Question

Divide Rs.20,816 between A and B so that A’s share at the end of 7 years is equal to B’s share at the end of 9 years with compound interest being $4\% p.a.$

Answer 1

Hint:
Here, we will first find the amount earned by A in 7 years and the amount earned by B in 9 years by using the compound interest formula. We will then equate both the amounts to find the ratio of A and B. Then by using the shares formula, we will find the share of A and B. Share is the amount which is divided into two halves which may or may not be equal from the total amount.

Formula Used:
We will use the following formula:
1) If the amount is compounded annually, then the amount is given by $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$ where $A$ is the amount, $P$ is the principal, $R$ is the rate of Interest and $t$ is the number of years.
2) Share is given by the formula Share $ = $ (Ratio of one share $ \div $ Total share) $ \times $ Amount

Complete step by step solution:
Let $A$ be the amount invested by A with compound interest being $4\% p.a.$ and $B$ be the amount invested by B with compound interest being $4\% p.a.$
Now, we will find the amount earned by A after being compounded annually at the end of 7 years at the rate of being $4\% p.a.$ by using the compound interest formula $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$.
Amount earned by A at the end of 7 years $ = A{\left( {1 + \dfrac{4}{{100}}} \right)^7}$
Now, we will find the amount earned by B after being compounded annually at the end of 9 years at the rate of being $4\% p.a.$ by using the compound interest formula $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$.
Amount earned by B at the end of 9 years $ = B{\left( {1 + \dfrac{4}{{100}}} \right)^9}$
We know that
 Amount earned by A $ = $ Amount earned by B
Substituting the values in the above equation, we get
$ \Rightarrow A{\left( {1 + \dfrac{4}{{100}}} \right)^7} = B{\left( {1 + \dfrac{4}{{100}}} \right)^9}$
By taking LCM on both the sides, we get
$ \Rightarrow A{\left( {1 \times \dfrac{{100}}{{100}} + \dfrac{4}{{100}}} \right)^7} = B{\left( {1 \times \dfrac{{100}}{{100}} + \dfrac{4}{{100}}} \right)^9}$
$ \Rightarrow A{\left( {\dfrac{{104}}{{100}}} \right)^7} = B{\left( {\dfrac{{104}}{{100}}} \right)^9}$
On cross multiplication, we get
$ \Rightarrow \dfrac{A}{B} = \dfrac{{{{\left( {\dfrac{{104}}{{100}}} \right)}^9}}}{{{{\left( {\dfrac{{104}}{{100}}} \right)}^7}}}$
$ \Rightarrow \dfrac{A}{B} = {\left( {\dfrac{{104}}{{100}}} \right)^2}$
By simplifying the equation, we get
$ \Rightarrow \dfrac{A}{B} = {\left( {\dfrac{{26}}{{25}}} \right)^2}$
Applying the exponent on the terms, we get
$ \Rightarrow \dfrac{A}{B} = \left( {\dfrac{{676}}{{625}}} \right)$
The ratio of $A:B$ is $676:625$
Now, we will find the share of A.
Substituting the values in the formula Share of A $ = $ (Ratio of A $ \div $ Total share) $ \times $ Amount, we get
Share of A$ = \dfrac{{676}}{{676 + 625}} \times 20816$
Adding the terms in the denominator, we get
$ \Rightarrow $ Share of A$ = \dfrac{{676}}{{1301}} \times 20816$
Dividing the terms, we get
$ \Rightarrow $ Share of A$ = 676 \times 16$
Multiplying the terms, we get
$ \Rightarrow $ Share of A$ = {\text{Rs}}.10,816$
Now, we will find the share of B.
Substituting the values in the formula Share of B $ = $ (Ratio of B $ \div $ Total share) $ \times $ Amount, we get
$ \Rightarrow $ Share of B$ = \dfrac{{625}}{{676 + 625}} \times 20816$
Adding the terms in the denominator, we get
$ \Rightarrow $ Share of B$ = \dfrac{{625}}{{1301}} \times 20816$
Dividing the terms, we get
$ \Rightarrow $ Share of B$ = 625 \times 16$
Multiplying the terms, we get
$ \Rightarrow $ Share of B$ = {\text{Rs}}.10,000$

Therefore, the amount ${\text{Rs}}.20,816$ is divided between A and B as ${\text{Rs}}.10,816$ and ${\text{Rs}}.10,000$ so that their shares are equal after certain years being compounded annually at a certain rate.

Note:
We are given that the share is invested in compound interest, so we are using the compound interest formula. We should not get confused between simple interest and compound interest. The main difference between simple interest and compound interest is that simple interest is based only on the principal amount whereas compound interest is based on the principal amount and the interest compounded for a period. We should also note that the Shares formula is similar to the Percentage formula.