Question

Calculate the amount and compound interest on ₹15,000 in 3 years when the rate of interest for successive years is 6%, 8% and 10% respectively.
A) ₹3,879.20
B) ₹3,889.20
C) ₹3,789.20
D) ₹3,689.20

Answer 1

Hint: Since three different rate of interests are given for the three years respectively, so we will use the formula for Amount as:
\[A = P\left( {1 + \dfrac{{{r_1}}}{{100}}} \right)\left( {1 + \dfrac{{{r_2}}}{{100}}} \right)\left( {1 + \dfrac{{{r_3}}}{{100}}} \right)\]
And Interest will be the difference between Amount and Principal.

Complete step-by-step solution:
Now, according to the statement of the question, the Principal given is 17000, time is of 3years and the rate of interest for three years is 10% for first year, 10% for second year and 14% for third year.
So now according to the formula, the Amount is given as:
\[A = P\left( {1 + \dfrac{{{r_1}}}{{100}}} \right)\left( {1 + \dfrac{{{r_2}}}{{100}}} \right)\left( {1 + \dfrac{{{r_3}}}{{100}}} \right)\]
And the Interest will be calculated as: Interest = Amount – Principal
Now, first we will calculate the Amount as per the information in the question, using the formula given above:
So,
$
  P = Rs.15000 \\
  {r_1} = 6\% ,{r_2} = 8\% ,{r_3} = 10\% \\
  t = 3years \\
 $
Then the amount will be:
$
  A = 15000\left( {1 + \dfrac{6}{{100}}} \right)\left( {1 + \dfrac{8}{{100}}} \right)\left( {1 + \dfrac{{10}}{{100}}} \right) \\
   = 15000\left( {\dfrac{{100 + 6}}{{100}}} \right)\left( {\dfrac{{100 + 8}}{{100}}} \right)\left( {\dfrac{{100 + 10}}{{100}}} \right) \\
   = 15000\left( {\dfrac{{106}}{{100}}} \right)\left( {\dfrac{{108}}{{100}}} \right)\left( {\dfrac{{110}}{{100}}} \right) \\
   = \dfrac{{15 \times 106 \times 108 \times 11}}{{100}} \\
   = 18889.20 \\
 $
Therefore, the Amount is ₹18889.20
$
  Interest = Amount - Principal \\
   = 18889.20 - 15000 \\
   = 3889.20 \\
 $
So, after 3 years the total amount earned is ₹18889.20 while the total interest earned is ₹3889.20
Therefore, the total interest earned in 3 years is ₹3889.20

Hence, the correct answer is option B.

Note: To calculate the interest earned we need to calculate the amount first. If there are n different rate of interests for n years, then, the formula for Amount will become:
\[A = P\left( {1 + \dfrac{{{r_1}}}{{100}}} \right)\left( {1 + \dfrac{{{r_2}}}{{100}}} \right)\left( {1 + \dfrac{{{r_3}}}{{100}}} \right).........\left( {1 + \dfrac{{{r_n}}}{{100}}} \right)\]
However, the Interest will always be calculated as the difference of Amount and Principal.. This is accumulated interest , compounded over n years. Hence called Compound Interest with variable rate of interests.