Compound Interest when Interest is Compounded Half-Yearly | Half Yearly Compounding Examples

Learn How to Calculate Compound Interest when Interest is Compounded Half-Yearly. Computation of Compound Interest by the growing principal can be complicated. Check out Solved Examples explaining the step by step process for finding the compound interest when compounded half-yearly. To help you better understand we have given the Compound Interest Formula when Interest Rate is Compounded Half-Yearly.

How to find Compound Interest when Interest is Compounded Half-Yearly?

If the rate of interest is annual and interest is compounded half-yearly then the annual interest rate is halved(r/2) and the number of years is doubled i.e. 2n. The Formula to Calculate the Compound Interest when Interest Rate is Compounded Half Yearly is given by

Let Principal = P, Rate of Interest = r/2 %, time = 2n, Amount = A, Compound Interest = CI then

A = P(1+r/2/100)2n

In the Case of the Half-Yearly Compounding, Rate Interest is divided by 2 and the number of years is multiplied by 2.

CI = A – P

= P(1+r/2/100)2n – P

=P{(1+r/2/100)2n-1}

If any three of the terms are given the fourth one can be found easily.

Problems on Compound Interest when Interest is Compounded Half-Yearly

1. Find the amount and the compound interest on $12,000 at 8 % per annum for 3 1/2 years if the interest is compounded half-yearly?

Solution:

Given Principal = $12, 000

r = 8% per annum

rate of interest half-yearly = 8/2 %

= 4%

n = 3 1/2

= 7/2 years

when compounded half yearly multiply by 2 i.e. 2n

= 7/2*2

= 7

We know Amount A = P(1+r/100)n

= 12,000(1+4/100)7

=12,000(1+0.04)7

= 12,000(1.04)7

A = Rs. 15791

We know CI = A – P

= 15791 – 12,000

= Rs. 3791

2. Find the compound interest on Rs 5000 for 3/2 years at 5% per annum, interest is payable half-yearly?

Solution:

A = P(1+r/100)n

P = 5000

n = 3/2

2n = 3/2*2

= 3

r = 5%

A = 5000(1+5/100)3

A = 5000(1.05)3

A = Rs. 5788

CI = A – P

= 5788 – 5000

= Rs. 788

Compound Interest when Interest is Compounded Yearly| How to Calculate Compound Interest Annually?

Let us learn How to Calculate Compound Interest when Interest Rate is Compounded Yearly in the coming modules. Check out Formula, Solved Examples on finding the Compound Interest Annually. We tried explaining each and every step on how to find Compound Interest. Computing the Compound Interest using Growing Principal can be difficult if the time period is long.

How to find Compound Interest when Interest is Compounded Annually?

If the Interest is Compounded Annually then Formula to Calculate the Compound Interest is given by

A = P(1+r/100)n

Where A is the Amount

P = Principal

r = rate of interest per unit time

n = Time Duration

CI can be obtained by subtracting the Principal from the Amount

CI = A – P

= P(1+r/100)n – P

= P{1+r/100)n – 1}

Solved Examples on Compound Interest when Interest is Compounded Annually

1. Find the amount and the compound interest on $8, 000 in 2 years and at 5% compounded yearly?

Solution:

Principal = $8, 000

r = 5%

n = 2

A = P(1+r/100)n

Substitute the Input Values in the formula of Amount

A = 8,000(1+5/100)2

= 8000(1+0.05)2

= 8000(1.05)2

= $8820

CI = A – P

= $8820 – $8000

= $820.

2. Find the amount of $12,000 for 2 years compounded annually, the rate of the interest being 5 % for the 1st year and 6 % for the second year?

Solution:

A = P*(1+p/100)*(1+q/100)

= 12,000(1+5/100)(1+6/100)

= 12,000(1.05)(1.06)

= $13356

Amount after 2 years is $13356.

3. Calculate the compound interest (CI) on Rs. 10, 000 for 3 years at 8% per annum compounded annually?

Solution:

Principal = Rs. 10,000

n = 3

r = 8%

A = P(1+r/100)n

= 10,000(1+8/100)3

= 10,000(1.08)3

= Rs. 12,597

CI = A – P

= 12, 597 – 10, 000

= Rs. 2, 597

Compound Interest by Using Formula | How to Calculate Compound Interest using Formula?

Do you feel any difficulty in calculating the Compound Interest? Not anymore after going through this article. Finding Compound Interest using the Formula is quite simple and you don’t have to do hectic calculations, unlike the manual methods. You just need to substitute the inputs and perform basic maths to obtain the Calculate Compound Interest instantly.

For the sake of your convenience, we have listed the Formulas for finding Compound Interest Annually, Half-Yearly, Quarterly along with Solved Examples. Refer to the Step by Step Solutions provided and understand the concept easily.

Compound Interest Formula in Different Cases

In general, Compound Interest is the Interest calculated on the Principal and the Interest accumulated over the previous period. We have listed the Compound Interest Formulas in various cases like When Interest Rate is Compounded Annually, Half-Yearly, Quarterly in the coming modules by taking enough examples.

  • Compound Interest Formula when Rate is Compounded Annually
  • Compound Interest when Rate is Compounded Half-Yearly
  • Compound Interest Quarterly Formula
  • When the Interest is Compounded Annually but rates are different for different years
  • Interest is compounded annually but time is a fraction

Annually Compound Interest Formula

We know the Formula to Calculate the Amount is A = P(1+r/n)nt

Where A= Amount

P= Principal

R= Rate of Interest

n= Number of times interest is compounded per year

If the Interest Rate is Compounded Annually we have the Formula as A = P(1+R/100)t

CI = A – P

Examples

1. Find the amount of $4000 for 2 years, compounded annually at 5% per annum. Also, find the compound interest?

Solution:

We know the formula to calculate Amount is A = P(1+R/100)n

P = $4000, R = 5%, n = 2 years

Substitute the input data we get the equation as such

A = 4000(1+5/100)2

= 4000(105/100)2

= $4410

CI = A – P

= $4410- $4000

= $410

Therefore, Compound Interest is $410.

2. Calculate the compound interest (CI) on Rs. 3000 for 1 year at 10% per annum compounded annually?

Solution:

The Formula to Calculate the Amount is A = P(1+R/100)n

P = Rs. 3000, n = 1 year R = 10%

Substituting the input data in the formula and we get

A = 3000(1+10/100)1

= 3000(1.1)

= Rs. 3300

CI = A – P

= 3300 – 3000

= Rs. 300

Half-Yearly Compound Interest Formula

Let us calculate the Compound Interest on a Principal P kept for 1 Year and at an Interest Rate R% compounded half-yearly

As the Interest is compounded half-yearly Interest Amount will vary after the first 6 months. The Interest for the next 6 months will be calculated on the remaining amount after the first 6 months.

Principal = P, Rate = R/2 %, Time = 2n

A = P(1+R/100)n

Substitute R/2 and 2n in terms of Rate and Time in the above formula

A = P(1+R/2*100)2n

CI = A – P

Example

Calculate the compound interest to be paid on a loan of Rs. 20,000 for 3/2 years at 10% per annum compounded half-yearly?

Solution:

From the given data

Principal = 20,000

R = 10%

n = 3/2

A= P(1+R/2*100)2n

Substitute the input values in the formula and we have

A = 20,000(1+10/200)2*3/2

= 20, 000(1+10/200)3

= 20,000(1.157)

= Rs. 23152

CI = A – P

= 23,152 – 20,000

= Rs. 3152

Compound Interest Quarterly Formula

Let us find the Compound Interest Kept on a Principal for 1 year and a Rate of R% compounded quarterly. Since CI is compounded quarterly principal amount will change after the first 3 months. The next 3 months(second quarter) interest will be calculated on the amount left after the first 3 months. Third-quarter Interest will be calculated on the amount left after the first 6 months. Last quarter will be found on the amount left after the first 9 months.

The formula of Compound Interest Compounded Quarterly is given as

A = P(1+R/4/100)4n

CI = A – P

Example

Find the compound interest on $12,000 if Nick took a loan from a bank for 6 months at 8 % per annum, compounded quarterly?

Solution:

From the given data P = $12, 000

R = 8% per Annum, (8/4)% per quarter = 2% per quarter

T = 6 months = 2 Quarters

A = P(1+R/100)n

= 12,000(1+2/100)2

= 12,000(1+0.02)2

= 12,000(1.02)2

= Rs. 12,484

CI = A – P

= 12,484 – 12,000

= Rs. 484

When the Interest is Compounded Annually but rates are different for different years

Suppose the Interest Compounded Annually be different in different years. In the first year if the Interest Rate is p % per annum and for the second year if it is q % then

Amount Formula is given by = P*(1+p/100)*(1+q/100)

This formula can be extended for any number of years.

To get the Compound Interest Subtract Principal from Amount i.e. CI = A – P

Example

Find the amount of $10, 000 after 2 years, compounded annually, if the rate of interest being 3 % p.a. during the first year and 4 % p.a. during the second year. Also, find the compound interest?

Solution:

Formula to Calculate the Amount is A = P*(1+p/100)*(1+q/100)

From the given data P = $10, 000, p = 3%, q = 4%

Substitute the inputs in the formula to calculate the Amount and the equation is as under

A = 10,000(1+3/100)(1+4/100)

= 10, 000(1.03)(1.04)

= $10712

CI = A – P

= $10712 – $10, 000

= $712

Interest is Compounded Annually but time is a fraction

For instance, if time is 5 3/4 years then Amount is given as under

A = P * (1 + R/100)5 * [1 + (3/4 × R)/100]

Example

Find the compound interest on $ 30,000 at 6 % per annum for 3 years. Solution Amount after  3 3/4 years?

Solution:

Amount after 3 3/4 years is given by A = $ [30,000 × (1 + 6/100)3 × (1 + (3/4 × 8)/100)]

= $[30,000 * (1 + 0.06)3 * (1 + 6/100)]

=$[30,000*(1.06)3*(1.06)]

= $37874

CI = A – P

= $37874 – $30, 000

= $874

Compound Interest with Periodic Deductions | How to Calculate Periodic Compound Interest?

In this article, you will learn about How to Calculate Compound Interest with Periodic Deductions or Additions to the Amount. Practice the Problems in Periodic Compound Interest and learn how to solve the related problems. To help you understand the concept better we even listed the Formula along with Step by Step Solutions. Check out the Solved Examples for finding the Compound Interest with Periodic Deductions and learn the concept behind them.

Example Questions on Compound Interest with Periodic Deductions

1. Jasmine borrows $ 10,000 at a compound interest rate of 4% per annum. If she repays $ 2,000 at the end of each year, find the sum outstanding at the end of the second year?

Solution:

From the Given Data

Principal = $ 10, 000

Interest Rate = 4% Per Annum

Time = 1 Year

Interest = PTR/100

= 10,000*1*4/100

= $400

Amount of the loan after 1 year = Principal + Interest

= $10,000 + $400

= $10, 400

Jasmine Repays $2,000 after the first year

Therefore, New Principal for the 2nd year = $10, 400 – $2,000

= $8,400

Thus for 2nd Year Principal = $8, 400

Interest = 4%

Time = 1 Year

Interest = PTR/100

= 8,400*1*4/100

= $336

Amount after 2nd Year = Principal + Interest

= $8400+$336

= $8736

Therefore, Jasmine needs to pay an outstanding amount of $8736 dollars by the end of the 2nd Year.

2. David invests $ 10,000 at the beginning of every year in a bank and earns 5 % annual interest, compounded at the end of the year. What will be his balance in the bank at the end of two years?

Solution:

From the Given Data

Principal = $10, 000

Rate of Interest = 5%

Time = 1 Year

Interest = PTR/100

= 10,000*1*5/100

= $500

Therefore Amount after 1 Year = Principal + Interest

= $10,000+$500

= $10, 500

David deposits 10,000 at the beginning of the second year

New Principal becomes = $10, 500+ $10, 000

= $20, 500

Thus, for the 2nd Year

Principal = $20, 500

Interest Rate = 5%

Time = 1 Year

Interest = PTR/100

= $20, 500*1*5/100

= $1025

Amount after the 2nd Year = Principal + Interest

= $20, 500 + $1025

= $21, 525

Therefore, David will have $21, 525 in the bank after the end of 2 Years.

3. John lent $ 5,000 at a compound interest rate of 10% per annum. If he repays $ 500 at the end of the first year and $ 1,000 at the end of the second year, find his outstanding loan at the beginning of the third year?

Solution:

From the Given Data

Principal = $ 5,000

Interest rate = 10%

Time = 1 Year

Interest = PTR/100

= 5000*1*10/100

= $500

Amount after 1 year = Principal + Interest

= $5000+$500

= $5500

John repays $500 at the end of first year thus new principal = $5500 – $500

= $5000

Thus, For 2nd Year

New Principal = $5000

Time = 1 Year

Interest Rate = 10%

Interest = PTR/100

= $5000*1*10/100

= $500

Amount after 2nd Year = Principal + Interest

= $5,000+$500

= $5500

John repays $1000 after the end of 2nd year

Thus for third year New Principal = $5500 – $1000

= $4500

Therefore the outstanding loan at the beginning of the third year is $4500.

Formula and Framing the Formula | Framing Formulas Solved Examples

Formula and Framing the Formula are the concepts used to explain the relationship between different variables. Let us discuss deeply every individual concept of Formula and also Framing the Formula. It is easy to find the relation between two variables by framing a formula with simple steps. Every concept is clearly explained with the solved examples in the below article. Therefore, completely read the entire article and follow every step to get complete knowledge on Formula and Framing the Formula.

List of Topics for Formula and Framing the Formula

  • Change the Subject of a Formula
  • Changing the Subject in an Equation or Formula
  • Practice Test on Framing the Formula

Formula

A formula is a relation between different variables. The formula can be written as an equation with the help of variables and also mathematical symbols. When you look at the equation, it is clearly stating how a variable is related to another variable.

Example:

1. Let us consider a square which is of side a and the perimeter of it is p, then the formula will be p = 4a.
Here the formula shows the exact relation between the perimeter of a square and also the side of a square. It is easy to find the unknown quantity using the known quantity when the values of all the quantities are known.
2. If the perimeter of a rectangle p is twice the sum of its breadth b and length l, then the formula will represent as p = 2(l + b)
3. The volume of a cube is V and its side is a. Then, the formula is V = a^3.
4. We can also write a formula to express the relation between force, mass, and acceleration. Force of an object F is the product of mass “m” and acceleration “a” of that object. The formula is F = ma.
5. If the sum of two unknown variables is 15, then the formula is a + b = 15, where a and b are unknown variables.

Framing a Formula

Framing a formula is arranging the formula of the given mathematical statements with the help of symbols and literals.

1. Firstly, select variables need to form an equation. Also, decide the symbols that need to use for the equation. Some of the symbols and letters are already in the use to represent certain quantities. For example, p is used to represent the principal.
2. Finally, understand the conditions to write an equation and frame the formula.

Subject of a formula

When a quantity is expressing in terms of other quantities, then that particular quantity expressed is defined as the Subject of a formula. Generally, the Subject of the formula is written on the left-hand side and other constants are written on the right-hand side of the equality sign in a formula.

Example:
If z = x + y, then z is expressed in terms of the sum of the x and y. Here, z is the subject of this formula.
x = z – y. Here x is the subject of this formula.

Substitution in a formula

If the variable of an algebraic expression is assigned with certain values, then the given expression gets a particular value. This process is known as substitution.
1. Note down the unknown quantity as the subject of the formula.
2. Substitute different values of the known quantity in the formula to find the value of the subject.

Examples of Framing of a Formula:

1. The total amount A is equal to the sum of the Interest (I) and Principal (P).

Solution:
Formula: A =  I + P

2. One-third of a number subtracted from 4 gives 2.

Solution:
Formula: 4 – 1/3 x = 2

3. The sum of the three angles (∠a, ∠b, ∠c) of a triangle is equal to two right angles.

Solution:
Formula: ∠a + ∠b + ∠c = 180°

4. The area of the rectangle (A) is equal to the product of the breadth (B) and length (L) of the rectangle.

Solution:
Formula: A = B × L

Solved Examples on Formula and Framing the Formula

1. Express the following as an equation. Arun’s father’s age is 3 years more than 4 times Arun’s age. Father’s age is 39 years.

Solution:
Given that Arun’s father’s age is 3 years more than 4 times Arun’s age. Father’s age is 39 years.
Let Arun’s age is s years.
Four times his age = 4s.
Father’s age 3 + 4s
Given father’s age = 39 years
3 + 4s = 39

The final equation is 3 + 4s = 39

2. Write the formula for the following statement. One-fourth of the weight of an apple (A) is equal to one-fifth of the difference between Orange (O) and 2.

Solution:
Given that One-fourth of the weight of an apple (A) = 1/4
Difference between Orange (O) and 2 = O – 2
One-fifth of the difference between Orange (O) and 2. = 1/5(O – 2)
One-fourth of the weight of an apple (A) is equal to one-fifth of the difference between Orange (O) and 2.
1/4 = 1/5(O – 2)

3. Change the following statement using expression into a statement in ordinary language.
(a) Cost of a Desk is Rs. X and Cost of a Box is Rs. 4X
(b) Sam’s age is p years. His brother’s age is (5p + 2) years.

Solution:
(a) Given that Cost of a Desk is Rs. X and Cost of a Box is Rs. 4X
The Cost of a Box is 4 times the Cost of a Desk.
(b) Sam’s age is p years. His brother’s age is (5p + 2) years.
Sam’s brother’s age is two years more than five times his age.

4. A rectangular box is of height h cm. Its length is 7 times its height and the breadth is 5 cm less than the length. Express the length, breadth, and height.

Solution:
Given that A rectangular box is of height h cm. Its length is 7 times its height and the breadth is 5 cm less than the length. Express the length, breadth, and height.
Height = h, length = l, breadth = b
The length of the rectangle is 7 times the height.
The Length of the rectangle = 7h
The breadth of the rectangle is 5 cm less than the length
The Breadth of rectangle = l – 5
As l = 7h, b = 7h – 5.

Height h
Length l = 7h
Breadth b = 7h – 5