Question

John deposited 10,000 to open a new saving that earned 4 percent annual interest, compounded quarterly. If there were no other transactions in the account, what was the amount of money in John’s account 6 months after the account was opened?
(a) 10,100
(b) 10,101
(c) 10,200
(d) 10,201
(e) 10,400

Answer 1

Hint: Use the formula for calculating the compound interest, that is (Amount – Principal), where Amount \[=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\]. Here substitute P with Principal that is 10000, r with the rate of the interest for each quarter and t for 6 months when interest is compounded quarterly.



Complete step-by-step answer:
Here, we are given that John deposited 10,000 to open a new saving that earned 4 percent annual interest, compounded quarterly. Now, we have to find the amount of money in John’s account 6 months after the account was opened if there were no transactions in the account. Before proceeding with this question, let us first see what compound interest is. Compound Interest is the interest calculated as the principal and the interest accumulated over the previous period. While calculating compound interest, the amount for a certain period of the time becomes principal for a further period of time. We can use the following formula to calculate the compound interest:
Compound Interest = Amount – Principal
where the amount is given by:
\[A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\]
Here, A = Amount for the desired time, P = Principal, r = rate of interest for the desired time, t = time for which the interest is to be calculated.

Now, let us consider our question. Here, we are given that John deposited 10,000, so we get our principal value as 10,000. Also, we are given that the interest is compounded quarterly which means 4 times in a year. We have a total of 12 months in a year. So, we get \[\dfrac{12\text{ months}}{4}=3\text{ months}\]. So, this means that the interest is getting compounded every 3 months in a year. Now, we are given that the annual rate of the interest is 4%. That means the quarterly rate of the interest would be
\[\dfrac{\text{annual rate of interest}}{4}=\dfrac{4}{4}=1%\]

Also, we are asked to find the amount in John’s account after 6 months. We already know that the interest is getting compounded every 3 months, so in 6 months, it would be compounded twice. So, we get the time period as two. So, we finally get P = 10,000, r = 1 %, t = 2. By using the formula for amount, that is,
\[A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\]
We get, \[A=10000{{\left( 1+\dfrac{1}{100} \right)}^{2}}\]
\[=10000{{\left( 1.01 \right)}^{2}}\]
\[=10201\]
So, the amount of money in John’s account will be $10201 after 6 months.

Hence, option (d) is the right answer.

Note: In this question, students often make a mistake while writing the ratio of the interest and time period. They write the rate of the interest as 4% which is wrong because we are given that this is the annual interest rate while we need to find the quarter interest rate. Also, some students write the time period as 6 months which is again wrong because interest is getting compounded every 3 months or quarterly. So, first of all, students must properly read whether interest is getting compounded annually or some other period of time and then only solve the question.