John requires 3000 in 6 months to pay off a loan that has no...

John requires $$ 3000$ in 6 months to pay off a loan that has no prepayment privileges. If he has the $$ 3000$ now, how much of it should he save in an account paying 3 % compounded monthly so that in 6 months he will have exactly 3000 ?

Key Concepts

Interest Rate Conversion

Interest rate conversion involves adjusting the annual interest rate to match the compounding period of the investment, such as converting an annual rate to a monthly rate. This conversion is necessary for accurately computing the growth of the investment over non-annual periods, ensuring that the compound interest formula is applied correctly.

Present Value Calculation

Present value is the current equivalent of a future amount of money given a specific rate of return. It involves discounting the future amount back to the present using the appropriate compounding interest formula. Understanding present value is crucial for making decisions about how much to invest now to achieve a desired amount in the future.

Time Value of Money

The time value of money is a financial principle that states money available now is worth more than the same amount in the future due to its potential earning capacity. This principle underlies the concept of present value and future value, as it helps in determining how much a current sum of money will grow over time with a specified interest rate.

Compound Interest

Compound interest refers to the process where interest is added to the principal amount of an investment, so that the added interest also earns interest from that point on. This concept is fundamental to understanding how investments grow over time with periodic interest calculations, and it is especially relevant when the interest is compounded at regular intervals such as monthly.

Transcript

00:01 Today we're going to be talking about paying off a loan, and so john requires $3 ,000 in six months to pay off his loan.

00:08 If he has the $3 ,000 now, how much of it should he save in an account paying 3 % compounded monthly so that in six months he will have exactly $3 ,000? and so we're going to use our compound interest formula, which is a equals p times 1 plus r over n to the n to the n.

00:34 T power.

00:37 So we want to know when he will have exactly $3 ,000.

00:42 We're going to say that a, which is our final amount, is going to be $3 ,000.

00:47 We want to know how much he would need to get exactly to $3 ,000.

00:51 So we're going to solve for p.

00:55 We're going to take our 1 plus hour rate.

00:59 In this question, our rate is going to be 3 % compounded monthly, which is going to be 0 .03.

01:06 You take 3 % you divide by 100.

01:09 You're going to get .03 divided by 12 or being compounded monthly, because there are 12 months in a year.

01:18 That same 12 is going to be used as an exponent as that's the same variable.

01:25 And then we want to figure out six months.

01:28 Well, since we're talking about per month and it being yearly, how many years is six months? and so that is half a year.

01:37 And so what we're going to put here is 0 .5.

01:42 And so what we're going to do here is we're going to kind of put all this stuff together on the right side to make it easier to simplify and solve for p...

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