What sum will amount to rupees 1000 in 2 years at 5% per annum compounded half yearly?

At what rate percent per annum will a sum of $Rs\,1000$ amount to $Rs\,1102.50$ in two years at compound interest?

Answer

Hint: The problem can be solved easily with the concept of compound interest. Compound interest is the interest calculated on the principal and the interest of the previous period. The amount in compound interest to be cumulated depends on the initial principal amount, rate of interest and number of time periods elapsed. The amount A after a certain number of time periods T on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ .

Complete step-by-step answer:
In the given problem,
Principal $ = P = Rs\,1000$
Rate of interest $ = R\% $
Time Duration $ = 2\,years$
In the question, the period after which the compound interest is compounded or evaluated is now given. So, we assume that the compound interest is compounded annually by default.
So, Number of time periods$ = n = 2$
Also, the amount after two years is given to us as $Rs\,1102.50$.
Now, The amount A to be paid after a certain number of time periods n on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ .
Hence, Amount $ = A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Now, substituting all the values that we have with us in the formula $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$, we get,
$ \Rightarrow 1102.50 = 1000{\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Shifting all the terms without variables to the left side of the equation, we get,
$ \Rightarrow \dfrac{{1102.50}}{{1000}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Now, simplifying the calculations, we get,
$ \Rightarrow {\left( {1 + \dfrac{R}{{100}}} \right)^2} = 1.1025$
Taking square root on both sides of equation, we get,
$ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = \sqrt {1.1025} $
Computing the square root of number,
$ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = 1.05$
Now, shifting all the constants to right side of equation, we get,
$ \Rightarrow \left( {\dfrac{R}{{100}}} \right) = 1.05 - 1$
Multiplying both sides by $100$, we get,
$ \Rightarrow R = 0.05 \times 100$
Simplifying the calculations,
$ \Rightarrow R = 5$
Therefore, the sum of $Rs\,1000$ will amount to $Rs\,1102.50$ in two years at $5\% $ per annum compound interest compounded annually.
So, the correct answer is “5%”.

Note: Time duration is not always equal to the number of time periods. The equality holds only when the compound interest is compounded annually. If the compound interest is compounded half yearly, then the number of time periods doubles in the given time duration and the rate of interest in each time period becomes half of the specified rate of interest.

What sum will amount to Rs 4913 in 18 months, if the rate of interest is \[12\frac{1}{2} \%\] per annum, compounded half-yearly?

Let the sum be Rs x.
Given: 
A = Rs 4913
R = 12 . 5 % 
n = 18 months = 1 . 5 years
We know that: 
\[A = P \left( 1 + \frac{R}{200} \right)^{2n} \]
\[4, 913 = P \left( 1 + \frac{R}{200} \right)^{2n} \]
\[4, 913 = x \left( 1 + \frac{12 . 5}{200} \right)^3 \]
\[4, 913 = x\left[ \left( 1 . 0625 \right)^3 \right]\]
\[x = \frac{4, 913}{1 . 1995}\]
\[ = 4, 096\]
Thus, the required sum is Rs 4, 096 .

Jump to

• Compound Interest Exercise 14.1
• Compound Interest Exercise 14.2
• Compound Interest Exercise 14.3
• Compound Interest Exercise 14.4
• Compound Interest Exercise 14.5

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RD Sharma Solutions Class 8 Mathematics Solutions for Compound Interest Exercise 14.2 in Chapter 14 - Compound Interest

Question 7 Compound Interest Exercise 14.2

What sum will amount to Rs. 4913 in 18 months, if the rate of interest is 12 ½ %per annum, compounded half-yearly?

Answer:

Given details are,

Rate = 12 ½% per annum = 25/2% = 25/2/2 = 25/4% half yearly

Amount = Rs 4913

Time (t) = 18months = 18/12years = 3/2 × 2 = 3 half years

By using the formula,

A = P (1 + R/100)^n

4913 = P (1 + 25/4 ×100)^3

P = 4913 / (1 + 25/400)^3

= 4913/1.19946

= 4096

∴ The principal amount is Rs 4096

Video transcript

"Hello students. Welcome Toledo's question-and-answer classroom. My name is Shai Sofia Rosie class. And today we will be finding the principal. So let's see the question and as you can see the question says what some will amount to rupees 4930. So that means amount is four thousand nine hundred and thirteen in 18 months. So the time period or n is 18 months if the rate of interest is to And a half percent per and so our rate of interest is 12 and a half percent per annum compounded half-yearly. So that means that this clear from depression that you have to find out for the half-yearly. Okay, so we will find out for half yearly. Let's quickly find out for half yearly. So first we will write down the details, but I'll get whatever are given at. This first is given that amount the amount is rupees 4 And 913 correct? Then you have given 18 months. So 18 months is the time. The time is 18 months. Now, as you know, that time is always considered to be in yours, but this is given in months. So first we will convert this into here. So it becomes 18 upon 12 years. Okay. Now since you're it has been specified that we are going to take it for half yearly. So I am going to convert this into half here. So first of all, we will calculate this eighteen upon well, which will be equal. 3 upon 2 years now to convert it into half yearly it becomes 3 upon 2 x 2 which is equal to three years. Okay. So we have got that our time which is equal to P is 3 years amount which is nothing but a is four thousand nine hundred and thirteen now it is being given the rate. Okay, so we will right now. Date, which is n is equal to 12 and a half for and so first we are going to convert this into Prop improper fraction that is 25 upon to now. This is for per annum we wanted for half-yearly. So I am just going to divide by 2. So I will get 25 upon 4 percent. Okay, so this is 25.4% / Anjali Okay, so Okay, so this is for half yearly. We have got 425 upon poor wittle half-yearly. Now we will quickly write down the amount formula because you want to find out the principal. I will just write down the formula for Amount so my amount formula Is equals to e that is principle 1 plus r upon hundred raise to n that this time period so as the specify that we don't know the principal amount is 4930. I am going to write down both thousand nine hundred and thirteen principal B1 plus rate of interest is 25. Upon for X hundred which is already present and n n is the rate of interest that is free. Okay. So if your you can see it is four thousand nine hundred and thirteen principal is P1 plus 25. Upon 400 raise to 3. Just remember a means amount time is nothing but N R means rate of interest. Okay, so four thousand nine hundred and thirteen principal now, since you are once again, you can see here, but you have to remove an LCM over here because the denominators are different. So I'm just going to take an LCM from which I'm going to get. Hundred plus Upon 400 raise to 3. So four thousand nine hundred and thirteen is equals to P 425 upon 400 the raise to 3 on which the calculation for the spot of the bracket. We'll give me. One point one nine nine four six. So therefore I'm going to divide both thousand nine hundred and thirteen divided by 1.1 9 9 4 6 will give me the principal. So therefore you can save that. I principle therefore principle. Is equals to on dividing you are going to add rupees 4096. Okay, so you please note down that the principal is rupees 4096 for half-yearly. So hope you all have understood. All right at the answer. See you all next time."

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