The compound interest for first and second years is 200 and 220 on a certain amount. Find the sum.

To find the principal amount based on the compound interest for the first and second years, we can break down the problem step by step. Let's denote the principal amount as P, the rate of interest as r, and the compound interest for the first year as CI1 and for the second year as CI2.

Understanding Compound Interest

In compound interest, the interest for each year is calculated on the principal plus any interest that has already been added. This means that the interest for the second year is calculated on the new total, which includes the interest from the first year.

Given Values

• CI1 (First Year Interest) = 200
• CI2 (Second Year Interest) = 220

Formulating the Equations

For the first year, the interest can be expressed as:

CI1 = P * (r / 100)

For the second year, the interest is calculated on the total amount after the first year, which is:

CI2 = (P + CI1) * (r / 100)

Substituting Known Values

From the first equation, we can express r in terms of P:

r = (CI1 * 100) / P

Substituting this value of r into the second equation gives us:

220 = (P + 200) * ((200 * 100) / P)

Simplifying the Equation

Now, let's simplify this equation step by step:

220 = (P + 200) * (20000 / P)

Multiplying both sides by P to eliminate the fraction:

220P = (P + 200) * 20000

Expanding the right side:

220P = 20000P + 4000000

Now, rearranging the equation:

220P - 20000P = 4000000

-19780P = 4000000

Dividing both sides by -19780:

P = -4000000 / -19780

P ≈ 202.01

Finding the Principal Amount

Now that we have the principal amount, we can round it to a reasonable figure. The principal amount P is approximately 202.01. To find the sum of the principal and the interest earned over the two years, we can add the total interest to the principal:

Total Interest = CI1 + CI2 = 200 + 220 = 420

Total Amount = P + Total Interest = 202.01 + 420 = 622.01

Final Result

The sum of the principal amount and the compound interest earned over the two years is approximately 622.01.