Important Formulas: Simple Interest and Compound Interest | CSAT Preparation - UPSC PDF Download

Simple Interest (SI)

(i) Let the Principal Amount = P
Rate of Interest = R
and Time for which the money is invested = T

Then , Simple Interest = (P x R x T) / 100

(ii)  Amount = Principal + Interest

(iii) SI of 1 Year = CI of 1 year

(iv) Difference between SI and CI for 2 years = PR²/ 100²​

(v) Difference between SI and CI for 3 years =

 Compound Interest (CI)

(ii)  When the interest is compounded Annually:
Amount = P (1 + R/100) ⁿ

(iii)When the interest is compounded Half-yearly:
Amount = P (1 + (R/2)/100)²ⁿ

(iv)  When the interest is compounded Quarterly:
Amount = P (1 + (R/4)/100)⁴ⁿ

(v) When the rates are different for different years, say R₁%, R₂% and R₃% for 1 year, 2 years and 3-year resp. Then,
Amount = P (1 + R₁/100) (1 + R₂/100) (1 + R₃/100)

(vi)Present worth of ₹ x due n years hence is given by:
Present worth = x/ (1 + R/100)ⁿ

(vii) If a certain sum becomes “x” times in n years, then the rate of compound interest will be R = 100(x^1/n – 1)

(viii) If a sum of money P amounts to A₁ after T years at CI and the same sum of money amounts to A₂ after (T + 1) years at CI, then
R = (A₂ – A₁)/ A₁ x 100

(ix) Depreciation formula 
if A₀ is the value at a certain time and r% per annum is the rate of depreciation then the value A₁ at the end of t years will be  
A₁ = A₀(1 - r/100)^t_.

_{(x) Population Based Questions}

Let P₁ = Final population after increase/decrease
P₀ = Initial Population or Present population
r% = rate at which increase or decrease is happening

When population increases,

P₁= P₀(1 + r/100)ⁿ

When population decreases,

P₁= P₀(1- r/100)ⁿ

Growth and Growth Rates 

(i) Absolute Growth = Final Value – Initial Value
(ii) Growth rate for one year period =
(iii) SAGR or AAGR =
(iv) CAGR =

DId You Know?
If the time period is more than a year, CAGR < AAGR. 
This can be used for approximating the value of CAGR instead of calculating it.

Solved Examples 

Example 1: Ram sells onions in the streets of Chandni Chowk. Due to recent shortfall in the supply of onions, he doubles his selling price despite the cost price remains same for him due to a fixed price contract. He realizes that his profit have tripled. Find the original profit percent.

Sol:​Let the C.P. be x and S.P. be y.

∴ Given,

3(y-x) = 2y – x

=> 3y – 3x = 2y – x

=> y = 2x

Original profit = Rs y – x

= Rs 2x – x (Since, y = 2x)

= Rs x

∴ Original profit % = x/x ∗ 100

= 100 %

Example 2: Ram deposits Rs. P with a bank at r% compound interest and sees it reach Rs.16P in 20 years. If he had invested the same amount at r% simple interest for 20 years, what would be the amount?
(a) Between Rs. 2P and 2.5P
(b) Between Rs. 2.5P and 3P
(c) Between 3P and 3.5P
(d) Between 3.5P and 4P

Sol: P (1 + r)²⁰ = 16P
(1 + r)²⁰ = 16
((1 + r)¹⁰)² = 4²
(1 + r)¹⁰ = 4
((1 + r)⁵)² = 2²
((1 + r)⁵) = 2
In order for the money to double, the approximate formula for r = 72/n = 14.4% .
Rate of interest should be roughly 14 – 15%
If he had invested this in simple interest,
A = P + P x 20 x(14 to 15 %)/100 = 3.8P to 4P

Example 3: Ram earns an interest of 600 over two years on a simple interest basis. On a compound interest basis, at the same interest rate, he would earn Rs. 630. What is the rate of interest?
(a) 5%
(b) 20%
(c) 30%
(d)10% 

Sol: Given that interest on a simple interest basis is 600 for two years => Interest for one year is equal to Rs 300.
If Ram invests the amount on a compound interest basis – then interest for the first year is the same as investing in simple interest basis and that is equal to Rs 300.
Now, for CI we calculate interest on interest earned in previous time periods, whereas SI is computed purely on the principal invested.
So, CI for second year is 630 – 300 = 330 and Rs30 is the interest earned on interest of 300 which amounts to 10%.
Therefore, rate of interest = 10%.

 Example 4: The tensile strength of a material A is a multiple of amount of materials a, b, c, d used. If the amount of material of a, b, c, d are changed by +30%, -30%, -25%, +25% respectively, what will be the overall change in tensile strength of A?
(a) No Change
(b) +14.68%
(c) –14.68%
(d) Depends on the initial amount of a, b, c, d

Sol: Since, tensile strength of A = a * b * c * d, If we assume initial tensile strength to be 100, we can apply successive % changes to arrive at the final figure
100 -30% ---> 70 (Any of the change can be carried out first, the result would be same)
70 + 30% ---> 91
91 + 25% ---> 113.75
113.75 – 25% --> ~85.31
Therefore, percent change in tensile strength = -14.68%