Question

A man lends Rs. 12,500 at 12% for the first year, at 15% for the second year and at 18% for the third year. If the rates of interest are compounded yearly. Find the difference between C.I. for the first year and the third year.

Answer 1

Hint: The principle is given, with the help of which we can calculate the amount and compound interest for each year to find the difference. The formulas to be used are:
Amount (A) = Principle (P) + Interest (I)
Interest (I) = Amount (A) – Principle (P)
${\text{Amount = P}}{\left( {1 + \dfrac{R}{{100}}} \right)^t}$

Complete step-by-step answer:
Calculating Amount for each year:
1st year:
P = 12500
R = 12% (Given)
t = 1
Substituting the values,
${\text{A = P}}\left( {1 + \dfrac{R}{{100}}} \right)$
${\text{A = 12500}}\left( {1 + \dfrac{{12}}{{100}}} \right)$
${\text{A = 12500}}\left( {\dfrac{{112}}{{100}}} \right)$
$A = 14000$
Amount for the first year Rs. 14000

2nd year:
P = 14000
R = 15% (Given)
t = 1
Substituting the values,
${\text{A = P}}\left( {1 + \dfrac{R}{{100}}} \right)$
${\text{A = 14000}}\left( {1 + \dfrac{{15}}{{100}}} \right)$
${\text{A = 14000}}\left( {\dfrac{{115}}{{100}}} \right)$
$A = 16100$
Amount for the second year Rs. 16100

This will be principle for the next year.

3rd year:
P = 16100
R = 18% (Given)
t = 1
Substituting the values,
${\text{A = P}}\left( {1 + \dfrac{R}{{100}}} \right)$
${\text{A = 16100}}\left( {1 + \dfrac{{18}}{{100}}} \right)$
${\text{A = 16100}}\left( {\dfrac{{118}}{{100}}} \right)$
$A = 18998$
Amount for the third year is Rs. 18898.
Now, calculating CI for the first and the third year.

1 Year
A = 14000, P = 12500
Therefore CI = A-P
     \[CI{\text{ }} = {\text{ }}14000 - 12500\]
     \[CI{\text{ }} = {\text{ }}1500\]

3 Year
A = 18998, P = 16100
     \[CI{\text{ }} = {\text{ }}18998 - 16100\]
     \[CI{\text{ }} = {\text{ }}2898\]

Difference between the two is given as:
\[2898 - 1500 = 1398\]
Therefore the required difference between the CI for the first and the third year is Rs. 1398.

Note: The amount of first year will become the principal for second year and so on, when we take CI for the subsequent years compounded annually.