# Compound Interest Formulas, Shortcuts, Example Problems

### Arithmetic Compound Interest

Compound Interest (C.I): Compound interest is interest added to the principal so that the added interest also earns interest from then on. This addition of interest to the principal is called compounding.

### Compound Interest Basic Formulas:

•  When the interest is compounded annually, then it is called Compound Interest. Then amount after 'n' years is
Amount = P * [(1 + R/100)n];     C.I. = P * [(1 + R/100)n - 1]
• If the interest is calculated on Half yearly basis, then
Amount = P * [(1 + (R/2)/100)2n]
• If the interest is calculated on quarterly basis, then
Amount = P * [(1 + (R/4)/100)4n]
• When the rates are different for different years, say R1 %, R2 %, R3 % for 1st, 2nd, 3rd years respectively. Then Amount is given by
Amount = P * [1 + (R1/100)] * [1 + (R2/100)] * [1 + (R3/100)]
Compound Interest = Amount - Principal
• Difference between S.I & C.I
D = P * (R/100)n ; for 2 years

D = (P * R2 * [100 + R])/1003 ; for 3 years
• If amount 'A' doubles in 'x' years with S.I, then in '2x' years of time amount gets thrice. i.e. 3A, in '3x' years of time amount gets four times. i.e. 4A.
• If amount 'A' doubles in 'x' years with C.I then in '2x' years of time amount gets four times. i.e. 4A, in '3x' years of time amount gets eight times. i.e. 8A.
Problems on Compound Interest:
1. Find the C.I. on Rs. 10,000 for 2 years at the rate interest is 10 % ?
A) Rs. 2,000          B) Rs. 2,100                 C) Rs. 2,150                 D) Rs. 2,200

Ans. B
Solution:
A = P * [1 + (R/100)]n
A = 10,000 * [1 + (10/100)]2
A = 10,000 * (1.1)2
A = 10,000 * 1.21
A = 12,100
C.I = A - P = 12,100 - 10,000
C.I = Rs. 2,100.

2. Find the difference between S.I & C.I. at 5 % per annum for 2 years on a principal of Rs. 2000 ?
A) Rs. 5                 B) Rs. 10                    C) Rs. 50                       D) Rs. 100

Ans. A
Solution:
A = P * [1 + (R/100)]n
A = 2,000 * [1 + (5/100)]2
A = 2,000 * (1.1025)2
A = 2,000 * 0.0025
A = Rs. 2,205
C.I = A - P = 2,205 - 2,000
C.I = Rs. 205

S.I = (P * R * T)/100
S.I = (2,000 * 5 * 2)/100
S.I = Rs. 200
C.T - S.I = 205 - 200 = Rs 5.

`       [OR]       `

Short Cut Method:
Difference between S.I & C.I, D = P * (R/100)n
D = 2,000 * (5/100)2
D = 2,000 * (25 / [100 * 100])
D = Rs. 5

3. Find the C.I. on Rs. 1,000 at the rate of 20 % per annum for 18 months when interest is compounded half yearly ?
A) Rs. 325            B) Rs. 328                 C) Rs. 330                     D) Rs. 331

Ans. D
Solution:
C.I = P * [(1 + (R/2)/100)2n - 1]
C.I = 1,000 * [1 + (20/2)/100)]2*1.5 - ]
C.I = 1,000 * [(1.1)3 - 1]
C.I = 1,000 * [1.331 - 1]
C.I = 1,000 * 0.331
C.I = Rs. 331.

4. find the rate of interest which gives Rs. 168 as C.I on the Principal of Rs. 800 ?
A) 5 %                 B) 8 %                        C) 10 %                         D) 12 %

Ans. C
Solution:
R = ($\sqrt{\mathrm{\left(A/P\right)}}$ - 1) * 100
R = ($\sqrt{\mathrm{\left(968/800\right)}}$ - 1) * 100
R = [(11/10) - 1] * 100
R = 10 %

5.The compound interest on Rs. 30,000 at 7 % per annum is Rs. 4347. The time period (in years) is
A) 2                     B) 2.5                         C) 3                                D) 4

Ans. A
Solution:
Amount, A = 30,000 + 4,347 = Rs. 34,347
Let the time be 'n' years.
P * (1 + [R/100])n = A;
30,000 * (1 + [7/100])n = 34,347
(107/100)n = 34,347/30,000 = 11,449/10,000
(107/100)n = (107/100)2
n = 2 years.

6.What will be the compound interest on a sum of Rs. 25,000 after 3 years at the rate of 12 % per annum ?
A) Rs. 9,000                                               B) Rs. 9,725
C) Rs. 10,123.20                                        D) Rs. 10,483.20

Ans. C
Solution:
Amount, A = P * (1 + [R/100])n
A = 25,000 * (1 + [12/100])3
A = 25,000 * (28/25) * (28/25) * (28/25)
A = Rs. 35,123.20
C.I = A - P = 35,123.20 - 25,000
C.I = Rs. 10,123.20

7.At what rate of compound interest per annum will a sum of Rs. 1200 become Rs. 1348.32 in 2 years?
A) 6 %                         B) 6.5 %                        C) 7 %                        D) 7.5 %

Ans. A
Solution:
R = ($\sqrt{\mathrm{\left(A/P\right)}}$ - 1) * 100
R = ($\sqrt{\mathrm{\left(1348.32/1200\right)}}$ - 1) * 100
R = ($\sqrt{\mathrm{\left(134832/120000\right)}}$ - 1) * 100
R = [(11236/10000) - 1] * 100
R = [(106/100) - 1] * 100
R = 6 %