Difference of Compound Interest and Simple Interest

We will discuss here how to find the difference of compound interest and simple interest.

If the rate of interest per annum is the same under both simple interest and compound interest  then for 2 years, compound interest (CI) - simple interest (SI) = Simple interest for 1 year on “Simple interest for one year”.

Compound interest for 2 years – simple interest for two years

= P{(1 + \(\frac{r}{100}\))\(^{2}\) - 1} - \(\frac{P × r × 2}{100}\)

= P × \(\frac{r}{100}\) × \(\frac{r}{100}\)

= \(\frac{(P × \frac{r}{100}) × r × 1}{100}\)

= Simple interest for 1 year on “Simple interest for 1 year”.

Solve examples on difference of compound interest and simple interest:

1. Find the difference of the compound interest and simple interest on $ 15,000 at the same interest rate of 12\(\frac{1}{2}\) % per annum for 2 years.

Solution:

In case of Simple Interest:

Here,

P = principal amount (the initial amount) = $ 15,000

Rate of interest (r) = 12\(\frac{1}{2}\) % per annum = \(\frac{25}{2}\) % per annum = 12.5 % per annum

Number of years the amount is deposited or borrowed for (t) = 2 year

Using the simple interest formula, we have that

Interest = \(\frac{P × r × 2}{100}\)

           = $ \(\frac{15,000 × 12.5  × 2}{100}\)

           = $ 3,750

Therefore, the simple interest for 2 years = $ 3,750

In case of Compound Interest:

Here,

P = principal amount (the initial amount) = $ 15,000

Rate of interest (r) = 12\(\frac{1}{2}\) % per annum = \(\frac{25}{2}\) % per annum = 12.5 % per annum

Number of years the amount is deposited or borrowed for (n) = 2 year

Using the compound interest when interest is compounded annually formula, we have that

A = P(1 + \(\frac{r}{100}\))\(^{n}\)

A = $ 15,000 (1 + \(\frac{12.5}{100}\))\(^{2}\)

   = $ 15,000 (1 + 0.125)\(^{2}\)

   = $ 15,000 (1.125)\(^{2}\)

   = $ 15,000 × 1.265625

   = $ 18984.375

Therefore, the compound interest for 2 years = $ (18984.375 - 15,000)

                                                             = $ 3,984.375

Thus, the required difference of the compound interest and simple interest = $ 3,984.375 - $ 3,750 = $ 234.375.

2. What is the sum of money on which the difference between simple and compound interest in 2 years is $ 80 at the interest rate of 4% per annum?

Solution:

In case of Simple Interest:

Here,

Let P = principal amount (the initial amount) = $ z

Rate of interest (r) = 4 % per annum

Number of years the amount is deposited or borrowed for (t) = 2 year

Using the simple interest formula, we have that

Interest = \(\frac{P × r × 2}{100}\)

           = $ \(\frac{z × 4  × 2}{100}\)

           = $ \(\frac{8z}{100}\)

           = $ \(\frac{2z}{25}\)

Therefore, the simple interest for 2 years = $ \(\frac{2z}{25}\)

In case of Compound Interest:

Here,

P = principal amount (the initial amount) = $ x

Rate of interest (r) = 4 % per annum

Number of years the amount is deposited or borrowed for (n) = 2 year

Using the compound interest when interest is compounded annually formula, we have that

A = P(1 + \(\frac{r}{100}\))\(^{n}\)

A = $ z (1 + \(\frac{4}{100}\))\(^{2}\)

   = $ z (1 + \(\frac{1}{25}\))\(^{2}\)

   = $ z (\(\frac{26}{25}\))\(^{2}\)

   = $ z × (\(\frac{26}{25}\)) × (\(\frac{26}{25}\))

   = $ (\(\frac{676z}{625}\))

So, the compound interest for 2 years = Amount – Principal

                                                    = $ (\(\frac{676z}{625}\)) - $ z

                                                    = $ (\(\frac{51z}{625}\))

Now, according to the problem, the difference between simple and compound interest in 2 years is $ 80

Therefore,

    (\(\frac{51z}{625}\)) - $ \(\frac{2z}{25}\) = 80

⟹ z(\(\frac{51}{625}\) - \(\frac{2}{25}\)) = 80

⟹ \(\frac{z}{625}\) = 80

⟹ z = 80 × 625

⟹ z = 50000

Therefore, the required sum of money is $ 50000

● Compound Interest

Compound Interest

Compound Interest with Growing Principal

Compound Interest with Periodic Deductions

Compound Interest by Using Formula

Compound Interest when Interest is Compounded Yearly

Compound Interest when Interest is Compounded Half-Yearly

Compound Interest when Interest is Compounded Quarterly

Problems on Compound Interest

Variable Rate of Compound Interest

Practice Test on Compound Interest

● Compound Interest - Worksheet

Worksheet on Compound Interest

Worksheet on Compound Interest with Growing Principal

Worksheet on Compound Interest with Periodic Deductions

8th Grade Math Practice 

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