A woman invests 1,000 in a fund for which interest is...

A woman invests $\$ 1,000$ in a fund for which interest is compounded annually at a rate $r .$ After one year, she deposits an additional $\$ 2,000$. After two years, the balance in the account is $\$ 1,000(1+r)^{2}+\$ 2,000(1+r) .$ If this amount is $\$ 3,368.10,$ find $r$.

Key Concepts

Compound Interest

Compound interest is the process of earning interest on both the initial principal and the accumulated interest from previous periods. This concept is fundamental in finance because it shows how investments can grow exponentially over time, as each period's interest builds upon the previous total amount.

Effect of Cash Flow Timing on Compound Interest

When additional deposits are made at different times, each deposit will compound for a different number of periods. This means that the timing of cash flows is crucial; for example, a deposit made earlier will have more time to accrue interest compared to one made later.

Solving Exponential Equations in Financial Contexts

Many financial problems involve equations where the unknown variable, such as the interest rate, appears in an exponent due to the compounding process. Solving these exponential equations often requires iterative numerical methods, estimation techniques, or logarithmic transformation to isolate the variable and determine its value.

Transcript

00:01 All right, for this word problem, they've given us the equation, we just have to solve it.

00:04 So remember here that this squared is 1 plus r times 1 plus r, and then i'd still have my 2 ,000 times 1 plus r.

00:16 So i'm going to need to foil out my brackets here first, and then i can just multiply my 2 ,000 into that one is already.

00:28 So i'd have 1 ,000, 1 times 1 is 1.

00:31 1 times r is r, r times 1 is r, r times r is r squared.

00:38 2 ,000 times 1 is 2 ,000, and 2 ,000 times r is 2 ,000 r.

00:45 So now i can foil this 10 ,000 in, or 1 ,000 in.

00:51 So i have 1 ,000 plus 1 ,000 r, plus 1 ,000 r, plus 8 ,000 r, r, r squared, plus 2 ,000 plus 2 ,000 r.

01:07 So now i can combine like terms.

01:09 So i've got 1 ,000r, 1 ,000 r, and 2 ,000 r, which gets me 4 ,000 r.

01:20 I've got 1 ,000 and 2 ,000, which gets me 3 ,000, and then i just have my 1 ,000 r squared.

01:33 And on my other side i have 3 ,368 .10.

01:38 So i'm almost in the form i need to get it into, which is that a x squared plus bx plus c equals 0.

01:46 So i'm going to move over this 3 ,368 and 10 cents in order to do that.

01:57 So what i have left over here and i'm just going to reorder it is i'd have a thousand an r squared plus 4 ,000 r minus 368 .10.

02:17 So now i can say that my a is 1 ,000, my b is 4 ,000, and my c is negative 368 .10, or just 0 .1...

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