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. Show
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-Yearly1. 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 Formula: Compound interest is defined as the interest on a certain sum or amount, where the interest gets accrued successively for every year from the previous periods. People may have noticed that when a certain sum of money in a bank is kept on a savings account basis, the money gets increased every year due to the addition of the annual interest amount. It is because the bank’s interest is calculated on the previous year’s amount. It is known as interest compounded or Compound Interest (C.I.). In this article, we have provided the compound interest formula along with some examples to help students become confident on this topic. Compound Interest Formula: OverviewCompound interest is the interest calculated on the principal and the interest earned previously. Compounding is when the interest is calculated not on the principal amount, but also the interest earned in the previous periods. So, the total interest for the successive period includes the interest on principal plus interest in the prior period. It is called “interest on interest”. It is different from Simple Interest (SI), in which previously accumulated interest is not added to the principal amount of the current period, so there is no compounding. Let us understand what compound interest with an example is: Case 1: Simple Interest Formula: Case 2: Compound Interest Formula: Now you can see that compound interest gives more return on the same principal amount for an extended period of time. Some of the real-life applications of compound interest are:(i) To calculate the growth of bacteria.(ii) To calculate the increase or decrease in population. (iii) To determine the rise or depreciation in the value of an item. Get Maths formulas below: Compounding Interest CalculatorHere we have provided the formula of CI. With the formula provided below, you can quickly know how to calculate compound interest for any principal amount for years.
The amount is calculated with the help of the following formula: The general formula of compound interest in maths is:
If the principal amount is annually compounded, the CI formula is:
We also have the CI formula for half-yearly and quarterly, which we will discuss in the subsequent sections. Compounding Interest Calculator Formula DerivationHere we have derived the compound interest formula when compounded annually. Let, Principal amount = P, Time = n years, Rate = r \(SI_1=(\frac{P\times r\times t}{100})\) Amount after first year = P + SI1 = P + (P × r × t)/100 = P(1+r/100) = P2 \(SI_2=(\frac{P_2\times r\times t}{100})\) Amount after second year = P2 + SI2 = P(1+r/100)2 Similarly, if we proceed further to n years, we can deduce:A = P(1 + r/100)n C.I. = (A – P)= P[(1 + r/100)n – 1] Compounding Interest Calculator Half Yearly FormulaWhen Compound Interest is calculated for a time duration of half-year, we divide the rate by two and multiply the time by 2 in the general formula. So, the compound interest formula half-yearly becomes:
Derivation: Here we calculate the compound interest half-yearly on a principal, P kept for one year at an interest rate r % compounded half-yearly. Since interest is compounded half-yearly, the principal amount will change at the end of the first 6 months. The interest for the next six months will be calculated on the amount remaining after the first six months. Simple interest at the end of the first six months, SI1 = (P × r × 1)/(100 × 2)Amount at the end of the first six months, A1 = P + SI1 = P + (P × r × 1)/(2 × 100) = P[1 + r/(2 × 100)]= P2 Simple interest for the next six months, now the principal amount has changed to P2 SI2 = (P2 × r × 1)/(100 × 2)Amount at the end of 1 year, A2 = P2 + SI2 = P2 + (P2 × r × 1)/(2 × 100) = P2[1 + r/(2 × 100)] = P(1 + r/2×100)(1 + r/2×100) = P[1 + r/(2 × 100)]2 Now we have the final amount at the end of 1 year:A = P[1 + r/(2 × 100)]2 Rearranging the above equation,A = P[1 + (r/2)/100)2×1 Let r/2 = r′; 2t = t′, the above equation can be written as, [for the above case t = 1 year]A = P(1 + r′/100)t′ Compound Interest Formula When Amount Is Compounded QuarterlyHere we have provided the Compound Interest Formula when the amount is compounded quarterly. When the rate is compounded quarterly, we divide the rate by four and multiply the time by 4 in the general formula. Compound Interest Quarterly Formula:
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Compounding Interest Calculator Formula ContinuousContinuously compounded interest is the mathematical limit of the general compound interest formula, with interest compounded many times each year. In other words, you are paid every possible time increment. Mathematically, the formula is: \(CCI=\lim_{n\rightarrow\infty}P\lbrack(1+\frac rn)^{nt}-1\rbrack\)
Now that we have provided the compound interest formulas, let’s have a summary of the formulas in the table below:
Also, Check How To Find Compound Interest?Compound interest can be found when we have the principal amount, rate of interest, time, and the number of times the interest is compounded. Using the formula for compound interest, we can substitute all the values in the formula and get the result. Sometimes, the value of compound interest is given, and we have to deduce other values such as the final amount, principal amount, or rate of interest. Solved Compound Interest Formula ExamplesWe have provided compound interest formula examples with solutions to help you understand the concepts in a better manner: Q.1: Rohit deposited Rs. 8000 with a finance company for 3 years at an interest of 15% per annum. What is the compound interest that Rohit gets after 3 years? A = P(1 + r/100)n = 8000 (1 + 15/100)3 = 8000 (115/100)3= Rs 12167∴ Compound Interest = (A – P) = Rs 12167 – Rs 8000 = Rs 4167 Q.2: Find the compound interest on Rs. 160000 for one year at the rate of 20% per annum, if the interest is compounded quarterly? A = P (1 + R/100) n = 160000 (1 + 5/100)4 = 160000 (105/100)4= Rs 194481∴ Compound Interest = (A – P) = Rs 194481 – Rs 160000 = Rs 34481 Q.3: The count of a certain breed of bacteria was found to increase at the rate of 2% per hour. Find the bacteria at the end of 2 hours if the count was initially 600000? Q.4: Roma borrowed Rs. 64000 from a bank for 1½ years at the rate of 10% per annum. Compare the total compound interest payable by Roma after 1½ years, if the interest is compounded half-yearly? A = P (1 + r/100) n = 64000 (1 + 10/2×100)3 = 64000 (210/200)3= Rs 74088∴ Compound Interest = (A – P) = Rs 74088 – Rs 64000 = Rs 10088 Q5: The price of a radio is Rs 1400 and it depreciates by 8% per month. Find its value after 3 months? Q6: Find the compound interest at the rate of 10% per annum for two years on that principal which in two years at the rate of 10% per annum given Rs. 200 as simple interest? A = P (1 + r/100) n = 1000 (1 + 10/100)2 = 1000 (110/100)2= Rs 1210∴ Compound Interest = (A – P) = Rs 1210 – Rs 1000 = Rs 210 Q7: Ramesh deposited Rs. 7500 in a bank which pays him 12% interest per annum compounded quarterly. What is the amount which he receives after 9 months? A = P (1 + r/100) n = 7500 (1 + 3/100)3 = 7500 (103/100)3= Rs 8195.45 ∴ Required amount is Rs 8195.45 Q8: A town had 10,000 residents in 2000. Its population declines at a rate of 10% per annum. What will be its total population in 2005? Get Algebra formulas from below: Practice Questions On Compound InterestHere we have provided some practice questions on compounding interest calculator Class 8 for you to practice:
FAQs On Compound Interest FormulaQ.1: How do you calculate compound interest? Q.2: Is compound interest good or bad? Q.3: Why is compound interest so powerful? Q.4: What is the formula for compounding interest calculator? Q.5: What is the compound interest formula used for? |