The difference in the compound interests on ₹ 10000 at 10 p.a. for 4 years and for 2 years is

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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The compound interest on Rs $10000$ in $2$ years at $4%$ per annum being compounded half yearly is A. $Rs\text{ 832}\text{.24}$B. $Rs\text{ 828}\text{.82}$C. $Rs\text{ 824}\text{.32}$D. $Rs\text{ 912}\text{.86}$

Answer

Verified

Hint: First we recall the definition and formula of compound interest and then calculate the compound interest. The formula used to calculate the compound interest is
Compound interest = Amount – Principal
And \[\text{Amount =}P{{\left( 1+\dfrac{R}{100} \right)}^{T}}\]
Where, $P=$ Principal
\[R=\] Rate of interest
$T=$Time period

Complete step by step answer:
Now, we have given that Principal sum $=10,000$
Rate of interest $=4%$ per annum
Time period \[=2\text{ years}\]
We have given that the compound interest being compounded half yearly, so the time period will be $4\text{ years}$and the rate of interest will be half i.e. $2%$ because when interest is compounded half yearly the rate of interest will be $\dfrac{R}{2}$.
Now, we have to calculate the Amount, so we put all values in the formula
\[\text{Amount =}P{{\left( 1+\dfrac{R}{100} \right)}^{T}}\]
$\Rightarrow 10000{{\left( 1+\dfrac{2}{100} \right)}^{4}}$
$\begin{align}
  & \Rightarrow 10000{{\left( 1+\dfrac{1}{50} \right)}^{4}} \\
 & \Rightarrow 10000\times \left( \dfrac{51}{50} \right)\times \left( \dfrac{51}{50} \right)\times \left( \dfrac{51}{50} \right)\times \left( \dfrac{51}{50} \right) \\
 & \Rightarrow 10.2\times 10.2\times 10.2\times 10.2 \\
 & \Rightarrow 10824.32 \\
\end{align}$
The Amount will be Rs. $10824.32$
Now we have to calculate compound interest.
We know that Compound interest = Amount – Principal
Putting the values, Compound interest will be
 $\begin{align}
  & =10824.32-10000 \\
 & =824.32 \\
\end{align}$
So, the compound interest on Rs $10000$ in $2$ years at $4%$ per annum being compounded half yearly is $Rs.824.32$.

So, the correct answer is “Option C”.

Note: Compound interest is interest on interest; it means compound interest is additional amount of interest to the principal sum. Before calculating compound interest students have to calculate the amount by using the formula and then subtract principal from amount. Students must read questions carefully about the compounding frequency i.e. interest compounded yearly, half-yearly, quarterly, monthly or weekly. The time period will be changed accordingly.

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