Question

A sum of Rs.44200 is divided between John and Smith, 12 years and 14 years respectively in such a way that if their portions be invested at 10% per annum compound interest , they will receive equal amounts on reaching 16 years of age. What is the share of each out of Rs. 44200?

Answer 1

Hint: We will consider that the amount is divided as \[x\] and \[44200 - x\]. Now we will apply the formula to find the amount on both the principle values.one of them will circulate the principle for 2 years and other will circulate this for 4 long years. But it is given that the amount obtained by both of them at the age of 16 is the same. So we will equate the amounts and then we will find the share of each of them.

Formula used:
\[Amount = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}\]

Step by step solution:
Let the portion of John be \[x\] and that of Smith be \[44200 - x\].
Now the formula for the amount is \[Amount = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}\]
We will tabulate the amounts for both of them.

Name of the person	John -12 years	Smith-14 years
Portion assigned (principle)	\[x\]	\[44200 - x\]
Time period	4 years	2 years
Rate of interest	10%	10%
Amount received	\[Amount = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}\]Putting the values we get,\[ \Rightarrow x{\left( {1 + \dfrac{{10}}{{100}}} \right)^4}\]Cancelling the zero,\[ \Rightarrow x{\left( {1 + \dfrac{1}{{10}}} \right)^4}\]Taking LCM,\[ \Rightarrow x{\left( {\dfrac{{11}}{{10}}} \right)^4}\]	\[Amount = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}\]Putting the values we get,\[ \Rightarrow \left( {44200 - x} \right){\left( {1 + \dfrac{{10}}{{100}}} \right)^2}\]Cancelling the zero,\[ \Rightarrow \left( {44200 - x} \right){\left( {1 + \dfrac{1}{{10}}} \right)^2}\]Taking LCM,\[ \Rightarrow \left( {44200 - x} \right){\left( {\dfrac{{11}}{{10}}} \right)^2}\]

Now we have the value of both of them but according to the condition these amounts are the same. So let’s equate them,
\[ \Rightarrow x{\left( {\dfrac{{11}}{{10}}} \right)^4} = \left( {44200 - x} \right){\left( {\dfrac{{11}}{{10}}} \right)^2}\]
\[ \Rightarrow \dfrac{{x{{\left( {\dfrac{{11}}{{10}}} \right)}^4}}}{{{{\left( {\dfrac{{11}}{{10}}} \right)}^2}}} = \left( {44200 - x} \right)\]
Now since the base is same we can operate on the power
\[ \Rightarrow x{\left( {\dfrac{{11}}{{10}}} \right)^{4 - 2}} = \left( {44200 - x} \right)\]
\[ \Rightarrow x{\left( {\dfrac{{11}}{{10}}} \right)^2} = \left( {44200 - x} \right)\]
Now taking the square
\[ \Rightarrow x\left( {\dfrac{{121}}{{100}}} \right) = \left( {44200 - x} \right)\]
Taking \[x\] on LHS from RHS,
\[ \Rightarrow x\left( {\dfrac{{121}}{{100}}} \right) + x = 44200\]
Again taking LCM on LHS,
\[ \Rightarrow x\left( {\dfrac{{221}}{{100}}} \right) = 44200\]
Now divide 44200 by 221,
\[ \Rightarrow x = 44200 \times \dfrac{{100}}{{221}}\]
\[ \Rightarrow x = 200 \times 100\]
On multiplying we get,
\[ \Rightarrow x = 20000\]
This is the portion that John will receive.
Now for the portion Smith will get is \[ \Rightarrow 44200 - x = 44200 - 20000 = 24200\]

So the share of John is Rs.20000 and that of Smith is Rs.24200.

Note:
Here note that they have given the amounts received are the same but asked to find the portion they will receive from 44200. So no need to further find the amount unless asked in the question. Also note that the time period for both of them is different since their ages are different. Even though we change the portion considered of the persons we will get the same answer.
The time period is from their age given upto the time the money is invested.