Introduction:-
"Compound Interest" is the one of the most important topic in quantitative aptitude. In "competitive exams" the most important thing is time management, if you know how to manage your time then you can do well in competitive exams. So here we are providing "shortcut tricks" on Compound Interest for your help.
First we know the following terms:-
Principal (P):
The original sum of money loaned or deposited. Also known as capital.
Interest (I):
The amount of money that you pay to borrow money or the amount of money that you earn on a deposit.
Time (T):
The duration for which the money is borrowed or deposited. The duration does not necessarily have to be years. The duration can be semi-annual, quarterly or any which may be deemed fit.
Rate of Interest (R):
The percent of interest that you pay for money borrowed, or earn for money deposited.
Read Simple Interest on click here
Compound Interest:-
Compound interest is an interest when you earn interest on both the money you have saved and the interest you earn.
Formula:-
A). If the principal is P, time is t and rate of interest per annum is R% , also amount is A.
Then we get,
1. Amount A = P\([1+\frac{R}{100}]^t\). When interest compounded annually.
And Compound Interest over the time t is
CI= A - P = P\([(1+\frac{R}{100})^t-1]\).
2. If interest compounded semi- annually, Then the amount is A= P\([1+\frac{R}{2×100}]^{2t}\).
3. If interest compounded quarterly, Then the amount is A= P\([1+\frac{R}{4×100}]^{4t}\).
4. If interest is compounded annually and time is in fraction say 2 \(\frac{3}{5}\) years.
Then amount = P\([1+\frac{R}{100}]^2[1+\frac{\frac{3}{5} R}{100}]\).
B). If population of a city is P and it increases by R % annually, then population after n years is given by: = P\([1+\frac{R}{100}]^t\).
C). If population of a city is P and it decreases by R % annually, then the population after n years is given by:
= P\([1-\frac{R}{100}]^t\).
D). When the rates of interest are different for different years, say R₁, R₂, R₃ percent for the first, second and third year, respectively, then
Amount= P\([1+\frac{R_1}{100}][1+\frac{R_2}{100}][1+\frac{R_3}{100}]\).
E). If difference between compound interest and simple interest is given for:
1) Two years
C.I. – S.I. = P\([\frac{R}{100}]^2\).
2) Three years
C.I. – S.I. =P\([\frac{R}{100}]^2[\frac{ (300 + R) }{100}]\).
F). A sum of money placed at compound interest becomes x time in ‘a’ years and y times in ‘b’ years. These two sums can be related by the following formula:
\([x]^{\frac{1}{a}}=[y]^{\frac{1}{b}}\).
G). If an amount of money grows up to Rs x in t years and up to Rs y in (t+1) years on compound interest, then
\(R\%=\frac{(y-x)\times 100}{x}\).
H). A sum at a rate of interest compounded yearly becomes Rs. \(A_1\) in n years and Rs. \(A_2\) in (n + 1) years, then
\(P=A_1(\frac{A_1}{A_2})^n\).
I). If a certain sum becomes x times of itself in t years, the rate of compound interest will be equal to
\(r=100[x^{\frac{1}{t}}-1]\).
J). If the compound interest on a certain sum for two years is CI and simple interest for two years is SI ,then rate of interest per annum is
\(r\%=2(\frac{CI-SI}{SI})\times 100\).
H). The rule of 72 is used to solve questions where a given sum of money needs to be doubled in a specific period of time with a certain rate of interest.
Formulas to remember:
Number of years invested = 72/ Annual Investment Rate
Investment Rate = 72/ Number of years Invested
Investment rate x Number of years invested = 72
For Example:
If Raju invested Rs.5000/- in a friend’s business, then how much time will it take to double Raju’s investment, if the rate of interest is 12%?
Solution:
So, according to rule of 72,
Time duration in which the amount will be doubled at 12% interest rate = 72/12 = 6 years.
Some Important Example:-
1. If the amount is 2.25 times of the sum after 2 years at compound interest (compound annually) , the rate of interest per annum is :
A. 25% B. 30% C. 45% D. 50%
2. A sum borrowed under compound interest doubles itself in 10 years. When will it become fourfold of itself at the same rate of interest?
A. 15 years B. 20 years
C. 24 years D. 40 years
3. Bharat took a loan of Rs. 20000 to purchase one LCD TV set from a finance company. He promised to make the payment after three years. The company charges compound interest at the rate of 10% per annum for the same. But, suddenly the company announces the rate of interest as 15% per annum for the last one year of the loan period. What extra amount does Rohit have to pay due to this announcement of the new rate of interest?
A) Rs. 7830 B) Rs. 4410
C) Rs. 6620 D) None of these
4. The difference between C.I. and S.I. on a certain sum at 10 % per annum for 2 years is Rs. 530. Find the sum.
A). 53000 B). 57500 C). 69800 D). 28090
5. Rs. 39030 is divided between ‘a’ and ‘b’ in such a way that the amount given to ‘a’ on C.I. in 7 years is equal to the amount given to ‘b’ on C.I. in 9 years. Find the part of ‘a’. If the rate of interest is 4%.
(A) 20200. (B) 20900. (C) 20280. (D) 20100
6. A sum of Rs. 2000 amounts to Rs. 4000 in two years at compound interest. In how many years does the same amount become Rs. 8000.
(A) 2. (B) 4. (C) 6. (D) 8
7. The compound interest on a certain sum at \(\frac{50}{3}\% \) for 3 years is Rs. 127. Find simple interest on same sum for same period and rate.
A). Rs. 205 B). Rs. 175
C). Rs. 152 D). Rs. 108
8. A sum of money is accumulating at compound interest at a certain rate of interest. If simple interest instead of compound were reckoned, the interest for first two years would be diminished by Rs 20 and that for the first three years by 61. Find the sum.
A). Rs 7000 B). Rs 47405
C). Rs 45305 D). Rs 8000
9. The difference between C.I. and S.I. accrued on an amount of Rs. 20,000 in 2 years was Rs. 392. Find the rate of interest per annum.
A). 11.5 % B). 13 % C). 14 % D). 12 %
10. There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
A). Rs. 2160 B). Rs. 3120
C). Rs. 3972 D). Rs. 6240
11. What is the difference between the compound interests on Rs. 5000 for \(1\frac{1}{2}\) years at 4% per annum compounded yearly and half-yearly?
A). Rs. 2.04 B). Rs. 3.06
C). Rs. 4.80 D). Rs. 8.30
12. If the simple interest on a sum of money for 2 years at 5% per annum is Rs. 50, what is the compound interest on the same at the same rate and for the same time?
A). Rs. 51.25 B). Rs. 52
C). Rs. 54.25 D). Rs. 60
13. The compound interest on a certain sum for 2 years at 10% per annum is Rs. 525. The simple interest on the same sum for double the time at half the rate percent per annum is:
A). Rs. 400 B). Rs. 500
C). Rs. 600 D). Rs. 800
14. The amount on ₹ 25000 in 2 years at annually compound interest. if the rate for the successive years be 4 % and 5 % per annum respectively is
A) ₹ 28500 B) ₹ 30000
C) ₹ 26800 D) ₹ 27300
15. A certain amount of money earns ₹ 540 as simple interest in 3 years. If it earns compound interest of ₹ 376.20 at the same rate of interest in 2 years, find the principal (in Rupees).
A) 2100 B) 2000 C) 1600 D) 1800
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