At what rate will a sum of Rs 40000 yield compound interest of Rs 5582.25 in 2 years?

Answer

Hint: To solve this question, first we will take all the given data from the question and then convert the rate of interest into numeric value by dividing the rate by 100, then we will substitute all the values in the formula of Compound interest $A=P{{\left( 1+\dfrac{r}{n} \right)}^{n}}$, then on solving an equation and on simplification, we will gate the value of n by comparing L.H.S and R.H.S.

Complete step by step answer:
Before we solve the question let us see what is meaning of compound interest and what the components of it are.
Compound interest just simply the addition of interest to the principal sum of a loan or deposits
The components of compound interest are as follows :
(a) Initial Principal Balance – It is the amount of loan or bond or the sum put into an investment.
(b) Interest Rate – The amount charged by a lender for the use of assets expressed as a percentage of the principal.
(c) Number of times interest is applied per time period.
(d) Number of time periods elapsed.

Now, in question, it is given that sum of Rs. 40,000 at compound interest of 5% p.m amounted to Rs.44,100.
So, we can say that the Initial Principal rate equals to Rs. 40,000. Rate of interest equals to 5% per annum and total compound interest calculated for time period ‘n’ equals to Rs. 44,100.
So, rate = $\dfrac{5}{100}$ .
We know that compound interest equals to,
$A=P{{\left( 1+\dfrac{r}{n} \right)}^{n}}$ , where A = compound Interest, P = Initial Principal balance, r = rate of interest, n = Number of times interest is applied per time period.
Putting value sin formula we get
$44,100=40,000{{\left( 1+\dfrac{5}{100} \right)}^{n}}$
On simplifying we get
$\dfrac{44,100}{40,000}={{\left( 1+\dfrac{5}{100} \right)}^{n}}$
On simplification, we get
$\dfrac{441}{400}={{\left( 1+\dfrac{5}{100} \right)}^{n}}$
Again, simplifying, we get
\[\dfrac{441}{400}={{\left( 1+\dfrac{1}{20} \right)}^{n}}\]
Solving bracket, we get
\[\dfrac{441}{400}={{\left( \dfrac{20+1}{20} \right)}^{n}}\]
\[\dfrac{441}{400}={{\left( \dfrac{21}{20} \right)}^{n}}\]
We know that ${{21}^{2}}=441$ and ${{20}^{2}}=400$
So, we can re – write above equation as
\[{{\left( \dfrac{21}{20} \right)}^{2}}={{\left( \dfrac{21}{20} \right)}^{n}}\]
On comparing, both sides L.H.S and R.H.S, we get
n = 2
So, the required time Rs 40,000 at compound interest of 5 p.m amounted to Rs. 44,100 is 2 years.

Note:
To solve such types of questions one must know the formula of compound interest which is $A=P{{\left( 1+\dfrac{r}{n} \right)}^{n}}$ and the meaning of its components. While solving, convert the rate of interest into numeric value by dividing the rate of interest by 100 and solve the equation such a way that we get the value of n by doing less calculation. Try not to make any calculation mistake.

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

• Rational Numbers
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• Playing with Numbers
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• Data Handling Probability
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RD Sharma Solutions Class 8 Mathematics Solutions for Compound Interest Exercise 14.2 in Chapter 14 - Compound Interest

Question 42 Compound Interest Exercise 14.2

At what rate percent per annum will a sum of Rs. 4000 yield compound interestof Rs. 410 in 2 years?

Answer:

Given details are,

Principal = Rs 4000

Time = 2years

CI = Rs 410

Rate be = R% per annum

By using the formula,

CI = P [(1 + R/100)^n - 1]

410 = 4000 [(1 + R/100)^2 - 1]

410 = 4000 (1 + R/100)^2 – 4000

410 + 4000 = 4000 (1 + R/100)^2

(1 + R/100)^2 = 4410/4000

(1 + R/100)^2 = 441/400

(1 + R/100)^2 = (21/20)^2

By cancelling the powers on both the sides,

1 + R/100 = 21/20

R/100 = 21/20 – 1

= (21-20)/20

= 1/20

R = 100/20

= 5

∴ Required Rate is 5% per annum

Video transcript

[Music] hello dear students i am sunita nair from leader learning i am here to show you how the sum is to be done the sum says at what rate percent per annum will a sum of rupees 4000 yield a compound interest of rupees 410 in two years right so now let's do the sum now the sum is about compound interest right and we know the formula to calculate compound interest is given by this so here the amount amount a is given by the principle multiplied by 1 plus r by 100 raised to the power of n all right now let's describe the terms here amount is a like i said the principle is p all right principle is the initial amount then we have r is the rate of interest and n is the time period all right so here's the formula a is equal to p into 1 plus r by 100 so let's substitute the values we know now what is a amount right so amount is given by the principal plus the interest accrued all right so that is p plus in this case the compound interest so here the principal amount was 4000 and the compound interest is 410. so this is equal to 4410. so let's substitute that in a so we have four thousand four hundred and ten is equal to the principle which is four thousand multiplied by one plus r which is the rate we have to find out by 100 raised to the power of 2 because it is a period of for 2 years period of 2 years so now let's do some um let's make this equation a bit easier so we will transfer 4 000 to the left hand side and 1 plus r by 100 raised to 2 remains on the right hand side so here we can cancel these zeros and we get 441 upon 400 right is equal to 1 plus r by 100 raised to 2. now what is 441 441 is the square of all right so 21 squared is 441 and 400 is the square of 20. so we have 21 over 20 raised to the power of 2 or squared is equal to 1 plus r by 100 raised to the power of 2. so now since the indices are the same on the left hand side and the right hand side we can now equate the terms without their indices so we have here 21 over 20 is equal to one plus r by hundred right in other words we are taking the square root on both the sides okay so we have 21 over 20 minus 1 we are removing the 1 from the right hand side and putting it in the left hand side is equal to r by 100 so 21 over 20 minus 1 is the same as 21 over 20 minus 20 over 20 which gives us 1 over 20 is equal to r over 100 then by cross multiplication we have 100 is equal to 20 r or r is equal to 100 upon 20 which is equal to 5 so this is our answer our rate uh which will yield a compound interest of 410 on a sum of rupees 4000 for a period of two years at compound interest is five percent so i hope you understood the explanation do drop in a comment if you have any doubt and visit our channel for more homework solutions and subscribe to it for updates as well thank you

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