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## Arithmetic - Solving Advanced Compound Interest

We know that simple and compound interest (compounded annually) in the first year is the same. In the second year, the only difference is that in compound interest, you earn interest on previous year’s interest too. Hence, the total two year interest in compound interest exceeds the two year interest in case of simple interest by an amount which is interest on year 1 interest.

So a question such as this one is very simple to solve:

Question 1: Bob invested one half of his savings in a bond that paid simple interest for 2 years and received \$550 as interest. He invested the remaining in a bond that paid compound interest (compounded annually) for the same 2 years at the same rate of interest and received \$605 as interest. What was the annual rate of interest?

(A) 5%
(B) 10%
(C) 12%
(D) 15%
(E) 20%

Solution:

Simple Interest for two years = \$550

So simple interest per year = 550/2 = 275

But in case of compound interest, you earn an extra 605 – 550 = \$55

This \$55 is interest earned on year 1 interest i.e. if rate of interest is R, it is

55 = R% of 275

R = 20

The question is – what happens in case you have 3 years here, instead of 2? How do you solve it then? Here is a small table of the difference between simple and compound interest to help you.

Say the Principal is P and the rate of interest if R

It gets a bit more complicated though not very hard to solve. All you need to do is solve a quadratic, which, if the values are well thought out, is fairly simple to solve. Let’s look at the same question adjusted for three years.

Question 2: Bob invested one half of his savings in a bond that paid simple interest for 3 years and received \$825 as interest. He invested the remaining in a bond that paid compound interest (compounded annually) for the same 3 years at the same rate of interest and received \$1001 as interest. What was the annual rate of interest?

(A) 5%
(B) 10%
(C) 12%
(D) 15%
(E) 20%

Simple Interest for three years = \$825

So simple interest per year = 825/3 = \$275

But in case of compound interest, you earn an extra \$1001 – \$825 = \$176

What all is included in this extra \$176? This is the extra interest earned by compounding.

This is R% of interest of Year1 + R% of total interest accumulated in Year2

This is R% of 275 + R% of (275 + 275 + R% of 275) = 176

(R/100) *[825 + (R/100)*275] = 176

Assuming R/100 = x to make the equation easier,

275x^2 + 825x – 176 = 0

25x^2 + 75x – 16 = 0

25x^2 + 80x – 5x – 16 = 0

5x(5x + 16) – 1(5x + 16) = 0

x = 1/5 or -16/5

Ignore the negative value to get R/100 = 1/5 or R = 20