# Illustration

This illustration shows how calculating figures to higher powers can be avoided.

**Illustration:**

Find compound interest on Rs. 1000 at 16% per annum for 2 years, compounded annually.

We can use the formula for compound interest, and then we get A = 1000(1.16)^{2}.

We see, however, that using the formula is quite cumbersome, especially if number of years are more.

Students should therefore calculate simple interest and compound it.

In this case, it becomes 16% on 1000 = Rs 160 for one year, Rs 320 for two years.

On first yearâ€™s interest we get an additional 16% of 160 = Rs 25.60.

Hence the compound interest on Rs 1000 for 2 years at 16% per annum would be Rs 160 + Rs 160 + Rs 25.60 = Rs 345.60.

By doing this, the student need not calculate the high power required.

**Illustration**:

If Rs 10,000 is invested at 10 percent annual interest, compounded semiannually, what is the balance after 1 year?

The balance after the first 6 months would be

10,000 + (10,000)(0.05) = 10,500 Rupees.

The balance after one year would be 10,500 + (10,500)(0.05) = Rs. 11025.

Note that the interest rate for each 6-month period is 5%, which is half of the 10% annual rate. The balance after one year can also be expressed as

10,000 [(1 + 10/200)^{2} ] = Rs. 11025.

**Illustration:**

An office-goer deposited a total of Rs 10,500 with a company in two different deposit schemes at 10% per annum, interest being compounded annually. As per the schemes, he gets the same amount after 2 years on the first deposit as he gets after 3 years on the second deposit. How much money did he deposit for 3 years?

1. Rs 5000 2. Rs 4500 3. Rs 5500 4. Rs 6000

Let the amount invested in the first scheme be x and the second scheme be (10,500 â€“ x). Amount after two years will be 1.21x. The second amount will be (10,500 â€“ x) Ã— 1.331.

By equating the two, we get 1.21x = (10,500 â€“ x) Ã— 1.331; on solving, x = 5500. So the amount invested in the second scheme will be (10,500 â€“ 5,500) = Rs 5000, which is choice (1).

Since the calculations are long, the shorter method of doing the sum would be to work from the choices given, as follows: Clearly, the amount invested in the second scheme is less than the first scheme, since the amounts at the end of 2 and 3 years respectively are same. So choices (3) and (4) can be knocked out. Looking at the balance two choices, the amounts in the scheme is:

First choice: 500 Ã— 1.331 = 5500 Ã— 1.21

Second choice: 4500 Ã— 1.331 = 6000 Ã— 1.21

A quick calculation shows that the second choice is wrong, since

600 Ã— 1.2 = 7200 and 4500 Ã— 1.3 = 5850.

The answer therefore has to be (1).

Such techniques help build up speed, crucial for the examination.

**Illustration:**

If Rs 500 amounts to Rs 583.20 in two years compounded annually, the rate of interest per annum will be:

1. 6% 2. 6.5% 3. 7% 4. 8%

The total interest is Rs 83.20 for two years. For one year, it should be around Rs 40 (approximation) i.e. (83.20 Ã· 2). Since 40/500 is 8%, this is given by the fourth choice.

The difference of Rs 3.20 is the interest on interest for the second year.

Note that the sum has been reduced to a mental calculation rather than solving by the formula.

**Illustration:**

If the simple interest is 10.5% annual and compound interest is 10% annual, compounded annually, find the difference between the interests after 3 years on a sum of Rs. 1000.

1. Rs. 15 2. Rs. 12 3. Rs. 16 4. Rs. 11

SI = 1000 Ã— 10.5 Ã— 3/100 = Rs. 315.

CI = 1000(1.10)^{3} = 1331 â†’ 1331 â€“ 1000 = 331.

Difference in the interest is Rs. 331 â€“ 315 = Rs. 16.

A sum of Rs. 8000 is borrowed at 5% p.a. compound interest and paid back in 3 equal annual instalments. What is the amount of each installment.

1. Rs. 2937.67 2. Rs. 3000 3. Rs. 2037.67 4. Rs. 2739.76

Let each of the installment be x.

Sum of the principal values of all the three instalments =

Solve to get the answer of x = Rs 2937.67