CI and SI - Difference Formulas

When the rate of interest is 12%, SI = 5000*3*12/100

In the case of compound interest, Amount = $$P(1+r)^n$$

CI = Amount - P = $$P(1+r)^n - P$$

CI = $$5000(1+12/100)^3 - 5000$$

CI - SI = $$5000(1+12/100)^3 - 5000 - 5000*3*12/100 = 224.64$$

Alternate Method:

Amount = $$P(1+r)^n$$

CI = Amount - P = $$P(1+r)^n - P$$

CI = $$5000(1+r/100)^3 - 5000$$

SI = 5000*3*r/100

CI - SI = $$5000(1+r/100)^3 - 5000 - 5000*3*r/100 =  224.64$$

Let R= r/100

CI - SI = $$5000(1+R)^3 - 5000 - 5000*3*R = 224.64$$

= $$(1+R)^3 - 1 - 3*R = 224.64/5000$$

=$$R^3+ 1+3R^2+3R - 1 - 3*R = 224.64/5000$$

=$$R^3+3R^2  = 224.64/5000$$

Solving the cubic equation we will get R=0.12

r= 12%

Let principal be P. SI for 2 years = P*2*10/100 = P/5. CI for 2 years = P(1+10/100)^2 - P = 21P/100. Therefore, difference in interest amount = P/100 = 1000 => P = 1,00,000. SI for 3 years = 30000. CI for 3 years = 33100. Difference = 3100.

In the 1st year the interst would be the same whether componded annually or on simple interest. The difference in the 2nd year comes due to the interest rate that is accrued on the interest gathered in the 1st year.
$$\frac{15x}{100}=21$$
where x=interst gathered in the 1st year
x=140
$$140=\frac{15P}{100}$$
P=2800/3=934