What would Rs 6000 amount to in 18 months at 20% pa if the interest is compounded every 6 months?

The sooner you start to save, the more you'll earn with compound interest.

How compound interest works

Compound interest is the interest you get on:

• the money you initially deposited, called the principal
• the interest you've already earned

For example, if you have a savings account, you'll earn interest on your initial savings and on the interest you've already earned. You get interest on your interest.

This is different to simple interest. Simple interest is paid only on the principal at the end of the period. A term deposit usually earns simple interest.

Save more with compound interest

The power of compounding helps you to save more money. The longer you save, the more interest you earn. So start as soon as you can and save regularly. You'll earn a lot more than if you try to catch up later.

For example, if you put $10,000 into a savings account with 3% interest compounded monthly:

• After five years, you'd have $11,616. You'd earn $1,616 in interest.
• After 10 years you'd have $13,494. You'd earn $3,494 in interest.
• After 20 years you'd have $18,208. You'd earn $8,208 in interest.

Compound interest formula

To calculate compound interest, use the formula:

A = P x (1 + r)n

A = ending balance
P = starting balance (or principal)
r = interest rate per period as a decimal (for example, 2% becomes 0.02)
n = the number of time periods

How to calculate compound interest

To calculate how much $2,000 will earn over two years at an interest rate of 5% per year, compounded monthly:

1. Divide the annual interest rate of 5% by 12 (as interest compounds monthly) = 0.0042

2. Calculate the number of time periods (n) in months you'll be earning interest for (2 years x 12 months per year) = 24

3. Use the compound interest formula

A = $2,000 x (1+ 0.0042)24
A = $2,000 x 1.106
A = $2,211.64

Lorenzo and Sophia compare the compounding effect

Lorenzo and Sophia both decide to invest $10,000 at a 5% interest rate for five years. Sophia earns interest monthly, and Lorenzo earns interest at the end of the five-year term.

After five years:

• Sophia has $12,834.
• Lorenzo has $12,500.

Sophia and Lorenzo both started with the same amount. But Sophia gets $334 more interest than Lorenzo because of the compounding effect. Because Sophia is paid interest each month, the following month she earns interest on 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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