You might have heard the term interest when it comes to banking. But this term is not particular to blanking only , It is widely used whenever one gets profit on the amount we invest. If you have even a basic understanding of Profit and Loss chapter,it is going to be easier for you to grasp Simple interest and compound interest.
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Amount = Principal + Simple Interest (Simple Interest)
Ans: Since the money is doubling under compound interest rate from $20,000 to $40,000.
Now we use 72 rule to calculate the years to take this amount to double.
Rate = 10%
\(\text{Time to double the amount (in years)} \approx \frac{72}{\text{10}}\)
= 7.2 Years.
It will take approx 7.2 years for $20,000 to double
Simple interest = \({£120 \times 10 \times 4} \over 100 \)
When Compounded half-yearly
(i) Given rate is divided by 2
(ii) Time is multiplied by 2
The formula in that case is : \( C.I = P ( 1+ {r \over 200})^{2t} - P \)
Example 1: A sum of $1000 is invested at a compound interest rate of 20% per annum, compounded half-yearly. Find the compound interest earned after 1 year.
Ans: Principal = $1000
Annual Interest rate = 20%
Time period = 1 year
Use this formula -- \( C.I = P ( 1+ {r \over 200})^{2t} - P \)
\( C.I = 1000 ( 1+ {20\over 200})^{2 \times 1} - 1000 \)
\( CI = 1000(1.1)^2 - 1000 \)
\( CI = 1000 \times 1.21 - 1000 \)
\( CI = 1210 - 1000 \)
\( CI = $ 210 \)
Compound interest after 1 Year is $ 210
The formula in that case is : \( C.I = P ( 1+ {r \over 400})^{4t} - P \)
Example 1 : William deposits $256 in a bank at a compound interest rate of 100% per annum, compounded quarterly. Find the compound interest after 1 year.
Ans: Principal = $ 256
Annual Interest rate = 100%
Time period = 1 year
We deposit our money in the Bank for a period of time with a fixed interest rate from the bank and when the period ends we get higher amount from the bank in return.So let's start with Simple interest.
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What is simple interest
When the interest is calculated only on the principal every time. It is called simple interest. Examples sound better so let's see it with an example.
Example 1 : Alex borrows $400 from Ann at a simple interest rate of 5% for 1 years.How much money will alex return to Ann after 1 year.
To find it ,let's first calculate 5% of $400
$400 × 5% = $400 × \( 5\over 100 \) = $20
⇒ Simple interest for 1 year = $20.
The amount of money that alex will return to Ann = Money Borrowed + 1 year simple interest.
= $400 + $20
= $420
Example 2 : Ann lends $420 to carry for 2 years then how much money will carry return back after 2 years?
Example 2 : Ann lends $420 to carry for 2 years then how much money will carry return back after 2 years?
Simple interest in 1 year = $420 × 5%
= $420×\( 5 \over 100 \)
= $21
Simple interest for 2 year = $420 × 5%
= $420× \( 5 \over 100 \)
= $21
Total simple interest = $21 +$21
=$42
The amount of money carry returns to Ann after 2 years
= $420+$42
= $462
What is Compound interest
When the interest is calculated on a new amount with a fixed/variable interest rate after every period of time. it is called compound interest.
Example 1 : Alexa deposits $500 in the bank for 2 years and the bank offers a 5% annual compound interest rate. How much money will Alexa be getting from the bank after 2 years.
Compound interest for first year
= $500 × 5%
= $500× \( 5 \over 100 \)
= $25
Money after 1st year
= $500+$25
= $525
Compound interest for 2nd year
= $525×5%
= $525× \( 5 \over 100 \)
= $26.25
Total Compound interest = $25+$26.25
=$51.25
Amount of money that Alexa gets from bank after 2 years
= $500+$51.25
= $551.25
We will be solving compound interest questions using formulas in the later part of this article.
We will be solving compound interest questions using formulas in the later part of this article.
Formulae related to Simple Interest and Compound interest
In this chapter whether it is simple or compound interest we will use the following terminology
Principal = P
Rate = R
Time = T
Amount = A
Principal = P
Rate = R
Time = T
Amount = A
The formulas are listed below to find different terms involving simple and compound interest.
1. \(\text{Simple Interest} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}\)
\(S.I\) = \({P \times R \times T} \over 100 \)
2. \(\text{Amount} = \text{Principal}\left(1+\frac{\text{Rate}}{100}\right)^{\text{Time}}\)
( Amount is the total sum after simple or compound interest add to the principal)
Amount = Principal + Compound Interest (compound Interest)
( Amount is the total sum after simple or compound interest add to the principal)
Amount = Principal + Compound Interest (compound Interest)
Amount = Principal + Simple Interest (Simple Interest)
3. \(\text{Compound interest} \) = \( ( 1+{Rate \over 100})^{Time}-Principal \)
\( C.I = P ( 1+ {R \over 100})^{Time} - P \)
4. \(Principal\) = \(Simple Interest \times 100 \over {Rate \times Time }\)
\( P\) = \(S.I \times 100\over{R \times T}\)
5. \(\text{Principal} = \frac{\text{Compound Interest}}{\left(1+\frac{\text{Rate}}{100}\right)^{\text{Time}} - 1}\)
\(\text{P} = \frac{\text{C.I}}{\left(1+\frac{\text{R}}{100}\right)^{\text{T}} - 1}\)
6. \(\text{Rate} = \frac{\text{Simple Interest} \times 100}{\text{Principal} \times \text{Time}}\)
7. \(\text{Time} = \frac{\text{Simple Interest} \times 100}{\text{Principal} \times \text{Rate}}\)
8. \(\text{Amount} = \text{Principal}\left(1+\frac{\text{Rate}}{100}\right)^{\text{Time}}\)
72 Rule In Compound Interest
72 Rule works in the Case of Compound Interest only and 72 is divisible by many numbers (2, 3, 4, 6, 8, 9, 12). If a certain amount of sum doubles under a compound interest rate then with the use of 72 rule we can easily calculate the time period in which the doubles .
This rule is mainly used when something doubles under compound growth . It gives a good approximation for rates between 4% and 20%.
This rule is mainly used when something doubles under compound growth . It gives a good approximation for rates between 4% and 20%.
\(\text{Time to double (in years)} \approx \frac{72}{\text{Rate of Interest}}\)
Example 1: $20,000 is invested at a compound interest rate of 10% per annum. Approximately how many years will it take to become $40,000?
Example 1: $20,000 is invested at a compound interest rate of 10% per annum. Approximately how many years will it take to become $40,000?
Ans: Since the money is doubling under compound interest rate from $20,000 to $40,000.
Now we use 72 rule to calculate the years to take this amount to double.
Rate = 10%
\(\text{Time to double the amount (in years)} \approx \frac{72}{\text{10}}\)
= 7.2 Years.
It will take approx 7.2 years for $20,000 to double
Compound interest And Simple interest Questions using Formulas
Example 1 : Wane borrowed £120 from Jane at the simple interest rate of 10% for 4 years. The amount of money Jane returned to Wane was?
Ans: Principal= £120
Rate =10%
Time =4
Amount Jane Returned to Wane =Principal + Simple Interest
\(S.I\) = \({P \times R \times T} \over 100 \)
= £48
Amount = £120+£48
=£168
Example 2 : Wane borrowed £1200 from Jane at the simple interest rate of 10% for 2 years. The amount of money Jane returned to Wane was?
Ans : \(\text{Amount} = \text{P}\left(1+\frac{\text{R}}{100}\right)^{\text{T}}\)
=\( 1200(1+10/100)^2 \)
= 1200×(11/10)(11/10)
= £1452
Jane returned £1452 to Wane after 2 years.
Example 3 : Emily deposits $500 in a bank at a compound interest rate of 10% per annum, compounded annually. Find the compound interest earned after 2 years.
Ans: Principal = $500
Interest rate = 10%
Time = 2 years
\( CI = 500(1.21) - 500 \)
\( CI = 605 - 500 \)
\( CI = 105 \)
The compound interest earned after 2 years is $105
Example 3 : Emily deposits $500 in a bank at a compound interest rate of 10% per annum, compounded annually. Find the compound interest earned after 2 years.
Ans: Principal = $500
Interest rate = 10%
Time = 2 years
\( C.I = P ( 1+ {R \over 100})^{Time} - P \)
\( C.I = 500 ( 1+ {10 \over 100})^{2} - 500\)
\( CI = 500(1.1)^2 - 500 \)
\( C.I = 500 ( 1+ {10 \over 100})^{2} - 500\)
\( CI = 500(1.1)^2 - 500 \)
\( CI = 500(1.21) - 500 \)
\( CI = 605 - 500 \)
\( CI = 105 \)
The compound interest earned after 2 years is $105
Compound Interest when compounded Half yearly or Quarterly
We know the formula to calculate the compound interest when the interest rate is compounded yearly
\( C.I = P ( 1+ {R \over 100})^{Time} - P \)
This formula undergoes a slight change when the interest rate is not compounded yearly , but instead is compounded half-yearly or quarterly
\( C.I = P ( 1+ {R \over 100})^{Time} - P \)
This formula undergoes a slight change when the interest rate is not compounded yearly , but instead is compounded half-yearly or quarterly
When Compounded half-yearly
(i) Given rate is divided by 2
(ii) Time is multiplied by 2
The formula in that case is : \( C.I = P ( 1+ {r \over 200})^{2t} - P \)
Example 1: A sum of $1000 is invested at a compound interest rate of 20% per annum, compounded half-yearly. Find the compound interest earned after 1 year.
Ans: Principal = $1000
Annual Interest rate = 20%
Time period = 1 year
Use this formula -- \( C.I = P ( 1+ {r \over 200})^{2t} - P \)
\( C.I = 1000 ( 1+ {20\over 200})^{2 \times 1} - 1000 \)
\( CI = 1000(1.1)^2 - 1000 \)
\( CI = 1000 \times 1.21 - 1000 \)
\( CI = 1210 - 1000 \)
\( CI = $ 210 \)
Compound interest after 1 Year is $ 210
When Compounded Quarterly
(i) Given rate is divided by 4
(ii) Time is multiplied by 4
(i) Given rate is divided by 4
(ii) Time is multiplied by 4
The formula in that case is : \( C.I = P ( 1+ {r \over 400})^{4t} - P \)
Example 1 : William deposits $256 in a bank at a compound interest rate of 100% per annum, compounded quarterly. Find the compound interest after 1 year.
Ans: Principal = $ 256
Annual Interest rate = 100%
Time period = 1 year
Use the formula -- \( C.I = P ( 1+ {r \over 400})^{4t} - P \)
\( C.I = 256 ( 1+ {100 \over 400})^{4 \times 1 } - 256 \)
\( CI = 256(1.25)^4 - 256 \)
\( CI = 256 \times \frac{625}{256} - 256 \)
\( CI = 625 - 256 \)
\( CI = $369 \)
The compound interest after 1 year is $369
\( C.I = 256 ( 1+ {100 \over 400})^{4 \times 1 } - 256 \)
\( CI = 256(1.25)^4 - 256 \)
\( CI = 256 \times \frac{625}{256} - 256 \)
\( CI = 625 - 256 \)
\( CI = $369 \)
The compound interest after 1 year is $369
FAQs Based on Simple Interest and compound interest
Q1: Which interest gives higher return, simple interest or compound interest?
Ans : Compound interest always gives higher return.
Q 2 : Could Simple Interest and Principal be Equal?
Ans : Yes, simple interest and Principal may sometimes be equal.
Q 3 : How to calculate interest rate If Principal and Simple interest are equal?
Rate = 100/T