The calculation of compound interest by using the growing principal is a complicated and lengthy process when the time period is long. Hence, compound interest when interest is compounded for 6 months is shown here. You can find the formula and its importance in the next sections. Also, you can find the derivation with solved examples of compound interest when interest is compounded half-yearly. 10th Grade Math Compound Interest concepts are explained in detail on our website for free.
How to find Compound Interest when Compounded Half Yearly?
To find the compound interest for half-year, Suppose that rate of interest is annual and interest is compounded half-yearly(i.e., 6 months), the annual interest rate is halved which is r/2, and years numbers are doubled i.e., 2n. To calculate the compound interest when the interest rate is for 6 months is given we use the formula:
Let Principal = P, Interest Rate = r/2%, Time – 2n, Compound Interest = CI, Amount = A then
A = P{(1 + \(\frac { r }{ 100 } \))2n}
In half-year compounding, the Number of years is multiplied by 2, and the interest rate is divided by 2.
Compound Interest = Amount – Principal
CI = P{(1 + \(\frac { r }{ 100 } \))2n} – P
CI = {(1 + \(\frac { r }{ 100 } \))2n}
If any three values of the terms are given, the fourth can be easily found.
Compound Interest Half Yearly Formula Derivation
In the procedure of derivation of formula, we consider the CI half-yearly on the principal P for 1 year at a rate of interest r% for 6 months. At the end of the first 6 months, the principal amount changes as it is compounded half-yearly. Then, the next 6 months’ interest is calculated based on the amount that is remained after the first 6 months.
Simple Interest is calculated at the end of first 6 months as:
SI = (P * r * 1)/(100 * 2)
At the end of first six months, the amount is
A = P + SI
A = P + (P * r * 1)/(100 * 2)
A = P[1 + r/(2 * 100)]
A = P2
Simple Interest for next 6 months, now principal amount is changed to P2
SI1 = (P2 * r * 1)/(100 * 2)
Amount at the end of 1 year
A2 = P2 + SI1
A2 = P2 + (P2 * r * 1)/(2 * 100)
A2 = P2[1 + r/(2 * 100)]2
Now, the final amount after 1 year:
A = P[1 + r/(2 * 100)]2
By rearranging the above equation we get,
A = P[1 + (r/2)/100)]2*1*t
Also, Read:
- Compound Interest with Periodic Deductions
- Compound Interest by Using Formula
- Compound Interest when Interest is Compounded Yearly
Half Yearly Compounding Examples | Compound Interest Half Yearly Questions
Here are a few solved examples of compound interest when it is compounded half-yearly.
Example 1:
Find the compound interest and amount on $10,000 at 10% per annum for 1 1/2 years if interest is compounded for 6 months?
Solution:
Given that, Principal (P) = $10,000
Number of years(n) = 1 * 1/2 = 3/2 * 2 = 3
Rate of interest compounded for 6 months = 10/2% = 5%
The formula used to calculate amount is
A = P{(1 + \(\frac { r }{ 100 } \))2n}
A = 10000{(1 + \(\frac { 5 }{ 100 } \))3
A = 10000{(1.05)3
A = 10000 * 1.05 * 1.05 * 1.05
A = 11,576.25
Hence, the amount is 11,576.25.
Compound Interest = Amount – Principal
CI = 11,576.25 – 10000
CI = 1,576.25
Hence, the compound interest = 1,576.25
Therefore, the compound interest and amount are 2,597.12 and 12,597.12
Example 2:
Find the compound interest and amount on $6,000 is 1 1/2 years at 8% per annum that is compounded 6 months(half-yearly)?
Solution:
Given that, Principal (P) = $6,000
Number of years(n) = 1* 1/2 = 3/2 * 2 = 3
Rate of interest compounded for 6 months = 8/2% = 4%
The formula used to calculate amount is
A = P{(1 + \(\frac { r }{ 100 } \))2n}
A = 6000{(1 + \(\frac { 4 }{ 100 } \))3
A = 6000{(1.04)3
A = 6000 * 1.04 * 1.04 * 1.04
A = 6,749.18
Hence, the amount is 6,749.18.
Compound Interest = Amount – Principal
CI = 6,749.18 – 6000
CI = 749.18
Hence, the compound interest = 749.18
Therefore, the compound interest and amount are 749.18 and 6,749.18.
Example 3:
The compound interest on Rs. 15,000 in 2 1/2 years at 8% per annum, the interest is compounded half-yearly. Calculate the amount and compound interest?
Solution:
Given that, Principal (P) = $15,000
Number of years(n) = 2* 1/2 = 2
Rate of interest compounded for 6 months = 8/2% = 4%
The formula used to calculate amount is
A = P{(1 + \(\frac { r }{ 100 } \))2n}
A = 15000{(1 + \(\frac { 4 }{ 100 } \))2
A = 15000{(1.04)}2
A = 15000 * 1.04 * 1.04
A = 16,224
Hence, the amount is 16,224.
Compound Interest = Amount – Principal
CI = 16,224 – 15,000
CI = 1,224
Hence, the compound interest = 1,224
Therefore, the compound interest and amount are 1,224 and 16,224.
See More:
Faqs on How to find Compound Interest when Interest is Compounded Half-Yearly
Various Frequently Asked Questions and Answers on Compound Interest are given here for your preparation. Read all the faqs and attempt your exams with perfect preparation.
1. What is the formula used to calculate compound interest when interest is compounded half-yearly?
The formula used to calculate the compound interest is CI = P{(1 + \(\frac { r }{ 100 } \))2t} – P
where CI is the compound interest
P is the initial principal amount
T is the time period
R is the rate of interest per annum
2. When the interest is compounded for 6 months, then the number of conversion periods in the year is?
The interest is compounded for 6 months, then the number of conversion periods in the year is 2.
3. What is the time period taken when interest is calculated half yearly?
In terms of interest for half a year, there are 2 conversion periods in the year. Hence, 2 is multiplied by the time period. The time period is taken when the interest calculated half yearly is twice as much as the years given.
4. For the calculation of compound interest for half-year and principal is the same, which among the following is true?
(a)Half the number of years and double the annual rate
(b)Double the annual rate and number of years
(c)Half the annual rate and number of years
(d)Double the number of years and half the annual rate
If the interest is compounded for 6 months, then
R = R/2 and T = 2T = 2n
Therefore, (d) double the number of years and half the annual rate.
Conclusion
Compound Interest when Interest is Compounded Half Yearly along with derivation is given in this article. Know what is the difference to find compound interest when it is compounded yearly, Half Yearly, and Quarterly. Know every difference and how to solve compound interest problems by referring to our articles.