### Theory:

Computing the compound interest $$C.I$$:
Example:
Let us assume Sheela deposited a sum of $$₹$$ 10000 in a bank for $$2$$ years at an interest of  4 $$\%$$ compounded annually. Then we will find out the compound interest (C.I) and the amount she has to pay at the end of $$2$$ years in a couple of steps.
Steps to determine the compound interest for $$2$$ years:

Step 1: Find out the interest for one year using the principal and rate.

Step 2: Calculate the amount for one year. This amount will be the principal for next year.

Step 3: Repeat the above two steps for the amount calculated.

Step 4: Add the interest of both years to determine the compound interest for $$2$$ years.
Step 1:

Calculate the interest for the first year.

We know that the formula to calculate simple interest is given by:

${I}_{1}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{{P}_{1}×n×r}{100}$

Here:

The principal $$P_{1}$$ $$=$$ $$₹$$10000.

The time period  $$n$$ $$=$$ $$1$$ year.

The rate of interest $$r$$ $$=$$ 4 $$\%$$

Substitute the known values in the formula.

$$I_{1}$$ $$=$$ $\frac{10000·1·4}{100}$

$$=$$ $$₹$$400

Step 2:

Find the amount at the end of the first year.

Amount at the end of the first year is given by the sum of the principal and the interest of the first year.

That is the amount at the end of the first year $$=$$ $$P_{1} + I_{1}$$.

$$=$$ $10000+400$

$$=$$ $$₹$$10400

Here $$₹$$10400 is the principal for the second year $$P_{2}$$.

Step 3:

Now, find the interest for the sum $$₹$$10400.

Interest at  4 $$\%$$ for the second year is given by ${I}_{2}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{{P}_{2}×n×r}{100}$.
$$=$$ $\frac{10400·1·4}{100}$

$$=$$ $$₹$$416.

Amount at the end of the second year is given by the sum of the principal and the interest of the second year.

That is the amount at the end of the second year $$=$$ $$P_{2} + I_{2}$$.

$$=$$ $10400+416$

$$=$$ $$₹$$10816

Step 4:

Calculate the compound interest for $$2$$ years.

The compound interest is given by the sum of the interest of the first and the second year.

That is, $$C.I$$ $$=$$ $$I_{1} + I_{2}$$.

$$=$$ $400+416$

$$=$$ $$₹$$816.

We can also find this compound interest in the following way.

Compound interest $$(C.I)$$ $$=$$ Final amount $$A_{2}$$ $$-$$ Initial principal $$P_{1}$$

$$=$$ $$₹$$10816 $$-$$ $$₹$$10000

$$=$$ $$₹$$816

Important!
Do you think is there will be any difference in rupees between simple and compound interest? Let's find out that too.

Simple interest for $$2$$ years at the rate of  4$$\%$$ for the principal of $$=$$ $$₹$$10000.

$$S.I$$ $$=$$ $$\frac{P \times n \times r}{100}$$

$$=$$ $\frac{10000·2·4}{100}$

$$=$$ $$₹$$800

Now, we can find the difference between the simple interest and compound interest.

$$S.I$$ after 2 years with rate of 4 $$\%$$ $$=$$ $$₹$$800.

$$C.I$$ after  2 years with rate of 4 $$\%$$ $$=$$ $$₹$$816.

Difference $$=$$ $$C.I$$ $$-$$ $$S.I$$

$$=$$ $$₹$$816 $$-$$ $$₹$$800

$$=$$ $$₹$$16

It is evident that the compound interest is higher than the simple interest.