What will be the compound interest on rupees 10000 for 1 year at 8% per annum if the interest is compounded half yearly?

A sum of 10000 is invested at the rate of 8% per year for 12 months. What is the compound interest if interest is compounded half yearly?

Answer

Hint:
 In this question, we are given the principal amount, rate of interest compounded half-yearly, and time period. We will first change the rate of interest per annum and then change the time period as per the number of times the amount increases. After that, we will use the formula of the compound amount and compound interest to find out the required answer. Formula for compound amount is given by $ A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}} $ where P is the principal amount, r is the numerical value of rate of interest and n is the time period.
The formula for compound interest is given by,
Compound interest = Compounds amount - principal amount.

Complete step by step answer:
Here we are given the principal amount as Rs.10000. Therefore, P = 10000.
Now we are given the rate of interest as 8% compounded half yearly but rate of interest is in per annum. So we will change it according to the half year that is interest becomes half so that it can be compounded after every half year. Hence, the rate of interest becomes $ \dfrac{8}{2}=4\% $ . Therefore, r = 4%.
Interest is compounded half-yearly so time period should also be half years. As the number of months are 12 so it means we have 2 half years. Therefore, n = 12.
Now we know compound amount is given by $ A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}} $ where A is compound amount, P is the principal amount, r is rate of interest and n is time period. Putting in all the values we get,
\[A=10000{{\left( 1+\dfrac{4}{100} \right)}^{2}}\]
Taking LCM as 100 we get,
\[\begin{align}
  & A=10000{{\left( \dfrac{100+4}{100} \right)}^{2}} \\
 & \Rightarrow A=10000{{\left( \dfrac{104}{100} \right)}^{2}} \\
 & \Rightarrow A=10000\times \dfrac{104}{100}\times \dfrac{104}{100} \\
\end{align}\]
Cancelling $ 100\times 100 $ with 10000 we get,
\[A=104\times 104=10816\]
Hence the amount after 12 months becomes Rs.10816. Now we know that compound interest can be calculated using the formula,
Compound interest = compound amount - principal amount.
Putting in the values we get,
Compound interest = Rs.10816 - Rs.10000 = Rs.816.
Hence the required compound interest is Rs.816

Note:
 Students should not forget to convert the rate of interest and time as for half-yearly. Students should note that in the formula for calculating the compound amount, we fill the numerical value of r only. For percentage, 100 is already divided.

Find the compound interest on ₹ 10,000 in 1 year at 5% per annum. the interest being compounded half-yearly.

1. 506.25
2. 606.25
3. 500.25
4. 500

Answer (Detailed Solution Below)

Option 1 : 506.25

Free

10 Questions 10 Marks 7 Mins

Given:

P = Rs. 10000

R = 5%

N = 1 year

Formula:

Let P = Principal, R = rate of interest and N = time

Compound interest calculated half yearly = P(1 + (R/2)/100)2n - P

Calculation:

Compound interest = 10000(1 + 5/200)2 - 10000 = Rs. 506.25

∴ Compound interest calculated half yearly is Rs. 506.25.

Last updated on Sep 21, 2022

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Solution:

What is known: Principal, Time Period, and Rate of Interest

What is unknown: Amount and Compound Interest (C.I.)

Reasoning:

A = P[1 + (r/100)]n

P = ₹ 10,000

n = \(1{\Large\frac{1}{2}}\) years

R = 10% p.a. compounded annually and half-yearly

where , A = Amount, P = Principal, n = Time period and R = Rate percent

For calculation of C.I. compounded half-yearly, we will take the Interest rate as 5% and n = 3

A = P[1 + (r/100)]n

A = 10000[1 + (5/100)]3

A = 10000[1 + (1/20)]3

A = 10000 × (21/20)3

A = 10000 × (21/20) × (21/20) × (21/20)

A = 10000 × (9261/8000)

A = 5 × (9261/4)

A = 11576.25

Interest earned at 10% p.a. compounded half-yearly = A - P

= ₹ 11576.25 - ₹ 10000 = ₹ 1576.25

Now, let's find the interest when compounded annually at the same rate of interest.

Hence, for 1 year R = 10% and n = 1

A = P[1 + (r/100)]n

A = 10000[1 + (10/100)]1

A = 10000[1 + (1/10)]

A = 10000 × (11/10)

A = 11000

Now, for the remaining 1/2 year P = 11000, R = 5%

A = P[1 + (r/100)]n

A = 11000[1 + (5/100)]

A = 11000[(105/100)]

A = 11000 × 1.05

A = 11550

Thus, amount at the end of \(1{\Large\frac{1}{2}}\)when compounded annually = ₹ 11550

Thus, compound interest = ₹ 11550 - ₹ 10000 = ₹ 1550

Therefore, the interest will be less when compounded annually at the same rate.

☛ Check: NCERT Solutions for Class 8 Maths Chapter 8

Video Solution:

Find the amount and the compound interest on ₹ 10,000 for \(1{\Large\frac{1}{2}}\) years at 10% per annum, compounded half yearly. Would this interest be more than the interest he would get if it was compounded annually?

NCERT Solutions Class 8 Maths Chapter 8 Exercise 8.3 Question 8

Summary:

The amount and the compound interest on ₹ 10,000 for \(1{\Large\frac{1}{2}}\) years at 10% per annum, compounded half-yearly is  ₹ 11576.25 and  ₹ 1576.25 respectively. The interest will be less when compounded annually at the same rate.

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