1}{{{{\left( {1 + \dfrac{5}{{100}}} \right)}^2}}} \\
 \]
\[
   = \dfrac{1}{{{{\left( {\dfrac{{105}}{{100}}} \right)}^2}}} \\
   =\dfrac{100}{105}\times\dfrac{100}{105} \\
   = \dfrac{{400}}{{441}} \;
 \]
Therefore, the ratio in which Pramod and Rohit should invest money is \[400:441\].

Note: Since the amount is calculated on compound interest, it is important that we use this formula and not the formula for calculating amount on SI. We can also find amount by calculating SI and amount for each year. The amount in one year becomes the principal for the next year.