The compound interest on ₹ 5000 at 10% per annum compounded annually for two years is equal to

What will be the compound interest on Rs. 5000 if it is compounded half-yearly for 1 year 6 months at 8 % per annum.

Answer

Hint: The amount can be calculated using the given data and the formula:
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$ where,
A = Amount
P = Principal
R = Rate
T = Time
Remember to half the rate and double the time as the principal is compounded half-yearly.
Then, the compound interest can be calculated using the relationship:
Amount = Principal + Compound Interest

Complete step-by-step answer:
Given:
Principal (P) = Rs. 5000
Rate (R) = 8 %
As it is compounded half yearly, the rate will reduce to half
$R = \dfrac{8}{2}\% $
R = 4 %
Time (t) = 1 year 6 months
= $\left( {1 + \dfrac{1}{2}} \right)yrs$
 $\left(
 \because 12m = 1yr \\
 6m = \dfrac{1}{{12}} \times 6 \\
 6m = \dfrac{1}{2}yrs \\
 \right)$
= $\dfrac{3}{2}yrs$
As it is compounded half yearly, the time will be doubled:
$t = 2 \times \dfrac{3}{2}yrs$
T = 3 yrs.
Substituting these values in the formula of Amount, we get:
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$
$A = 5000{\left( {1 + \dfrac{4}{{100}}} \right)^3}$
$A = 5000{\left( {\dfrac{{104}}{{100}}} \right)^3}$
$A = 5000 \times \dfrac{{104}}{{100}} \times \dfrac{{104}}{{100}} \times \dfrac{{104}}{{100}}$
A = 5624.32
The amount is equal to Rs. 5624.32
Now,
Amount = Principal + Compound Interest
Compound Interest = Amount – Principal
Substituting the values, we get:
Compound Interest = 5624.32 – 5000
Compound Interest = 624.32
Therefore, the compound interest is Rs. 624.32 on Rs. 5000 if it is compounded half yearly for 1 year 6 months at 8 % per annum.

Note: We make the respective changes when compounded half-yearly because:
Rate is halved. The rate is generally given for a year (per annum) but when we require for half-yearly, the per annum rate is also reduced to half.
Time is doubled.When we talk about half a year (6 months), it occurs twice every year (6 + 6 = 12) and hence the time is doubled.

Simple Interest: A = P(1+rt)

Ex1: If $1000 is invested now with simple interest of 8% per year. Find the new amount after two years.

Compound Interest:

a) simple : A = P(1+rt)

b) compounded annually, n = 1:

c) compounded semiannually, n =2:

d) compounded quarterly, n = 4:

e) compounded monthly, n =12:

f) compounded daily, n =365:

Continuous Compound Interest:

As the table shows, as n increases in size, the limiting value of A is the special number

e = 2.71828

If the interest is compounded continuously for t years at a rate of r per year, then the compounded amount is given by:

A = P. e rt

Ex3: Suppose that $5000 is deposited in a saving account at the rate of 6% per year. Find the total amount on deposit at the end of 4 years if the interest is compounded continuously. (compare the result with example 2)

Ex4: If $8000 is invested for 6 years at a rate 8% compounded continuously, find the new amount:

Equivalent Value:

• If the interest rate is compounded continuously at an annual interest rate r, then Effective interest rate: = er - 1

• If the interest rate is compounded n times per year at an annual interest rate r, then Effective interest rate = (1+r/n)n - 1

Ex6: An amount is invested at 7.5% per year compounded continuously, what is the effective annual rate?

Ex7: A bank offers an effective rate of 5.41%, what is the nominal rate?

Present Value:

If the interest rate is compounded continuously at an annual rate r, the present value of a A dollars payable t years from now is

P = A. e-rt

4.3: The Growth, Decline Model:

Growth: P(t) = Po . ekt

Decline: P(t) = Po . e-kt

• P(t): the new balance the new population the new price ...
• Po: the original balance the original population the original price
• k: the rate of change the growth, decline rate the interest rate
• t: the time (years, days...)

For compounded continuously, the time T it takes to double the price, population or balance using k as the rate of change, the growth rate or the interest rate is given by:

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Ex9: The growth rate in a certain country is 15% per year. Assuming exponential growth :

Ex10: If an amount of money was doubled in 10 years, find the interest rate of the bank.

Ex11: In 1965 the price of a math book was $16. In 1980 it was $40. Assuming the exponential model :

Ex12: How long does it take money to triple in value at 6.36% compounded daily?

Ex13: A couple want an initial balance to grow to $ 211,700 in 5 years. The interest rate is compounded continuously at 15%. What should be the initial balance?

Ex14: The population of a city was 250,000 in 1970 and 200,000 in 1980 (Decline). Assuming the population is decreasing according to exponential-decay, find the population in 1990.

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