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# In how much time would Rs.5000 amounts to Rs.6655 at 10% per annum compound interest?

Last updated date: 20th Jun 2024
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Hint:: We need to have concepts & know the formulas related to compound interest. Note down the given information promptly to see what remaining they asked for. Put the formula of compound interest & form an equation. Solving the equation for the unknown terms, we will have to get no. of years to produce the principal to amount & that will be the ultimate answer.

Complete step by step solution:
Given: Final amount ($A$) is $Rs.{\text{ }}6655$
Principal ($P$) is $Rs.{\text{ }}5000$
Compound interest rate ($r$) is $10{\text{ }}\%$
To find: Required time (in no. of years) to produce a given amount
To find this, there is a formula for finding the amount that will be produced after a certain time i.e, the amount produced will be the product of Principal with the one plus rate of compound interest per annum divided by 100 along with the power of no. of years (for which Principal will be compounded.)
The formula we need to apply here is
A = P${\left( {1 + \dfrac{{10}}{{100}}} \right)^t}$
Putting the values of A, P, r from the above question,
$6655$=${\text{ }}5000$ ${\left( {1 + \dfrac{{10}}{{100}}} \right)^t}$
$\Rightarrow \dfrac{6655}{5000}=$ ${\left( {1 + \dfrac{{10}}{{100}}} \right)^t}$
By dividing both numerator & denominator by $5$
$\Rightarrow $$\dfrac{{1331}}{{1000}}= {\left( {1 + \dfrac{1}{{10}}} \right)^t} Simplifying numerical part for covenience in comparision \Rightarrow$${\left( {\dfrac{{11}}{{10}}} \right)^3}$ = ${\left( {\dfrac{{11}}{{10}}} \right)^t}$
Comparing both sides of the equation,
As the base of indices are same, by laws of indices powers also should be equal.
$t=3$

$\therefore$ Hence, required time is 3 years to produce $Rs.{\text{ }}5000$ to $Rs.{\text{ }}6655$. At compound interest rate of $10\%$.

Note: To solve this problem, we should have the knowledge of compound interest in which basically interest of a year gets added with the Principal in the next year & then we find interest on the new principle that is the summation of Principal & interest of previous year. First of all, read the question very carefully to get the idea of what has been asked as an answer in the question (here which is no. of years required to produce the amount). Do the calculations carefully & always keep in mind that to find ‘$t$’ here that is in power form, you need to focus on bringing the same base in both sides of the equation otherwise it would be almost impossible to get the value of ‘$t$’ in any other way even after knowing all other procedures.