What amount of money will amount to 750 in 6 years and 800 in 8 years at simple interest?
Calculating Interest and Excel Functions:The simplest type of interest is called, appropriately, “simple interest.” If you were to loan a friend $1000 to start a business and you both agreed that in one year, the friend would repay you $1060, you would earn $60 interest on that investment. Show
You would earn $60 simple interest. Simple interest is calculated as a percentage of the initial amount borrowed and does not include any additional earnings. The percentage of the original loan amount that is paid as simple interest is called the interest rate. In our example, the interest rate on a $1000 loan that requires a $60 interest payment in one year is: \(\frac{60}{1000}=0.06 \:\text{or}\:6\%\) The simple interest rate on this loan would be 6%.
Video Source (06:53 mins) | Transcript The following video shows another example of simple interest and explains how this works with a little more detail. Video Source (03:44 mins) | Transcript There are two perspectives on interest. Someone is the lender, and someone is the borrower. From the viewpoint of the lender, interest is great. It allows them to make money on their investment. From the perspective of the borrower, interest can be a big problem. In addition to paying back the original amount of the loan, the borrower must also pay the interest. The higher the interest rate, the greater the interest payment will be. Interest Equation: I = Prt
Amount Owed = P + I Additional Resources
Practice Problems
Learning Outcomes
We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for retirement, or need a loan, we need more mathematics. Simple InterestDiscussing interest starts with the principal, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $100(0.05) = $5. The total amount you would repay would be $105, the original principal plus the interest.  Simple One-time Interest[latex]\begin{align}&I={{P}_{0}}r\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\end{align}[/latex]
ExamplesA friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn? The following video works through this example in detail. One-time simple interest is only common for extremely short-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly. For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value. ExercisesSuppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn? Further explanation about solving this example can be seen here. We can generalize this idea of simple interest over time. Simple Interest over Time[latex]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex]
The units of measurement (years, months, etc.) for the time should match the time period for the interest rate. APR – Annual Percentage RateInterest rates are usually given as an annual percentage rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up. For example, a 6% APR paid monthly would be divided into twelve 0.5% payments. A 4% annual rate paid quarterly would be divided into four 1% payments. ExampleTreasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn? This video explains the solution. Try ItTry ItA loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging. Try ItCompound InterestWith simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding. Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow? The 3% interest is an annual percentage rate (APR) – the total interest to be paid during the year. Since interest is being paid monthly, each month, we will earn [latex]\frac{3%}{12}[/latex]= 0.25% per month. In the first month,
In the first month, we will earn $2.50 in interest, raising our account balance to $1002.50. In the second month,
Notice that in the second month we earned more interest than we did in the first month. This is because we earned interest not only on the original $1000 we deposited, but we also earned interest on the $2.50 of interest we earned the first month. This is the key advantage that compounding interest gives us. Calculating out a few more months gives the following:
We want to simplify the process for calculating compounding, because creating a table like the one above is time consuming. Luckily, math is good at giving you ways to take shortcuts. To find an equation to represent this, if Pm represents the amount of money after m months, then we could write the recursive equation: P0 = $1000 Pm = (1+0.0025)Pm-1 You probably recognize this as the recursive form of exponential growth. If not, we go through the steps to build an explicit equation for the growth in the next example. ExampleBuild an explicit equation for the growth of $1000 deposited in a bank account offering 3% interest, compounded monthly. View this video for a walkthrough of the concept of compound interest. While this formula works fine, it is more common to use a formula that involves the number of years, rather than the number of compounding periods. If N is the number of years, then m = N k. Making this change gives us the standard formula for compound interest. Compound Interest[latex]P_{N}=P_{0}\left(1+\frac{r}{k}\right)^{Nk}[/latex]
The most important thing to remember about using this formula is that it assumes that we put money in the account once and let it sit there earning interest. In the next example, we show how to use the compound interest formula to find the balance on a certificate of deposit after 20 years. ExampleA certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a higher interest rate, but you cannot access your investment for a specified length of time. Suppose you deposit $3000 in a CD paying 6% interest, compounded monthly. How much will you have in the account after 20 years? A video walkthrough of this example problem is available below. Let us compare the amount of money earned from compounding against the amount you would earn from simple interest
As you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth. Try ItEvaluating exponents on the calculatorWhen we need to calculate something like [latex]5^3[/latex] it is easy enough to just multiply [latex]5\cdot{5}\cdot{5}=125[/latex]. But when we need to calculate something like [latex]1.005^{240}[/latex], it would be very tedious to calculate this by multiplying [latex]1.005[/latex] by itself [latex]240[/latex] times! So to make things easier, we can harness the power of our scientific calculators. Most scientific calculators have a button for exponents. It is typically either labeled like: ^ , [latex]y^x[/latex] , or [latex]x^y[/latex] . To evaluate [latex]1.005^{240}[/latex] we’d type [latex]1.005[/latex] ^ [latex]240[/latex], or [latex]1.005 \space{y^{x}}\space 240[/latex]. Try it out – you should get something around 3.3102044758. ExampleYou know that you will need $40,000 for your child’s education in 18 years. If your account earns 4% compounded quarterly, how much would you need to deposit now to reach your goal? Try ItRoundingIt is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to keep at least 3 significant digits (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 will usually give you a “close enough” answer, but keeping more digits is always better. ExampleTo see why not over-rounding is so important, suppose you were investing $1000 at 5% interest compounded monthly for 30 years.
If we first compute r/k, we find 0.05/12 = 0.00416666666667 Here is the effect of rounding this to different values:
If you’re working in a bank, of course you wouldn’t round at all. For our purposes, the answer we got by rounding to 0.00417, three significant digits, is close enough – $5 off of $4500 isn’t too bad. Certainly keeping that fourth decimal place wouldn’t have hurt. View the following for a demonstration of this example. Using your calculatorIn many cases, you can avoid rounding completely by how you enter things in your calculator. For example, in the example above, we needed to calculate [latex]{{P}_{30}}=1000{{\left(1+\frac{0.05}{12}\right)}^{12\times30}}[/latex] We can quickly calculate 12×30 = 360, giving [latex]{{P}_{30}}=1000{{\left(1+\frac{0.05}{12}\right)}^{360}}[/latex]. Now we can use the calculator.
Using your calculator continuedThe previous steps were assuming you have a “one operation at a time” calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter: 1000 × ( 1 + 0.05 ÷ 12 ) yx 360 = Solving For TimeNote: This section assumes you’ve covered solving exponential equations using logarithms, either in prior classes or in the growth models chapter. Often we are interested in how long it will take to accumulate money or how long we’d need to extend a loan to bring payments down to a reasonable level. ExamplesIf you invest $2000 at 6% compounded monthly, how long will it take the account to double in value? Get additional guidance for this example in the following: What amount of money will amount to 750 in 6 years and to 880 years at simple interest?150 = Total SI . therefore, → Principal = 6 years Amount - 6 years simple interest = 750 - 150 = Rs. 600 (Ans.)
What sum will amount to $600 in 5 years at 4% simple interest per annum?Amount=Principal+Interest=600+120=Rs720.
How do you find the principal in simple interest?The formula for calculating Principal amount would be P = I / (RT) where Interest is Interest Amount, R is Rate of Interest and T is Time Period.
How long will an amount of money triple at a simple interest rate of 1% per annum?⇒T=x×152x×100=340=13. 3 years.
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