Compound Interest & Effective Rate of Interest | Quantitative Aptitude for CA Foundation PDF Download

Compound Interest

Compound interest, also known as compounding interest, refers to interest computed on the initial principal amount, encompassing all accumulated interest from previous periods of a deposit or loan. The resulting interest is termed Compound Interest.

Conversion Period

The period at the end of which the interest is computed is called Conversion period.

Formula: Compound Amount =

= P(1 + i)^m

100m
Compound Interest = p[(1 + i)^m – 1]

Type – I (To find Amount & Compound Interest) Working Rule:-

• If rate of interest compounded yearly then divide r by 100 i.e i = r/100
• If rate of interest compounded 1/2 yearly Then divide r by 200 i.e = r/200
• If rate of interest compounded 1/4 yearly then i = r/400
• If rate of interest compounded monthly then i = r/1200 and so on.

Calculator Tricks:

• Step-I Type r then push + button then type 100 if r is yrly.; 200 if r is half yearly and so on.
• Step-II Press + 1.
• Step-Ill Press x button
• Step-IV Continue pressing “=” button (n – 1 = power – 1) times.
• Step -V Press × button then type principal value then push = button. We will get the result of amount on calculator screen.
• Step -VI Press – button then type principal value, we will get the value of compound Interest on the screen.

Example 1: ₹ 2000 is invested at annual rate of interest of 10%. What is the amount after 2 years if compounding is done.
(a) Annually
(b) Semi-annually
(c) Quarterly
(d) Monthly.
Ans:
(a) Given that P = ₹ 2000; r = 10% yearly.; t = 2 years.
i = r/100=10/100 = 1
n = mt = 1 × 2 = 2
[∵ m = No. of conversion periods in a year = 1]
∴ A = P(1 + i)ⁿ

= ₹ 2420.
Calculator Trick
Work As
Type 10 ÷ 100 + 1 × button then press = button (2 – 1 = 1) time then press × button type 2000 then press = button. (This is the required result).

(b) i = 10/200; m = 2
n = mt = 2 × 2 = 4

= 2000 × 1.21550625 = Rs. 2431.0125
= Rs. 2431.

Calculator Trick
Type 10 ÷ 200 then + 1 thenx button then press = button (4 – 1 = 3) times then type × 2000 = button (we get the required result)
[This Trick will take minimum 5 seconds and saves lots of time]
(c) n = mt = 4 × 2 = 8

= 2000 × [1 + 0.025]⁸ = 2000 × 1.2184
= ₹ 2436.81

Calculator Trick
Type 10 ÷ 400 + 1 × Then press = button (8 – 1 = 7) times then × 2000 = button (we get the result)
(d) i = 101/200, m = 12;n = mt = 12 × 2 = 24

Calculator Trick:
Type 10 ÷ 1200 + 1 × button then press = button (24 – 1 = 23) times then press × 2000 = button (we get the result) = ₹ 2440.78.

Example 2: Determine the compound amount and compound interest on ₹ 1000 at 6% compounded semi-annually for 6 years. Given that (1 + i)ⁿ = 1.42576 for i = 3% and n = 12.
(a) ₹ 1425.76; ₹ 425.76
(b) ₹ 1420.76 ; ₹ 420.76
(c) ₹ 1525.76 ; ₹ 525.76
(d) None
Ans:
(a) is correct.
Given that P = ₹ 1000; t = 6 years
r = 6 % compounded half -yearly
∴ m = 2 ; i = 6/200 = 0.03
n = mt = 2 × 6 = 12
∴ A = 1000(1 + 0.03)¹² = 1000 × 1.42576
= ₹ 1425.76
C.I = A-P = 1425.76 – 1000 = ₹ 425.76
[Note:- If (1 + i)ⁿ value is given in the question then use that given value]
Calculator Trick [If values not given]
Type 6 ÷ 200 + 1 × Then press = button (12 – 1 = 11) time then × 1000 = button; we will get the required result = 71425.76 = Amount. Press -1000 = button we will get compound interest value.

Effective Rate of Interest:
Type – II
The equivalent annual rate of interest compounded annually if interest is compounded more than once in a year is called EFFECTIVE RATE of INTEREST. It is denoted by E or r_e.

Formula:

where r = Nominal rate of interest, m = No. of conversion periods in a year.Calculator Trick
Type r ÷ 100m then push × button then push = button (m – 1) times then – 1 then push × 100 = we get re in %.

Ex- Find effective rate of interest of the nominal rate of interest 6% compounded quarterly.
Ans:
r = 6% compounded quarterly,
m = 4

Calculator Trick
Type 6 ÷ 400 + 1 × button then press = button (4 – 1 = 3) times then – 1 then × 100 then press = button we get the required result.

Example 3: Find effective rate of interest of 12% compounded monthly.
Ans:
r = 12% compounded monthly, m = 12

Type 12 ÷ 1200+ 1 × button then press=button 11 times then – 1 then × 100 = button.[we get the required result ]

Type-III (To find Present Value)
∵ A = P(1 + i)ⁿ

or P = A(1 + i)^-n

Calculator Trick
Step-I Type i = r ÷ 100 m + 1
Step-II Press ÷ button
Step-III Press = buttons n times i.e. (power times)
Step-IV Press × button then type A value then press = button (we get the required result)

Example 4: A certain sum invested at 4% per annum compounded semi annually amounts to ₹ 78030 at the end of one year. Find the sum.
(a) ₹ 75000
(b) ₹ 70,000
(c) ₹ 72000
(d) None
Ans:
option (a) is correct
A = ₹ 78030; r = 4% compounded semi – annually
m = 2, t = 1 yr.
n = mt = 2 × 1 = 2
= 74999.999….= ₹ 75000
Calculator Trick
Type 4 ÷ 200 + 1 then press ÷ button then press = button (power=2) times then press × button then type A value = 78030 then press = button we get the required result.