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Find the compound interest on $6000$ for one year. If the rates of interest for the first and second years are $5\% $ and $10\% $per annum respectively.

Last updated date: 22nd Jul 2024
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Hint: We have to find the value of compound interest. Already we know the formula of compound interest. Using the given values and using the formula and substitute the values in the formula. And simply find the value of compound interest.
Formula used:
  \[A = P\left[ {\left( {1 + \dfrac{{{r_1}}}{{100}}} \right)\left( {1 + \dfrac{{{r_2}}}{{100}}} \right)} \right]\];
$A = $ Amount;
$P = $ Principle amount;
$r = $Rate of interest per year;
$C.I = amount - principle$

Complete step-by-step solution:
In this problem we have to find the value of compound interest.
In this problem the given values are,
Principal amount is six thousand.
Rate of interests are five percent and ten percent.
$P = 6000$
\[{r_1} = 5\% \]
\[{r_2} = 10\% \]
We know that the formula,
\[A = P\left[ {\left( {1 + \dfrac{{{r_1}}}{{100}}} \right)\left( {1 + \dfrac{{{r_2}}}{{100}}} \right)} \right]\]
Substitute the values in the formula, we get,
$A = 6000\left[ {\left( {1 + \dfrac{5}{{100}}} \right)\left( {1 + \dfrac{{10}}{{100}}} \right)} \right]$
Cancel the common terms in the above equation,
Here, hundred is the multiplication of five and twenty and
Hundred is the multiplication of ten and ten.
Therefore, we have,
 $ = 6000\left[ {\left( {1 + \dfrac{1}{{20}}} \right)\left( {1 + \dfrac{1}{{10}}} \right)} \right]$
Take Least common multiple inside the bracket,
Least common multiple for one and twenty is twenty and
Least common multiple for one and ten is ten.
Therefore we have,
$ = 6000\left[ {\left( {\dfrac{{20 + 1}}{{20}}} \right)\left( {\dfrac{{10 + 1}}{{10}}} \right)} \right]$
Add the values, we have,
$ = 6000\left[ {\left( {\dfrac{{21}}{{20}}} \right)\left( {\dfrac{{11}}{{10}}} \right)} \right]$
Simplify the values we have,
$ = 60\left[ {\left( {\dfrac{{21}}{2}} \right)\left( {\dfrac{{11}}{1}} \right)} \right]$
Now cancel the value sixty by two, we have
$ = 30\left[ {\left( {21} \right)\left( {11} \right)} \right]$
Now multiply the values, we have
$A = 6930$
Now we find the amount for the given problem,
Now we have to find the value of compound interest.
We know that the formula,
$C.I = amount - principle$
We already know the value of the principal amount and we find the amount value. Now substitute the values already we knows,
We get,
$C.I = 6930 - 6000$
Subtracting the values we get,
$ = 930$
Therefore the compound interest is,
$C.I = 930$

Note: Compound interest is an interest calculated on the principal amount which we have. This also includes all the accumulated interest. In mathematics, compound interest gives each and every time interest is paid onto the principal amount. In compound interest for every time we have to work a new calculation on it.