Mẹo Compute the compound interest on Rs 12000 for 1 year at 20 per annum when compounded half yearly
Mẹo về Compute the compound interest on Rs 12000 for 1 year 20 per annum when compounded half yearly 2022
Hoàng Thị Thanh Mai đang tìm kiếm từ khóa Compute the compound interest on Rs 12000 for 1 year 20 per annum when compounded half yearly được Update vào lúc : 2022-10-14 16:44:06 . Với phương châm chia sẻ Bí kíp về trong nội dung bài viết một cách Chi Tiết Mới Nhất. Nếu sau khi Read nội dung bài viết vẫn ko hiểu thì hoàn toàn có thể lại Comments ở cuối bài để Admin lý giải và hướng dẫn lại nha.Compound interest is the interest imposed on a loan or deposit amount. It is the most commonly used concept in our daily existence. The compound interest for an amount depends on both Principal and interest gained over periods. This is the main difference between compound and simple interest.
Nội dung chính- Compound Interest
in MathsCompound Interest FormulaInterest Compounded for Different YearsDerivation of Compound Interest FormulaCompound Interest when the Rate
is Compounded half YearlyQuarterly Compound Interest FormulaFormula for Periodic Compounding RateHow to Calculate Compound Interest?Related ArticlesCompound Interest Solved ExamplesCompound Interest and Simple InterestCompound Interest Word ProblemsCompound Interest Practice ProblemsFrequently Asked Questions on Compound InterestWhat is Compound interest?How do you calculate compound interest?Who benefits from compound interest?What is interest compounded quarterly formula?How do you find the compound interest rate?What is the formula of compound interest with an example?What
is the compounded daily formula?What is the difference between the compound interest on 12000 20% pa for one year when compounded yearly and half yearly?What will be the compound interest earned on an amount of Rs 12000?How do you calculate interest compounded half yearly?What will be the compound interest on a sum of 12000 for 2 years the rate of 20 per annum when the interest compounded yearly?
Suppose we observe our bank statements, we generally notice that some interest is credited to our account every year. This interest varies with each year for the same principal amount. We can see that interest increases for successive years. Hence, we can conclude that the interest charged by the bank is not simple interest; this interest is known as compound interest or CI. In this article, you will learn what is compound interest, the formula and the derivation to calculate compound interest when compounded annually, half-yearly, quarterly, etc. Also, one can understand why the return on compound interest is more than the return on simple interest through the examples given based on real-life applications of compound interest here.
Table of Contents:
- DefinitionFormulaDerivationCI
When rate is compounded half-yearlyCI Quarterly FormulaPeriodic compoundingHow to calculate CIExamplesCI vs SIQuestions and AnswersPractice problemsFAQs
Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is different from simple interest, where interest is not added to the principal while calculating the interest during the next period. In Mathematics, compound interest is usually denoted by C.I.
Also, try out: Compound Interest Calculator.
Compound interest finds its usage in most of the transactions in the banking and finance sectors and other areas. Some of its applications are:
Increase or decrease in population.The growth of bacteria.Rise or Depreciation in the value of an item.Compound Interest in Maths
In Maths, Compound interest can be calculated in different ways for different situations. We can use the interest formula of compound interest to ease the calculations. To calculate compound interest, we need to know the amount and principal. It is the difference between amount and principal.
Compound Interest Formula
As we have already discussed, the compound interest is the interest-based on the initial principal amount and the interest collected over the period of time. The compound interest formula is given below:
Compound Interest = Amount – Principal
Here, the amount is given by:
Where,
- A = amountP = principalr = rate of interestn = number of times interest is compounded
per yeart = time (in years)
Alternatively, we can write the formula as given below:
CI = A – P
And
(beginarraylCI=Pleft ( 1+fracrn right )^nt-Pendarray )
This formula is also called periodic compounding formula.
Here,
- A represents the new principal sum or the total amount of money after compounding
periodP represents the original amount or initial amountr is the annual interest raten represents the compounding frequency or the number of times interest is compounded in a yeart represents the number of years
It is to be noted that the above formula is the general formula for the number of times the principal is compounded in a year. If the interest is compounded annually, the amount is given as:
(beginarraylA = P left (1 + fracR100 right )^tendarray )
Thus, the compound interest rate formula can be expressed for different scenarios such as the interest rate is compounded yearly, half-yearly, quarterly, monthly, daily, etc.
Interest Compounded for Different Years
Let us see, the values of Amount and Interest in case of Compound Interest for different years-
Time (in years) Amount Interest 1 P(1 + R/100)
(beginarraylfracPR100endarray )
2(beginarraylPleft (1+fracR100 right )^2endarray )
(beginarraylPleft (1+fracR100 right )^2-Pendarray )
3(beginarraylPleft (1+fracR100 right )^3endarray )
(beginarraylPleft (1+fracR100 right )^3-Pendarray )
4(beginarraylPleft (1+fracR100 right )^4endarray )
(beginarraylPleft (1+fracR100 right )^4-Pendarray )
n(beginarraylPleft (1+fracR100 right )^nendarray )
(beginarraylPleft (1+fracR100 right )^n-Pendarray )
The above formulas help determine the interest and amount in case of compound interest quickly.
NOTE:
From the data, it is clear that the interest rate for the first year in compound interest is the same as that in simple interest. PR/100.
Other than the first year, the interest compounded annually is always greater than that in simple interest.
Derivation of Compound Interest Formula
To derive the formula for compound interest, we use the simple interest formula as we know SI for one year is equal to CI for one year (when compounded annually).
Let, Principal amount = P, Time = n years, Rate = R
Simple Interest (SI) for the first year:
(beginarraylSI_1 = fracP~times~R~times~T100endarray )
Amount after first year:
(beginarrayl= P~+~SI_1endarray )
(beginarrayl= P ~+~ fracP~times ~R~times ~T100endarray )
(beginarrayl=P left(1+ fracR100right)= P_2endarray )
Simple Interest (SI) for second year:
(beginarraylSI_2 = fracP_2~times ~R~times ~T100endarray )
Amount after second year:
(beginarrayl= P_2~+~SI_2endarray )
(beginarrayl= P_2 ~+~ fracP_2~times ~R~times ~T100endarray )
(beginarrayl=P_2left(1~+~fracR100right)endarray )
(beginarrayl=Pleft(1~+~fracR100right) left(1~+~fracR100right)endarray )
(beginarrayl= P left(1~+~fracR100right)^2endarray )
Similarly, if we proceed further to n years, we can deduce:
(beginarraylA = Pleft(1~+~fracR100right)^nendarray )
(beginarraylCI = A-P = P left[left(1~+~ fracR100right)^n- 1right]endarray )
Compound Interest when the Rate is Compounded half Yearly
Let us calculate the compound interest on a principal, P for 1 year an interest rate R % compounded half-yearly.
Since interest is compounded half-yearly, the principal amount will change the end of the first 6 months. The interest for the next six months will be calculated on the total amount after the first six months. Simple interest the end of first six months,
(beginarraylSI_1 = fracP~times ~R~times ~1100~times ~2endarray )
Amount the end of first six months,
(beginarraylA_1 = P~ + ~SI_1endarray )
(beginarrayl= P ~+~ fracP~times ~R~times ~12~times ~100endarray )
(beginarrayl= P left(1~+~fracR2~times ~100right)endarray )
(beginarrayl= P_2endarray )
Simple interest for next six months, now the principal amount has changed to P2
(beginarraylSI_2 = fracP_2~times ~R~times ~1100~times ~2endarray )
Amount the end of 1 year,
(beginarraylA_2 = P_2~ +~ SI_2endarray )
(beginarrayl= P_2 ~+~ fracP_2~times ~R~times ~12~times ~100endarray )
(beginarrayl=P_2left(1~+~fracR2~times ~100right)endarray )
(beginarrayl= P left(1~+~fracR2~times ~100right)left(1~+~fracR2~times ~100right)endarray )
(beginarrayl= P left(1~+~fracR2~times ~100right)^2endarray )
Now we have the final amount the end of 1 year:
(beginarraylA = Pleft(1~+~fracR2~times ~100right)^2endarray )
Rearranging the above equation,
(beginarraylA = Pleft(1~+~fracfracR2100right)^2~times ~1endarray )
Let R/2 = R’ ; 2T = T’, the above equation can be written as, [for the above case T = 1 year]
(beginarraylA = Pleft(1~+~fracR’100right)^T’endarray )
Hence, when the rate is compounded half-yearly, we divide the rate by 2 and multiply the time by 2 before using the general formula for compound interest.
Quarterly Compound Interest Formula
Let us calculate the compound interest on a principal, P kept for 1 year an interest rate R % compounded quarterly. Since interest is compounded quarterly, the principal amount will change the end of the first 3 months(first quarter). The interest for the next three months (second quarter) will be calculated on the amount remaining after the first 3 months. Also, interest for the third quarter will be calculated on the amount remaining after the first 6 months and for the last quarter on the remaining after the first 9 months. Thus the interest compounded quarterly formula is given by:
(beginarraylA=P(1+fracfracR4100)^4Tendarray )
CI = A – P
Or
(beginarraylCI =P(1+fracfracR4100)^4T-Pendarray )
Here,
A = Amount
CI = Compound interest
R = Rate of interest per year
T = Number of years
Formula for Periodic Compounding Rate
The total accumulated value, including the principal P plus compounded interest I, is given by the formula:
P’ = P[1 + (r/n)]nt
Here,
P = Principal
P’ = New principal
r = Nominal annual interest rate
n = Number of times the interest is compounding
t = Time (in years)
In this case, compound interest is:
CI = P’ – P
How to Calculate Compound Interest?
Let us understand the process of calculating compound interest with the help of the below example.
Example: What amount is to be repaid on a loan of Rs. 12000 for one and half years 10% per annum compounded half yearly?
Solution:
For the given situation, we can calculate the compound interest and total amount to be repaid on a loan in two ways. In the first method, we can directly substitute the values in the formula. In the second method, compound interest can be obtained by splitting the given time bound into equal periods.
This can be well understood with the help of the table given below.
Related Articles
- Simple Interest and Compound InterestMonthly compound interest formulaDaily compound interest formulaSimple interest formula
Compound Interest Solved Examples
As mentioned above, compound interest has many applications in real-life. Let us solve various examples based on these applications to understand the concept in a better manner.
Increase or Decrease in Population
Examples 1:
A town had 10,000 residents in 2000. Its population declines a rate of 10% per annum. What will be its total population in 2005?
Solution:
The population of the town decreases by 10% every year. Thus, it has a new population every year. So the population for the next year is calculated on the current year population. For the decrease, we have the formula A = P(1 – R/100)n
Therefore, the population the end of 5 years = 10000(1 – 10/100)5
= 10000(1 – 0.1)5 = 10000 x 0.95 = 5904 (Approx.)
The Growth of Bacteria
Examples 2:
The count of a certain breed of bacteria was found to increase the rate of 2% per hour. Find the bacteria the end of 2 hours if the count was initially 600000.
Solution:
Since the population of bacteria increases the rate of 2% per hour, we use the formula
A = P(1 + R/100)n
Thus, the population the end of 2 hours = 600000(1 + 2/100)2
= 600000(1 + 0.02)2 = 600000(1.02)2 = 624240
Rise or Depreciation in the Value of an Item
Examples 3:
The price of a radio is Rs. 1400 and it depreciates by 8% per month. Find its value after 3 months.
Solution:
For the depreciation, we have the formula A = P(1 – R/100)n.
Thus, the price of the radio after 3 months = 1400(1 – 8/100)3
= 1400(1 – 0.08)3 = 1400(0.92)3 = Rs. 1090 (Approx.)
Compound Interest and Simple Interest
Now, let us understand the difference between the amount earned through compound interest and simple interest on a certain amount of money, say Rs. 100 in 3 years . and the rate of interest is 10% p.a.
Below table shows the process of calculating interest and total amount.
Compound Interest Word Problems
Question 1: A sum of Rs.10000 is borrowed by Akshit for 2 years an interest of 10% compounded annually. Find the compound interest and amount he has to pay the end of 2 years.
Solution:
Given,
Principal/ Sum = Rs. 10000, Rate = 10%, and Time = 2 years
From the table shown above it is easy to calculate the amount and interest for the second year, which is given by-
(beginarraylAmount(A_2) = Pleft (1+fracR100 right )^2endarray )
Substituting the values,
(beginarraylA_2 = 10000 left ( 1 + frac10100 right )^2 = 10000 left ( frac1110 right )left ( frac1110 right )= Rs.12100endarray )
Compound Interest (for 2nd year) = A2 – P = 12100 – 10000 = Rs. 2100
Question 2: What is the compound interest (CI) on Rs.5000 for 2 years 10% per annum compounded annually?
Solution:
Principal (P) = Rs.5000 , Time (T)= 2 year, Rate (R) = 10 %
We have, Amount,
(beginarraylA = P left ( 1 + fracR100 right )^Tendarray )
(beginarraylA = 5000 left ( 1 + frac10100 right )^2\ = 5000 left ( frac1110 right )left ( frac1110 right )\ = 50 times 121\ = Rs. 6050endarray )
Interest (Second Year) = A – P = 6050 – 5000 = Rs.1050
OR
Directly we can use the formula for calculating the interest for the second year, which will give us the same result.
(beginarraylInterest (I1) = Ptimes fracR100 = 5000 times frac10100 =500endarray )
(beginarraylInterest (I2) = Ptimes fracR100left (1 + fracR100 right )\ = 5000 times frac10100left ( 1 + frac10100 right )\ = 550endarray )
Total Interest = I1+ I2 = 500 + 550 = Rs. 1050
Question 3: What is the compound interest to be paid on a loan of Rs.2000 for 3/2 years 10% per annum compounded half-yearly?
Solution: From the given,
Principal, P = Rs.2000,
Time, T’=2 (3/2) years = 3 years,
Rate, R’ = 10% / 2 = 5%
Amount, A can be given as:
(beginarraylA = P(1+fracR’100)^T’endarray )
(beginarraylA = 2000~times ~left(1~+~frac5100right)^3endarray )
(beginarrayl=2000~times ~left(frac2120right)^3 \ = Rs.2315.25endarray )
CI = A – P = Rs.2315.25 – Rs.2000 = Rs.315.25
Compound Interest Practice Problems
Try solving the below questions on compound interest.
What is the least number of complete years in which a sum of money put out 20% compound interest will be more than doubled? Heera invests Rs. 20,000 the beginning of every year in a bank and earns 10 % annual interest, compounded the end of the year. What will be her balance in the bank the end of three years? What is the difference between the compound interests on Rs. 5000 for one and half years 4% per annum compounded yearly and half-yearly?For a detailed discussion on compound interest, tải về BYJU’S -The learning app.
Frequently Asked Questions on Compound Interest
What is Compound interest?
Compound interest is the interest calculated on the principal and the interest accumulated over the previous period.
How do you calculate compound interest?
Compound interest is calculated by multiplying the initial principal amount (P) by one plus the annual interest rate (R) raised to the number of compound periods (nt) minus one. That means, CI = P[(1 +
R)nt – 1] Here,
P = Initial amount
R = Annual rate of interest as a percentage
n = Number of compounding periods in a given time
Who benefits from compound interest?
The investors benefit from the compound interest since the interest pair here on the principle plus on the interest which they already earned.
What is interest compounded quarterly formula?
The formula for interest compounded quarterly is given by:
A = P(1 + (R/4)/100)4T
How do you find the compound interest rate?
The compound interest rate can be found using the formula,
A = P(1 + r/n)nt
A = Total amount
P = Principal
r = Annual nominal interest rate as a decimal
n = Number of compounding periods
t = Time (in years)
Thus, compound interest (CI) = A – P
What is the formula of compound interest with an example?
The compound interest formula is given below:
Compound Interest = Amount – Principal
Where the amount is given by:
A = P(1 + r/n)nt
P = Principal
r = Annual nominal interest rate as a
decimal
n = Number of compounding periods
t = Time (in years)
For example, If Mohan deposits Rs. 4000 into an account paying 6% annual interest compounded quarterly, and then the money will be in his account after five years can be calculated as:
Substituting, P = 4000, r = 0.06, n = 4, and t = 5 in A = A = P(1 + r/n)nt, we get A = Rs. 5387.42
What is the compounded daily formula?
The compound interest formula when the interest is compounded daily is given by:
A = P(1 + r/365)365 * t
What is the difference between the compound interest on 12000 20% pa for one year when compounded yearly and half yearly?
∴ The difference between the compound interest compounded yearly and compounded half yearly for 18 months 20% per annum on a sum of Rs. 12,000 is Rs. 132.What will be the compound interest earned on an amount of Rs 12000?
[or directly=Rs. 14520 - Rs. 12000 = Rs. 2520]How do you calculate interest compounded half yearly?
The formula for calculation of compound interest for half year is CI = p(1 + r/2/100)2t. - p. Here in this formula 'A' is the final amount, 'p' is the principal, and 't' is the time in years.What will be the compound interest on a sum of 12000 for 2 years the rate of 20 per annum when the interest compounded yearly?
Hence, the compound interest is Rs. 2,520. Tải thêm tài liệu liên quan đến nội dung bài viết Compute the compound interest on Rs 12000 for 1 year 20 per annum when compounded half yearly Compound interest calculator
Post a Comment