In either case, whether you will be collecting the interest or paying the interest, the amount of the principal is generally symbolized by the variable P. [2] X Research source For example, if you have made a loan to a friend of $2,000, the principal loaned would be $2,000.
For example, suppose you loaned money to a friend under the understanding that at the end of 6 months your friend would pay you back the $2,000 plus 1. 5%. The one-time interest rate is 1. 5%. But before you can use the rate of 1. 5% you must convert it to a decimal. To change percent to a decimal, divide by 100: 1. 5% ÷ 100 = 0. 015.
It is important that the length of the term match the interest rate, or at least be measured in the same units. For example, if your interest rate is for a year, then your term should be measured in years as well. If the rate is advertised as 3% per year, but the loan is only six months, then you would calculate a 3% annual interest rate for a term of 0. 5 years. As another example, if the rate is agreed to be 1% per month, and you borrow the money for six months, then the term for calculation would be 6.
I=P∗r∗t{\displaystyle I=Prt} Using the above example of the loan to a friend, the principal (P{\displaystyle P}) is $2,000, and the rate (r{\displaystyle r}) is 0. 015 for six months. Because the agreement in this example was for a single term of six months, the variable t{\displaystyle t} in this case is 1. Then calculate the interest as follows: I=Prt=(2000)(0. 015)(1)=30{\displaystyle I=Prt=(2000)(0. 015)(1)=30}. Thus, the interest due is $30. If you want to calculate the amount of the full payment due (A), with the interest and the return of the principal, then use the formula A=P(1+rt){\displaystyle A=P(1+rt)}. This calculation would look like: A=P(1+rt){\displaystyle A=P(1+rt)} A=2000(1+. 015∗1){\displaystyle A=2000(1+. 015*1)} A=2000(1. 015){\displaystyle A=2000(1. 015)} A=2,030{\displaystyle A=2,030}
A=P(1+rt){\displaystyle A=P(1+rt)} A=5000(1+. 03∗0. 25){\displaystyle A=5000(1+. 03*0. 25)} A=5000(1. 0075){\displaystyle A=5000(1. 0075)} A=5037. 5{\displaystyle A=5037. 5} In three months, you would earn $37. 50 interest. Note that t=0. 25 here, because three months is one-fourth (0. 25) of the original one year term.
The formula for calculating the value (A) of compounding interest is: A=P(1+rn)nt{\displaystyle A=P(1+{\frac {r}{n}})^{nt}}
For example, a credit card may advertise interest of 15% per year. However, interest is generally applied each month, so you may want to know the monthly interest rate. In that case, divide by 12, to find the monthly interest rate of 1. 25% per month. These two rates, 15% per year or 1. 25% per month, are equivalent to each other.
If interest is compounded annually, then n=1. If interest is compounded quarterly, for example, then n=4.
For example, for a loan of one year, then t=1{\displaystyle t=1}. But, for a term of 18 months, then t=1. 5{\displaystyle t=1. 5}.
First, identify the variables that you need to solve the problem. In this case: P=$5,000{\displaystyle P=$5,000} r=0. 05{\displaystyle r=0. 05} n=12{\displaystyle n=12} t=3{\displaystyle t=3}
For the problem started above, this would look as follows: A=P(1+rn)nt{\displaystyle A=P(1+{\frac {r}{n}})^{nt}} A=5000(1+0. 0512)12∗3{\displaystyle A=5000(1+{\frac {0. 05}{12}})^{12*3}} A=5000(1+0. 00417)36{\displaystyle A=5000(1+0. 00417)^{36}} A=5000(1. 00417)36{\displaystyle A=5000(1. 00417)^{36}} A=5000(1. 1616){\displaystyle A=5000(1. 1616)} A=5808{\displaystyle A=5808} Thus, after three years, compound interest will have amounted to $808, in addition to the original $5,000 deposit.
Using some calculus, mathematicians have developed a formula that simulates interest that is compounded and added back to the account in a continuous stream. This formula, which is used to calculate continuously compounding interest, is: A=Pert{\displaystyle A=Pe^{rt}}
A{\displaystyle A} is the future value (or Amount) of money that the loan will be worth after compounding the interest. P{\displaystyle P} is the principal. e{\displaystyle e}. Although this looks like a variable, it is actually a constant number. The letter e{\displaystyle e} is a special number called “Euler’s constant,” named for the mathematician Leonard Euler who discovered its properties. Most advanced graphing calculators have a button for ex{\displaystyle e^{x}}. If you press this button, with the number 1, to represent e1{\displaystyle e^{1}}, you will learn that the value of e{\displaystyle e} is approximately 2. 718. r{\displaystyle r} is the interest rate per year. t{\displaystyle t} is the term of the loan, measured in years.
P=200,000{\displaystyle P=200,000} e{\displaystyle e}, again, is not a variable but is the constant 2. 718. r=0. 042{\displaystyle r=0. 042} t=30{\displaystyle t=30}
A=Pert{\displaystyle A=Pe^{rt}} A=200000∗2. 718(0. 042)(30){\displaystyle A=2000002. 718^{(0. 042)(30)}} A=200000∗2. 7181. 26{\displaystyle A=2000002. 718^{1. 26}} A=200000∗3. 525{\displaystyle A=200000*3. 525} A=705000{\displaystyle A=705000} Notice the enormous value of compounding interest continuously.
