Answer Show  Hint: To find the number of years in which Compound Interest amounts to eight times given that a sum of money placed at Compound Interest (C.I.) doubles itself in 5 years. We shall take the sum of the money, i.e. the principal amount be ‘X’. Now, according to the question, after 5 years, X becomes 2X, and after next five years, i.e. after 10 years, 2X will become 4X because of the same, and so on. We will use \[A\ \ =\ \ P\ {{\left( 1\ \ +\ \ \dfrac{R}{100} \right)}^{n}}\] . Let us take \[A\] as $2X$ , $P$ as $X$ and $n$ as $5$ . We shall put these values in the equation to find the answer. To find the number of years in which the sum that is given to us will become 8 times of itself, take \[A\] as $8X$ , $P$ as $X$ . After substitution and comparing both the equations, we will get the number of years. Complete step by step answer: We need to find the number of years in which Compound Interest amounts to eight times.We know that \[A\ \ =\ \ P\ {{\left( 1\ \ +\ \ \dfrac{R}{100} \right)}^{n}}\] is the formula for finding compound interest, where$A$ is the final amount, $P$ is the initial principal balance, \[R\] is the interest rate, $n$ is the of times interest applied per time period.Let us take \[A\] as $2X$ , $P$ as $X$ and $n$ as $5$ . We shall put these values in the equation to find the answer.So, now the equation that we get is:-\[\begin{align} & 2X\ \ =\ \ X\ {{\left( 1\ \ +\ \ \dfrac{R}{100} \right)}^{5}} \\ & \\ \end{align}\] Cancelling $X$ on both sides, we will get\[2\ \ =\ \ {{\left( 1\ \ +\ \ \dfrac{R}{100} \right)}^{5}}\]Removing the power of $5$ from RHS, we will get\[{{2}^{\dfrac{1}{5}}}\ \ =\ \ 1\ \ +\ \ \dfrac{R}{100}\ \,\ \ \ \ .....(a)\]Now, as mentioned in the question, we have to find the number of years in which the sum that is given to us will become 8 times of itself.Let us take \[A\] as $8X$ , $P$ as $X$\[8X\ \ =\ \ X\ \ \times \ \ {{\left( 1\ \ +\ \ \dfrac{R}{100} \right)}^{n}}\]Cancelling $X$ on both sides, we will get\[8\ \ =\ \ {{\left( 1\ \ +\ \ \dfrac{R}{100} \right)}^{n}}\]Now, using equation (a) that we have found before, we can find the value of ‘n’.\[8\ \ =\ \ {{\left( {{2}^{\dfrac{1}{5}}} \right)}^{n}}\] $8$ can be written as\[{{2}^{3}}\ \ =\ \ {{2}^{\dfrac{n}{5}}}\]Now, on comparing the exponents of both the sides, we get the equation as follows:- \[\begin{align} & \dfrac{n}{5}\ \ =\ \ 3 \\ &\Rightarrow n\ \ =\ \ 3\ \ \times \ \ 5 \\ & \Rightarrow n\ \ =\ \ 15 \\ \end{align}\] Hence, as the value of ‘n’ is 15 years, the answer of this question is also 15 years as it is the number of years in which the sum amounts to eight times itself at the same rate of interest.So, the correct answer is “Option C”. Note: Be careful with the equation of CI \[A\ \ =\ \ P\ {{\left( 1\ \ +\ \ \dfrac{R}{100} \right)}^{n}}\] . There can be a chance of making an error in this equation. We can also use the following method to find the answer to this question. According to the question, the sum of money doubles itself in five years, therefore, after five years, $X$ will become $2X$ .$X$ after five years = $2X$$X$ after next five years ,i.e., after 10 years = $2X\times 2=4X$$X$ after next five years ,i.e., after 15 years = $4X\times 2=8X$We need to find the year in which CI amounts to eight times itself. From the above step, we get the answer as 15 years.The Rule of 72 is a quick and simple technique for estimating one of two things: The rule states that an investment or a cost will double when: [Investment Rate per year as a percent] x [Number of Years] = 72. When interest is compounded annually, a single amount will double in each of the following situations: The Rule of 72 indicates than an investment earning 9% per year compounded annually will double in 8 years. The rule also means if you want your money to double in 4 years, you need to find an investment that earns 18% per year compounded annually. You can confirm the rationality of the Rule of 72 as follows: Find factors on the FV of 1 Table that are close to 2.000. (The factor of 2.000 tells you that the present value of 1.000 had doubled to the future value of 2.000.) When you find a factor close to 2.000, look at the interest rate at the top of the column and look at the number of periods (n) in the far left column of the row containing the factor. Multiply that interest rate times the number of periods and you will get the product 72. To use the Rule of 72 in order to determine the approximate length of time it will take for your money to double, simply divide 72 by the annual interest rate. For example, if the interest rate earned is 6%, it will take 12 years (72 divided by 6) for your money to double. If you want your money to double every 8 years, you will need to earn an interest rate of 9% (72 divided by 8). Here's another way to demonstrate that the Rule of 72 works. Assume you make a single deposit of $1,000 to an account and wish for it to grow to a future value of $2,000 in nine years. What annual interest rate compounded annually will the account have to pay? The Rule of 72 indicates that the rate must be 8% (72 divided by 9 years). Let's verify the rate with the format we used with the FV Table: To finish solving the equation, we search only the "n = 9" row of the FV of 1 Table for the FV factor that is closest to 2.000. The factor closest to 2.000 in the row where n = 9 is 1.999 and it is in the column where i = 8%. An investment at 8% per year compounded annually for 9 years will cause the investment to double (8 x 9 = 72). Page 2The future value of a present amount can be expressed as: We will illustrate how this mathematical expression works by using the amounts from the three accounts in Part 1. Account #1: Annual Compounding A single deposit of $10,000 will earn interest at 8% per year and the interest will be deposited at the end of one year. Since the interest is compounded annually, the one-year period can be represented by n = 1 and the corresponding interest rate will be i = 8% per year: The formula shows that the present value of $10,000 will grow to the FV of $10,800 at the end of one year when interest of 8% is earned and the interest is added to the account only at the end of the year. Account #2: Semiannual Compounding In Account #2 the $10,000 deposit will earn interest at 8% per year, but the interest will be deposited at the end of each six-month period for one year. With semiannual compounding, the life of the investment is stated as n = 2 six-month periods. The interest rate per six-month period is i = 4% (8% annually divided by 2 six-month periods). The present value of $10,000 will grow to a future value of $10,816 (rounded) at the end of two semiannual periods when the 8% annual interest rate is compounded semiannually. Account #3: Quarterly Compounding In Account #3 the $10,000 deposit will earn interest at 8% per year, but the interest earned will be deposited at the end of each three-month period for one year. With quarterly compounding, the life of the investment is stated as n = 4 quarterly periods. The annual interest rate is restated to be the quarterly rate of i = 2% (8% per year divided by 4 three-month periods). The present value of $10,000 will grow to a future value of $10,824 (rounded) at the end of one year when the 8% annual interest rate is compounded quarterly. Future Values for Greater Than One Year To be certain that you understand how the number of periods, n, and the interest rate, i, must be aligned with the compounding assumptions, we prepared the following chart. Note that the chart assumes an interest rate of 12% per year. To be certain you understand the information in the chart, let's assume that a single amount of $10,000 is deposited on January 1, 2021 and will remain in the account until December 31, 2025. This will mean a total of five years: 2021, 2022, 2023, 2024, and 2025. If the account will pay interest of 12% per year compounded quarterly, then n = 20 quarterly periods (5 years x 4 quarters per year), and i = 3% per quarter (12% per year divided by 4 quarters per year). The mathematical expression will be: Let's try one more example. Assume that a single amount of $10,000 is deposited on January 1, 2021 and will remain in the account until December 31, 2022 (a total of two years). If the account will pay interest of 12% per year compounded monthly, then n = 24 months (2 years x 12 months per year), and i = 1% per month (12% per year divided by 12 months per year). The mathematical expression will be: Page 3The mathematics for calculating the future value of a single amount of $10,000 earning 8% per year compounded quarterly for two years appears in the left column of the following table. In the right column is the formula which uses a future value factor. Future value factors are available in future value tables, such as the abbreviated version shown here: We highlighted the factor used in our computation. As you can see, the future value factor of 1.172 is located where n = 8, and i = 2%. Our future value of 1 table is unique in that we have an additional row: n = 0. Most FV of 1 tables omit the row for n = 0, and begin with the row n =1. There should be no difference in FV factors other than minor rounding differences. To appreciate the usefulness of the FV of 1 table, focus on the column with the heading of i = 10%. This column tells you that the present value of 1.000 is 1.000 at time period 0—the present time. As you move down the 10% column, the next row (where n = 1) shows that the future value will increase by 10% to 1.100. Continuing down the 10% column, you see that at the end of two periods (n = 2) the future value is 1.210, an increase of 0.110 (1.100 x 10%). The next figure down indicates that at the end of three periods the future value is 1.331, which is an increase of 0.121 (1.331 – 1.210; and 1.210 x 10%). The FV of 1 table provides the future amounts at compound interest for a single amount of 1.000 at various interest rates. These factors should make the future calculations a bit simpler than calculations using exponents. The 10% column of the future value table can be used to determine the future value of a single $1.00 invested today at 10% interest compounded annually. The single $1.00 amount will grow to $3.138 at the end of 12 years. The FV table also provides some insight as to the future cost of items that are expected to increase at a constant rate. For example, if a cup of coffee presently costs $1.00 and the cost is expected to increase by 10% per year compounded annually, then a cup of coffee will cost $3.138 per cup at the end of 12 years. We can also use the factors for amounts greater than $1. For example, if the monthly cost of a family's health insurance plan is $1,000 at the present time and it is expected to increase by 10% per year compounded annually, then the monthly cost at the end of 12 years will be $3,138. Throughout the remainder of this topic we will use a more complete Future Value of 1 Table: Page 4If we know the single amount (PV), the interest rate (i), and the number of periods of compounding (n), we can calculate the future value (FV) of the single amount. Calculations #1 through #5 illustrate how to determine the future value (FV) through the use of future value factors. You make a single deposit of $100 today. It will remain invested for 4 years at 8% per year compounded annually.
What will be the future value of your single deposit at the end of 4 years? The following timeline plots the variables that are known and unknown: Calculation using an FV factor: At the end of 4 years, you will have $136 in your account. Paul makes a single deposit today of $200. The deposit will be invested for 3 years at an interest rate of 10% per year compounded semiannually. What will be the future value of Paul's account at the end of 3 years? The following timeline plots the variables that are known and unknown: Because the interest is compounded semiannually, we convert 3 years to 6 semiannual periods, and the annual interest rate of 10% to the semiannual rate of 5%. Calculation using an FV factor: At the end of 3 years, Paul will have $268 in his account. Sheila invests a single amount of $300 today in an account that will pay her 8% per year compounded quarterly. Compute the future value of Sheila's account at the end of 2 years. The following timeline plots the variables that are known and unknown: Because interest is compounded quarterly, we convert 2 years to 8 quarters, and the annual rate of 8% to the
quarterly rate of 2%. Calculation using an FV factor: At the end of 2 years, Sheila will have $351.60 in her account. You invest $400 today in an account that earns interest at a rate of 12% per year compounded monthly. What will be the future value at the end of 2 years? The following timeline plots the variables that are known and unknown: Because the interest is compounded monthly, we convert 2 years to 24 months, and the annual rate of 12% to the monthly rate of 1%. Calculation using an FV factor: At the end of 2 years, you will have $508 in your account. Page 5If we know the present value (PV), the future value (FV), and the interest rate per period of compounding (i), the future value factors allow us to calculate the unknown number of time periods of compound interest (n). Calculations #5 through #8 illustrate how to determine the number of time periods (n). An airplane ticket costs $500 today and it is expected to increase at a rate of 5% per year compounded annually.
Determine the number of years it will take for the $500 airplane ticket to have a future cost of $700. The following timeline plots the variables that are known and unknown: Because the rate of increase is compounded annually, we use the given annual rate of 5%. The answer (n) will be stated in annual time periods (years). Calculation using the FV of 1 Table: To finish solving the equation, we search only the i = 5% column of the FV of 1 Table for the future value factor that is closest to 1.400. In this case, the factor we find is 1.407, and we see it is located in the row where n = 7. This tells us that it will take approximately 7 annual time periods (7 years) for an airplane ticket to go from its present cost of $500 to the future cost of $700. Lorenzo put $600 today in an account that earns an annual rate of 8% compounded semiannually. How many years will it take for Lorenzo's single investment of $600 to have a future value of $900? The following timeline plots the variables that are known and unknown: Because the interest is compounded semiannually, we converted the annual interest rate of 8% to the semiannual rate of 4%. Calculation using the FV of 1 Table: To finish solving the equation, we search only the 4% column of the FV of 1 Table for the future value factor that is closest to 1.500. In this case, the factor we find is 1.480, and we see it is located in the row where n = 10. This means it will require 5 years (10 semiannual time periods divided by 2 semiannual periods in each year) for Lorenzo's $600 to reach a future value of $900. Nancy invests a sum of $700 at a fixed rate of 8% per year with quarterly compounding. How many years will it take her $700 investment to reach a future value of $1,000? The following timeline plots the variables that are known and unknown: Because the interest is compounded quarterly (every 3 months), the annual interest rate is converted to 2% per quarter. Calculation using the FV of 1 Table: To finish solving the equation, we search only the 2% column of the FV of 1 Table for the future value factor that is closest to 1.429. In this case, the factor is 1.428, and we see it is located in the row where n = 18. To convert n = 18 quarters to years, we simply divide the 18 quarters by 4, the number of quarterly periods in a year. The answer is that it will take approximately 4.5 years for Nancy's $700 investment to reach a future value of $1,000. You invest $787 today in an account that will return an annual interest rate of 12% with interest compounded monthly. How many years will it take for the $787 investment to have a future value of $1,000? The following timeline plots the variables that are known and unknown: Because the interest is compounded monthly, we convert the annual rate of 12% to i = 1% per month. Calculation using the FV of 1 Table: To finish solving the equation, we search only the i = 1% column in the FV of 1 Table for the FV factor that is closest to 1.270. In this case, there is a factor of exactly 1.270, and it is located in the row where n = 24. Since n = 24 monthly time periods, we need to divide the 24 months by 12 months in a year in order to get the answer in years. It will take approximately 2 years for your $787 investment to reach a future value of $1,000. Page 6If we know the present value (PV), the future value (FV), and the number of time periods of compound interest (n), future value factors will allow us to calculate the unknown interest rate (i). Calculations #9 through #12 illustrate how to determine the interest rate (i). A single investment of $500 is made today and will remain invested for 5 years. At the end of the 5th year, the future value will be $669. Assuming that the interest is compounded annually, calculate the annual interest rate earned on this investment. The following timeline plots the variables that are known and unknown: Calculation using the FV of 1 Table: To finish solving the equation, we search only the "n = 5" row of the FV of 1 Table for the FV factor that is closest to 1.338. In this case, there is a factor of 1.338, and it is located in the column with the heading i = 6%. Since the time periods are one-year periods, the interest rate is 6% per year compounded annually. Calculation #10A basket of goods has a cost today of $100. Assume that the cost is estimated to increase to $180 at the end of 6 years. What is the annual rate of increase if the cost increases are compounded semiannually? The following timeline plots the variables that are known and unknown: Because the rate of increase (the "interest") is compounded semiannually, we convert the 6 years to 12 semiannual time periods. Calculation using the FV of 1 Table: To finish solving the equation, we search only the row "n = 12" of the FV of 1 Table for the FV factor that is closest to 1.800. In this case, there is a factor of 1.796 located in the column where i = 5%. Since (n) represents semiannual time periods, the rate of 5% is the semiannual rate, or the rate for a six-month period. To convert the semiannual rate to an annual rate, we multiply 5% x 2, the number of semiannual periods in a year. This means that the rate of increase for the basket of goods is 10% per year compounded semiannually. Calculation #11Assume you invest $100 today and intend to keep it invested for 6 years. You are told that at the end of the 6th year, the future value of your account will be $161. Assuming that the interest is compounded quarterly, compute the annual interest rate you are earning on this investment. The following timeline plots the variables that are known and unknown: Because the interest is compounded quarterly, we convert the 6 years to 24 quarterly time periods. In other words, we will refer to n = 24 when using the FV of 1 Table. Calculation using the FV of 1 Table:To finish solving the equation, we search only the row where n = 24 in the FV of 1 Table for the future value factor. We look for the FV factor that is closest to 1.610. In this case, a factor of 1.608 is located in the column where i = 2%. Since 2% is the interest rate per quarter, we multiply the quarterly rate of 2% x 4, the number of quarterly periods in a year. Hence the investment is earning an interest rate of 8% per year compounded quarterly. Calculation #12Aaron has a sum of $500 and he needs for it to grow to a future value of $634 by the end of one year. Assuming that the interest rate is compounded monthly, what interest rate does Aaron need for his investment? The following timeline plots the variables that are known and unknown: Because the interest is compounded monthly, we convert the 1 year time period to n = 12 monthly time periods. Calculation using the FV of 1 Table:To finish solving the equation, we search only the row where n = 12 in the FV of 1 Table for the FV factor that is closest to 1.268. In this case, the factor of 1.268 is located in the column where i = 2%. Since i = 2% is the monthly rate, we multiply 2% x 12, the number of monthly periods in a year in order to determine the annual rate. In this case, Aaron needs to find an interest rate of 24% per year compounded monthly in order to reach his future value goal of $634 in one year. Page 7If we know the future value (FV), the number of time periods of compound interest (n), and the interest rate (i), we can use future value factors to calculate the unknown amount that was originally deposited (the "present value," or PV). Calculations #13 through #16 illustrate how to determine the present value (PV). (Note: The single amount can also be calculated by using present value factors. This is discussed in the AccountingCoach topic Present Value of a Single Amount.) Joan wishes to make one deposit today into an individual retirement account (IRA) that is guaranteed to earn 6% per year compounded annually. She wants the amount deposited to grow to $10,000 at the end of 12 years. How much will she need to deposit today? The following timeline plots the variables that are known and unknown: Calculation using the FV of 1 Table: In order to have a future value of $10,000 in 12 years, Joan must deposit $4,970.18 today in her IRA. What amount will you need to invest today in order to have $15,000 at the end of 10 years? Assume your amount will earn 10% per year compounded semiannually. The following timeline plots the variables that are known and unknown: Because the interest is compounded semiannually, we convert the 10 annual time periods to 20 semiannual time periods. Similarly, the interest rate is converted from 10% per year to 5% per semiannual period. Calculation using the FV of 1 Table: You need to invest $5,653.98 today in order to have it grow to $15,000 in 20 six-month periods with interest at 10% per year compounded semiannually. Calculation #15What amount today will grow to $30,000 at the end of 7 years if the amount earns 8% per year compounded quarterly? The following timeline plots the variables that are known and unknown: Because the interest is compounded quarterly, we convert the 7 one-year time periods to 28 quarters. Similarly, the interest rate is converted from 8% per year to 2% per quarter. In other words, n = 28 quarters, and i = 2% per quarter. Calculation using the FV of 1 Table: A single deposit of $17,231.48 will grow to $30,000 if it remains invested at 8% per year compounded quarterly for 7 years. Calculation #16The number of visitors to Bill's website is increasing at an annual rate of 36% compounded monthly. At the end of one year Bill expects the number of visitors to his site to reach 50,000 per day. What is the present number of visitors per day? The following timeline plots the variables that are known and unknown: Because the rate is compounded monthly, we convert the one-year time period to 12 monthly time periods. Similarly, the rate is converted from 36% per year to 3% per month. Calculation using the FV of 1 Table: The present amount of visitors per day must be 35,063 if a 3% per month compounded increase results in 50,000 visitors per day after 12 months. (You can verify the answer 35,063 by using the table below.) If our future value factors were not rounded to 3 decimal places, the present number of visitors per day at December 31, 2020 would have been 35,069 and that would result in 50,000 at Dec 31, 2021. Page 8The future value of multiple amounts is determined by calculating, and then adding together, the future value for each single amount. We illustrate this with Calculations #17 and #18. You are asked to determine the total future value on December 31, 2024 of a $1,000 deposit made on January 1, 2020 plus a $5,000 deposit made on January 1, 2022. Both amounts will earn 8% per year compounded annually. The timeline for this information is: The total future value on December 31, 2024 is the sum of these two calculations: Future value calculation of the $1,000 deposited on Jan 1, 2020: Future value calculation of the $5,000 deposited on Jan 1, 2022: The total future value on December 31, 2024 for these two deposits will be $7,769. You can verify the future value of $7,769 with the following table: You are asked to determine the total future value on December 31, 2024 of a $1,000 deposit made on January 1, 2020 plus a $5,000 deposit made on December 31, 2021. Both amounts will earn 8% per year compounded quarterly. Because the interest is compounded quarterly, we convert the first deposit from 5 years to 20 quarterly periods, and the second deposit from 3 years to 12 quarterly periods. We convert the interest rate of 8% per year to the rate of 2% per quarter. The following calculations reflect the restatement to quarters. Again, the sum of the answers to these two equations will be the future value on December 31, 2024. Future value calculation of the $1,000 deposited on Jan 1, 2020: Future value calculation of the $5,000 deposited on Dec 31, 2021: The total future value on December 31, 2024 for these two deposits will be $7,826. You can verify the future value of $7,826 with the following table:
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Approximately at what rate of compound interest would an amount double itself in 6 years
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