Question

A sum of money is borrowed and paid back in two equal annual installments of Rs. 882 allowing 5% compound interest. Find the sum.

Answer 1

Hint: We know the formula, \[\text{Amount=Principal}{{\left( \text{1+}\dfrac{\text{rate}}{\text{100}} \right)}^{\text{time}}}\] . Transform this formula and get the formula of the principal. The amount for the first year is Rs. 882. Now, put the value of the amount equal to Rs. 882 and get the value of the principal for the first year. Similarly, get the value of the principal for the second year. Since the sum of the money borrowed is paid in two years. So, the summation of the principal for the first year and the second year is equal to the sum of the money borrowed. Now, solve it further and get the value of the sum of the money borrowed.

Complete step by step answer:
According to the question, it is given that a sum of money is borrowed and paid back in two equal annual installments of Rs. 882 allowing 5% compound interest.
The amount paid at the end of the first year = Rs. 882 …………………………..(1)
The amount paid at the end of the second year = Rs. 882 ……………………………. (2)
The rate of the compound interest = 5% ………………………………….(3)
We know the formula, \[\text{Amount=Principal}{{\left( \text{1+}\dfrac{\text{rate}}{\text{100}} \right)}^{\text{time}}}\] ………………………………..(4)
Now, simplifying equation (4), we get
\[\dfrac{\text{Amount}}{{{\left( \text{1+}\dfrac{\text{rate}}{\text{100}} \right)}^{\text{time}}}}\text{=Principal}\] ……………………………(5)
From equation (1) and equation (3), we have the amount and rate for the first year.
Using equation (5), we can calculate the principal for the first year.
Since, we are calculating for the first year so, the time is equal to 1 year, time = 1 year …………………………(6)
Now, putting the amount from equation (1) ,the rate from equation (3), and the time from equation (6) in the formula shown in equation (5), we get
Principal for the first year = \[\dfrac{882}{{{\left( 1+\dfrac{5}{100} \right)}^{1}}}\] ………………………………(7)
Similarly, from equation (2) and equation (3), we have the amount and rate for the second year.
Using equation (5), we can calculate the principal for the first year.
Since, we are calculating for the second year so, the time is equal to 2 years, time = 2 year …………………………(8)
Now, putting the amount from equation (2), the rate from equation (3), and the time from equation (8) in the formula shown in equation (5), we get
Principal for the second year = \[\dfrac{882}{{{\left( 1+\dfrac{5}{100} \right)}^{2}}}\] ………………………………(9)
Since the sum of the money borrowed is paid in two years. So, the summation of the principal for the first year and the second year is equal to sum of the money borrowed.
From equation (7) and equation (9), we have the principal for the first year and the second year.
The sum of money borrowed = \[\dfrac{882}{{{\left( 1+\dfrac{5}{100} \right)}^{1}}}+\dfrac{882}{{{\left( 1+\dfrac{5}{100} \right)}^{2}}}=\dfrac{882}{\dfrac{105}{100}}+\dfrac{882}{{{\left( \dfrac{105}{100} \right)}^{2}}}=\dfrac{882}{\left( \dfrac{21}{20} \right)}+\dfrac{882}{{{\left( \dfrac{21}{20} \right)}^{2}}}\]
\[=\dfrac{882\times 20}{21}+\dfrac{882\times 400}{441}=42\times 20+2\times 400=840+800=1640\] .
Therefore, the sum of money borrowed is Rs. 1640.

Note:
We can also solve this question by another method.
Let us assume the sum of money be Rs. x.
The principal money for the first year = Rs. x ……………………………(1)
The rate of interest = 5% ………………………………(2)
The interest at the end of the first year = Rs. 5% of x = Rs. \[\dfrac{5}{100}\times x\] = Rs. \[\dfrac{x}{20}\] …………………………(3)
We know the formula, \[\text{Amount=Sum+Interest}\] ……………………………………(4)
From equation (1), equation (3), and equation (4), we get
\[\text{Amount=}x+\dfrac{x}{20}=\dfrac{21x}{20}\] …………………………………(5)
At the end of the first year the man pays Rs. 882.
So, the amount remaining to be paid = Rs. \[\dfrac{21x}{20}-882\] ……………………………(6)
The man clears this sum of money at the end of the second year.
The principal money for the second year = Rs. \[\dfrac{21x}{20}-882\] ……………………………………(7)
The rate of interest = 5%.
The amount that the man pays at the end of the second year = Rs. 882 ……………………………..(8)
The interest at the end of the second year = Rs. \[5\%\,of\,\left( \dfrac{21x}{20}-882 \right)\] ………………………….(9)
Now, from equation (4), equation (7), equation (8) and equation (9), we get
\[\begin{align}
  & \Rightarrow 882=\left( \dfrac{21x}{20}-882 \right)+5\%\,of\,\left( \dfrac{21x}{20}-882 \right) \\
 & \Rightarrow 882=\left( \dfrac{21x}{20}-882 \right)+\dfrac{5}{100}\times \,\left( \dfrac{21x}{20}-882 \right) \\
 & \Rightarrow 882=\left( \dfrac{21x}{20}-882 \right)\left( 1+\dfrac{5}{100} \right) \\
 & \Rightarrow 882=\left( \dfrac{21x}{20}-882 \right)\left( \dfrac{105}{100} \right) \\
 & \Rightarrow 882=\left( \dfrac{21x}{20}-882 \right)\left( \dfrac{21}{20} \right) \\
 & \Rightarrow 882=\left( \dfrac{21\times 21x}{20\times 20}-882\times \dfrac{21}{20} \right) \\
 & \Rightarrow 882+882\times \dfrac{21}{20}=\dfrac{441x}{400} \\
 & \Rightarrow 882\left( 1+\dfrac{21}{20} \right)=\dfrac{441x}{400} \\
 & \Rightarrow 882\times \dfrac{41}{20}\times \dfrac{400}{441}=x \\
 & \Rightarrow 2\times 41\times 20=x \\
 & \Rightarrow 1640=x \\
\end{align}\]
Therefore, the sum of money borrowed is Rs. 1640.