At what rate per annum compounded half yearly will Rs 10000 amount to Rs 13310 in 18 months?

• Aptitude
• Simple and compound interest

Correct Answer:

Description for Correct answer:
\( \Large A=P \left(1+\frac{1}{100}\right)^{3}\)

\( \Large \frac{1331}{1000}= \left(1+\frac{R}{100}\right)^{3}\)

\( \Large\left(\frac{11}{10}\right)^{3}= \left(1+\frac{R}{100}\right)^{3} \)

\( \Large \frac{11}{10}-1=\frac{R}{100} \)

\( \Large \frac{R}{100}=\frac{1}{10} \)

R= 10%

Part of solved Simple and compound interest questions and answers : >> Aptitude >> Simple and compound interest

Nagma invested RS. 6000 in a company at a compound interest compounded semi - annually. She receives RS. 7986 after 18 month from the company, then the rate of interest per annum is-

Answer

Verified

Hint: We solve this problem by using the formula of amount in a compound interest.
The formula for amount is given as
\[A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\]
Where, \['P'\] is the principal amount, \['r'\] is the rate of interest of a certain period and \['n'\] is the number of times the period repeated.
By using the above formula we calculate the rate of interest for semi – annual then we can find the interest per annum by multiplying the semi – annual interest by 2.

Complete step by step answer:
We are given that Nagma invested RS. 6000 in a company at a compound interest
Let us assume that the invested money which is the principal amount as
\[\Rightarrow P=6000\]
Let us assume that the rate of interest semi – annually as \['r'\]
We know that the time period for semi – annual is 6 months.
But, we are given the total time period as 18 months.
Let us assume that the 6 months is repeated \['n'\] times to get 18 months then we get
\[\begin{align}
  & \Rightarrow 6\times n=18 \\
 & \Rightarrow n=3 \\
\end{align}\]
We are given that the amount that Nagma gets after 18 months is RS. 7986
Let us assume that the amount she gets after 18 months as
\[\Rightarrow A=7986\]
We know that the formula for amount is given as
\[A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\]
Where, \['P'\] is the principal amount, \['r'\] is the rate of interest of a certain period and \['n'\] is the number of times the period is repeated.
Now, by substituting the required values by using the above formula we get
\[\begin{align}
  & \Rightarrow 7986=6000{{\left( 1+\dfrac{r}{100} \right)}^{3}} \\
 & \Rightarrow {{\left( 1+\dfrac{r}{100} \right)}^{3}}=\dfrac{1331}{1000} \\
 & \Rightarrow {{\left( 1+\dfrac{r}{100} \right)}^{3}}={{\left( \dfrac{11}{10} \right)}^{3}} \\
\end{align}\]
We know that the standard formula that is if \[{{a}^{x}}={{b}^{x}}\] then \[a=b\]
By using this formula to above equation we get
\[\begin{align}
  & \Rightarrow \left( 1+\dfrac{r}{100} \right)=\dfrac{11}{10} \\
 & \Rightarrow \dfrac{r}{100}=\dfrac{11-10}{10} \\
 & \Rightarrow r=10\% \\
\end{align}\]
Therefore, we can say that the rate of interest semi – annually is 10%
We know that we can find the interest per annum by multiplying the semi – annual interest by 2.
Let us assume that the interest per annum as \['R'\] then we get
\[\Rightarrow R=r\times 2\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow R=10\%\times 2 \\
 & \Rightarrow R=20\% \\
\end{align}\]

Therefore, we can conclude that the rate of interest per annum is 20%.

Note: Students may make mistakes in the formula of amount.
We have the formula of amount as
\[A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\]
Where, \['P'\] is the principal amount, \['r'\] is the rate of interest of a certain period and \['n'\] is the number of times the period is repeated.
But students may do mistake and take the formula as
\[A=P\left( 1+\dfrac{nr}{100} \right)\]
Where, \['P'\] is the principal amount, \['r'\] is the rate of interest of a certain period and \['n'\] is the number of times the period repeated.
This formula is wrong because we apply the rate of interest to the principal value at the start of the period.
The above formula represents that the rate of interest \['r'\] is applies for 18 months directly but, we need to apply the rate of interest of semi – annual in three parts for 6 months then the formula will be
\[A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\]
This point needs to be taken care of.

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