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

Compound Interest

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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.