The difference between simple interest and compound interest for 2 years at 4

The formula given below can be used to find the difference between compound interest and simple interest for two years.

The above formula is applicable only in the following conditions.

1. The principal in simple interest and compound interest must be same.

2. Rate of interest must be same in simple interest and compound interest.

3. In compound interest, interest has to be compounded annually.

Example 1 :

The difference between the compound interest and simple interest on a certain investment at 10% per year for 2 years is $631. Find the value of the investment.

Solution :

The difference between compound interest and simple interest for 2 years is 631.

Then we have,

P(R/100)2 = 631

Substitute R = 10.

P(10/100)2 = 631

P(1/10)2 = 631

P(1/100) = 631

Multiply both sides by 100.

P = 631 x 100

P = 63100

So, the value of the investment is $63100.

Example 2 :

The compound interest and simple interest on a certain sum for 2 years is $ 1230 and $ 1200 respectively. The rate of interest is same for both compound interest and simple interest and it is compounded annually. What is the principal ?

Solution :

To find the principal, we need rate of interest. So, let us find the rate of interest first.

Step 1 :

Simple interest for two years is $1200. So interest per year in simple interest is $600.

So, C.I for 1st year is $600 and for 2nd year is $630.

(Since it is compounded annually, S.I and C.I for 1st year would be same)

Step 2 :

When we compare the C.I for 1st year and 2nd year, it is clear that the interest earned in 2nd year is 30 more than the first year.

Because, in C.I, interest $600 earned in 1st year earned this $30 in 2nd year.

It can be considered as simple interest for one year.

That is, principle = 600, interest = 30

I = PRT/100

30 = (600 x R x 1)/100

30 = 6R

Divide both sides by 6.

5 = R

So, R = 5%.

Step 3 :

The difference between compound interest and simple interest for two years is

= 1230 - 1200

= 30

Then we have,

P(R/100)2 = 30

Substitute R = 5.

P(5/100)2 = 30

P(1/20)2 = 30

P(1/400) = 30

Multiply both sides by 400.

P = 30 x 400

P = 12000

So, the principal is $12000.

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Simple Interest vs. Compound Interest: An Overview

Interest is the cost of borrowing money, where the borrower pays a fee to the lender for the loan. The interest, typically expressed as a percentage, can be either simple or compounded. Simple interest is based on the principal amount of a loan or deposit. In contrast, compound interest is based on the principal amount and the interest that accumulates on it in every period. Simple interest is calculated only on the principal amount of a loan or deposit, so it is easier to determine than compound interest.

Key Takeaways

• Interest is the cost of borrowing money, where the borrower pays a fee to the lender for the loan.
• Generally, simple interest paid or received over a certain period is a fixed percentage of the principal amount that was borrowed or lent.
• Compound interest accrues and is added to the accumulated interest of previous periods, so borrowers must pay interest on interest as well as principal.

Simple Interest

Simple interest is calculated using the following formula:

Simple Interest = P × r × n where: P = Principal amount r = Annual interest rate n = Term of loan, in years \begin{aligned} &\text{Simple Interest} = P \times r \times n \\ &\textbf{where:} \\ &P = \text{Principal amount} \\ &r = \text{Annual interest rate} \\ &n = \text{Term of loan, in years} \\ \end{aligned} ​Simple Interest=P×r×nwhere:P=Principal amountr=Annual interest raten=Term of loan, in years​

Generally, simple interest paid or received over a certain period is a fixed percentage of the principal amount that was borrowed or lent. For example, say a student obtains a simple-interest loan to pay one year of college tuition, which costs $18,000, and the annual interest rate on the loan is 6%. The student repays the loan over three years. The amount of simple interest paid is:

$ 3 , 240 = $ 18 , 000 × 0.06 × 3 \begin{aligned} &\$3,240 = \$18,000 \times 0.06 \times 3 \\ \end{aligned} ​$3,240=$18,000×0.06×3​

and the total amount paid is:

$ 21 , 240 = $ 18 , 000 + $ 3 , 240 \begin{aligned} &\$21,240 = \$18,000 + \$3,240 \\ \end{aligned} ​$21,240=$18,000+$3,240​

Compound Interest

Compound interest accrues and is added to the accumulated interest of previous periods; it includes interest on interest, in other words. The formula for compound interest is:

Compound Interest = P × ( 1 + r ) t − P where: P = Principal amount r = Annual interest rate t = Number of years interest is applied \begin{aligned} &\text{Compound Interest} = P \times \left ( 1 + r \right )^t - P \\ &\textbf{where:} \\ &P = \text{Principal amount} \\ &r = \text{Annual interest rate} \\ &t = \text{Number of years interest is applied} \\ \end{aligned} ​Compound Interest=P×(1+r)t−Pwhere:P=Principal amountr=Annual interest ratet=Number of years interest is applied​

It is calculated by multiplying the principal amount by one plus the annual interest rate raised to the number of compound periods, and then minus the reduction in the principal for that year. With compound interest, borrowers must pay interest on the interest as well as the principal.

Simple Interest vs. Compound Interest Examples

Below are some examples of simple and compound interest.

Example 1

Suppose you plunk $5,000 into a one-year certificate of deposit (CD) that pays simple interest at 3% per annum. The interest you earn after one year would be $150:

$ 5 , 0 0 0 × 3 % × 1 \begin{aligned} &\$5,000 \times 3\% \times 1 \\ \end{aligned} ​$5,000×3%×1​

Example 2

Continuing with the above example, suppose your certificate of deposit is cashable at any time, with interest payable to you on a prorated basis. If you cash the CD after four months, how much would you earn in interest? You would receive $50: $ 5 , 0 0 0 × 3 % × 4 1 2 \begin{aligned} &\$5,000 \times 3\% \times \frac{ 4 }{ 12 } \\ \end{aligned} ​$5,000×3%×124​​

Example 3

Suppose Bob borrows $500,000 for three years from his rich uncle, who agrees to charge Bob simple interest at 5% annually. How much would Bob have to pay in interest charges every year, and what would his total interest charges be after three years? (Assume the principal amount remains the same throughout the three years, i.e., the full loan amount is repaid after three years.) Bob would have to pay $25,000 in interest charges every year:

$ 5 0 0 , 0 0 0 × 5 % × 1 \begin{aligned} &\$500,000 \times 5\% \times 1 \\ \end{aligned} ​$500,000×5%×1​

or $75,000 in total interest charges after three years:

$ 2 5 , 0 0 0 × 3 \begin{aligned} &\$25,000 \times 3 \\ \end{aligned} ​$25,000×3​

Example 4

Continuing with the above example, Bob needs to borrow an additional $500,000 for three years. Unfortunately, his rich uncle is tapped out. So, he takes a loan from the bank at an interest rate of 5% per year compounded annually, with the full loan amount and interest payable after three years. What would be the total interest paid by Bob?

Since compound interest is calculated on the principal and accumulated interest, here's how it adds up:

After Year One, Interest Payable = $ 2 5 , 0 0 0 , or  $ 5 0 0 , 0 0 0  (Loan Principal) × 5 % × 1 After Year Two, Interest Payable = $ 2 6 , 2 5 0 , or  $ 5 2 5 , 0 0 0  (Loan Principal + Year One Interest) × 5 % × 1 After Year Three, Interest Payable = $ 2 7 , 5 6 2 . 5 0 , or  $ 5 5 1 , 2 5 0  Loan Principal + Interest for Years One and Two) × 5 % × 1 Total Interest Payable After Three Years = $ 7 8 , 8 1 2 . 5 0 , or  $ 2 5 , 0 0 0 + $ 2 6 , 2 5 0 + $ 2 7 , 5 6 2 . 5 0 \begin{aligned} &\text{After Year One, Interest Payable} = \$25,000 \text{,} \\ &\text{or } \$500,000 \text{ (Loan Principal)} \times 5\% \times 1 \\ &\text{After Year Two, Interest Payable} = \$26,250 \text{,} \\ &\text{or } \$525,000 \text{ (Loan Principal + Year One Interest)} \\ &\times 5\% \times 1 \\ &\text{After Year Three, Interest Payable} = \$27,562.50 \text{,} \\ &\text{or } \$551,250 \text{ Loan Principal + Interest for Years One} \\ &\text{and Two)} \times 5\% \times 1 \\ &\text{Total Interest Payable After Three Years} = \$78,812.50 \text{,} \\ &\text{or } \$25,000 + \$26,250 + \$27,562.50 \\ \end{aligned} ​After Year One, Interest Payable=$25,000,or $500,000 (Loan Principal)×5%×1After Year Two, Interest Payable=$26,250,or $525,000 (Loan Principal + Year One Interest)×5%×1After Year Three, Interest Payable=$27,562.50,or $551,250 Loan Principal + Interest for Years Oneand Two)×5%×1Total Interest Payable After Three Years=$78,812.50,or $25,000+$26,250+$27,562.50​

It can also be determined using the compound interest formula from above:

Total Interest Payable After Three Years = $ 7 8 , 8 1 2 . 5 0 , or  $ 5 0 0 , 0 0 0  (Loan Principal) × ( 1 + 0 . 0 5 ) 3 − $ 5 0 0 , 0 0 0 \begin{aligned} &\text{Total Interest Payable After Three Years} = \$78,812.50 \text{,} \\ &\text{or } \$500,000 \text{ (Loan Principal)} \times (1 + 0.05)^3 - \$500,000 \\ \end{aligned} ​Total Interest Payable After Three Years=$78,812.50,or $500,000 (Loan Principal)×(1+0.05)3−$500,000​

This example shows how the formula for compound interest arises from paying interest on interest as well as principal.

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