Question

The difference of compound interest and simple interest for 3 years and for 2 years are in ratio 23 : 7 respectively. what is rate of interest per annum (in %)?

Answer 1

Let P is the initial amount of money,
r is the rate of interest per annum
T is the time,

Then, compound interest for 3 years = P(1 + r/100)³ - P
simple interest for 3 years = P × r × 3/100
then, CI - SI = P(1 + r/100)³ - P - 3Pr/100
= P[1 + r³/100³ + 3r²/100² + 3r/100 - 1 - 3r/100]
= P[ r³/100³ + 3r²/100² ]
= P(r/100)²[r/100 + 3]

again,
Compound interest for 2 years = P[ (1 + r/100)² - 1]
simple interest for 2 years = P × r × 2/100
C.I - S.I = P[1 + r²/100² + 2r/100 - 1 - 2r/100]
= P(r/100)²

Now, A/C to question,
{(CI-SI) for 3 years }/{(CI - SI) for 2 years} = 23/7
P(r/100)² [r/100 + 3]/P(r/100)² = 23/7
⇒r/100 + 3 = 23/7
⇒ r/100 = 23/7 - 3 = 2/7
r = 200/7 %

Hence, answer is 200/7 %

Answer 2

formula for compound interest (CI): (1+p)^n - 1

formula for simple interest (SI): (1+np) - 1

Equation:

(CI(3 years) - SI(3 years)) / (CI(2 years) - SI(2 years)) = 31/10

or

solve for p in

(((1+p)^3–1) - ((1+3p)-1)) / (((1+p)^21) - ((1+2p)-1)) = 31/10

((1+p)^3 - (1+3p)) / ((1+p)^2 - (1+2p)) = 31/10

=> p = 10%