An amount is deposited into an account accruing interest ann

Question says in 3rd year, not after 3rd year. In 3rd year interest would have been compounded twice.
Hope it clarifies ur doubt.

Above image explains the total amount at different times..

Amount in third year = Amount at the end of second year = P(1+r)^2
Amount in the fourth year = Amount at the end of third year = P(1+r)^3

Given, P(1+r)^2 = 900
P(1+r)^3 = 1080

dividing second eq by first eq will give
(1+r) = 1080/900 = 1.2

From first eq, P = 900/(1+r)^2 = 900/1.44 = 625

Answer is D

correct me if i am wrong

R= 6/5

Now , 900 = P (1+x/100)^3

900 = P {6/5 *6/5 *6/5}

P = 521

The interest is compounded once a year. Why are you divide r by 2 and 3?

Vips0000 solution is 100% correct.

After interest has compounded twice the amount is 900: \(p(1+\frac{r}{100})^2=900\).

After interest has compounded thrice the amount is 1,080: \(p(1+\frac{r}{100})^3=1,080\).

Divide the second equation by the first one: \(1+\frac{r}{100}=\frac{1,080}{900}\) --> \(1+\frac{r}{100}=\frac{6}{5}\) --> \(r=20\).

Substitute the value of r in either equations above: \(p(1+\frac{20}{100})^2=900\) --> \(p=625\).

Answer: D.

Hope it's clear.

$900 in the third year

t= 3 year .....why have you taken t= 2 years

interest compounded in the fourth year is 180. 180 is 20% of 900. if interest is the same then at the beginning of third year, we had 900*5/6. then ad the beginning of second year, we had 900*5/6 * 5/6. and at the beginning of first year, 900*5*5*5/6*6*6.
let's split 900 into 3*3*10*10.
now..do some simplifications.
we get to 625

At the end of the second year/ beginning of third year, the value was 900. At the end of the third year/beginning of the fourth year, the value was 1080.

The difference between the amounts: 1080-900 = 180

Beginning of year three divided into the difference 180/900 = 0.2 = 20% rate

plug into the formula x(1+r)^2 = 900

x(1+0.2)^2 = 900

x(1.44) = 900

x = 625

or into the other formula:

x(1+0.2)^3 = 1080

x(1.728) = 1080

x = 625

The easier approach would be the x(1+0.2)^2 = 900 since there is less multiplication.

A more efficient approach to find the Rate of interest can be

1080 - 900 = 180

Now: 900 * R% = 180
R% = 0.2
i.e. R = 20%

I think that's more intuitive and easy to solve.

Am I missing anything ?
Compound interest formula is -> p(1+r/n)^(nt)
where t is time in years , r is the rate in decimal and n is the number of times interest is compounded in a year

based on this formula -> equation should be ->900 = p(1+r/2)^2
and ->1080 = p(1+r/3)^3

if these are the correct equations, how do we solve these 2 to get the correct value of P?

Sorry couldn't get you, can you elaborate?

Bunuel can you have a look at this? I believe vips0000 has some things missing in the equation,Thank you.

oh I see !
When it said in the third year after interest was compounded twice , I thought in the third year interest was compounded twice , but that is not so.In this question interest is compounded annually.

Yes its clear now, thank you.

For interest problems you can either use the compound interest approach, as vips did, or my approach, repeating percentages.

First, you want to know the annual interest rate. As you yield 1080 in the 4th year and 900 in the third year, you have 900 * x = 1080. Hence 1080/900 = x = 1.2

This is our interest rate, our repeating percentage.

Thus, since we know that after the interest has compunded twice, the amount of the deposit is 900, we can say that

X (original amount) * 1.2 * 1.2 = 900.

You could compute and write 1.44x = 900 but in my opinion it is easier to divide 900 by 1.2 and the result by 1.2 again.

For my computing approach:

900/1.2 looks difficult at first. BUT you can simplify this by shifting the decimal point. E.g. 900/12 = something. still too difficult. Better: 90/12 = 7.5 But since you have shifted the decimal point in the numerator one to the left and the decimal point in the denominator one to the right you have to shift the decimal point in the result two to the right. hence 7.5 -> 75 -> 750.

Then you get x * 1.2 = 750 --> x = 750/1.2. Repeat the steps above and get X!

Hope it's clearer now!

Greets

Please refer to the explanation by Vips0000 or Bunuel.

For the last part one may want to do the following. 1.44x =900. x - 900/1.44. Now take square root from both sides hence 30/1.2=25. Nos sqrt (x) = 25, so x must equal 625. Therefore D is the correct answer choice

Hope this adds
Cheers
J

Hi,

It is hard to understand that why the rate is not divided by 2 in the third year and is not divided by 3 in the fourth year, while in the question it is said that the amount is compounded twice in the third year and compounded three times in the fourth year. So, according to the formula- for third year it should be p(1+r/2)^2 and for fourth year it should be p(1+r/3)^3. Kindly tell me where I went wrong.

took me more than 2.5 minutes to solve this.is there a shorter way ?

It came me to as a Daily Question under DS. Just thought to bring it up.
My Approach:
You figured out it is compound interest. And you figured out it is 20% because 180 over 900 is 20%.
900/(1.2*1.2) results in 625 which is the original amount.
Hence D

can someone please explain this query please:

Why the rate is not divided by 2 in the third year and is not divided by 3 in the fourth year, while in the question it is said that the amount is compounded twice in the third year and compounded three times in the fourth year. So, according to the formula- for third year it should be p(1+r/2)^2 and for fourth year it should be p(1+r/3)^3.

MathRevolution