Question

Find the amount and the compound interest on Rs. $1600$ for three years if the rates for three years are $3\%,4\%,5\%\text{ and 8 }\!\!\%\!\!\text{ }$ , respectively, the interest being payable annually.

Answer 1

Hint: We will apply the formula for calculating the amount for compound interest is:
$A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$, then we will apply this formula for all the four years with different interest rates and then finally we will subtract the principal from final amount to get the compound interest.

Complete step-by-step answer:
First let’s see what is a compound interest, so basically a compound interest is the addition of interest to the principal sum of a deposit, or interest on interest. It is the result of reinvesting interest, rather than paying it, so that interest in the next period is then earned on the principal plus previously accumulated interest.
The formula for calculating the amount for compound interest is:
$A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$
Where,
$A$ = amount generated
$P$ = initial principal
$r\%$ = rate of interest
$t$ = time period

Now, it is given that for the first year on interest being compounded at $r=3\%$ , now we have
$P=1600$ , $t=1$
Therefore,
$A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\Rightarrow A=1600{{\left( 1+\dfrac{3}{100} \right)}^{1}}\Rightarrow A=1600\times 1.03\Rightarrow A=1648$
Now for the second year, the principal amount will be the amount generated in the first year, therefore, $P=1648$ and the rate of interest at which principal is being compounded is $4\%$ , therefore: $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\Rightarrow A=1648{{\left( 1+\dfrac{4}{100} \right)}^{1}}\Rightarrow A=1648\times 1.04\Rightarrow A=1713.92$
Similarly , for the third year, the principal amount will, therefore, $P=1713.92$ and the rate of interest at which principal is being compounded is $5\%$ , therefore: $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\Rightarrow A=1713.92{{\left( 1+\dfrac{5}{100} \right)}^{1}}\Rightarrow A=1713.92\times 1.05\Rightarrow A=1799.61$
Now, finally for the last year, the principal amount will be $P=1799.61$ and the rate of interest at which principal is being compounded is $8\%$ , therefore: $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\Rightarrow A=1799.61{{\left( 1+\dfrac{8}{100} \right)}^{1}}\Rightarrow A=1799.61\times 1.08\Rightarrow A=1943.5788$
So, the total amount is Rs. $1943.58$ ,
And the total compound interest will be $\left( A-P \right)=1943.58-1600=343.58$ .
Therefore, $A=1943.58$ and $C.I.=343.58$ is the answer.

Note: Students may make the mistake while applying for the formula as the principal amount is different every time and also the rate of interest is different for four years. Also, in compound interest it is also possible to have yearly interest but with several compoundings within the year, which is called Periodic Compounding.