Word Problem. l

Question

Monica deposits $760 into a savings account that pays 6.2% per year. If the account compounds quarterly, find the time it takes to triple her investment.

Answer 1

The word problem is asking for the time (t) it takes to triple her investment, so we need to solve for t using the compound interest formula:

A = Amount = 3 x $760 = $2,280 (we are multiplying the Initial Principal Balance by 3 because the problem says that we are tripling her investment)

P = Initial Principal Balance = $760

r = interest rate = 0.062 (remember to use the decimal form a percentage)

n = number of times that interest is applied each time period = 4 (which represents quarterly)

t = time = the variable we need to solve for

Plug in all of your known variables into the formula:

$2,280 = $760 (1 + (0.062/4))^(4t)

Simplify by dividing both sides by $760

3 = (1+(0.062/4))^(4t)

Simplify inside the parentheses

3 = 1.0155^(4t)

Take log of both sides

log3 = log(1.0155^(4t))

Using property of logs (the power rule), you can move the 4t in front

log3 = 4t log1.0155

Isolate t by dividing both sides by 4log1.0155

(log3)/(4log1.0155) = t

Plugging into your calculator, you should get t = 17.8565 years or t ≈ 18 years

Answer 2

This looks like a compound interest problem. Why? We are given an interest rate, a rate of time that interest is compounded (quarterly), and an initial amount of money put towards compounding (aka the principle).

For compound interest, recall the formula:

A = P*(1 + r/n)^nt

A is the amount of money that the savings account will earn after compounding. P is the principle. r is the rate of interest as a decimal. n is the number of times the investment is compounded in a year. t is the time in years. The problem asks to find the time after Monica's investment is tripled. This would make the amount, A, equal to three times the value of the principle:

A = 3P

A = 3*(760) = 2280

Since we are looking for time, t, we need to isolate the compound interest equation. See below:

A = P*(1 + r/n)^nt

A/P = (1 + r/n)^nt

Take the natural log of both sides to begin isolating the t:

ln(A/P) = ln( (1+r/n)^nt)

RECALL: ln(x^a) = a * ln(x). Here, the a in the equation would relate to nt, and x would relate to (1+r/n)

ln(A/P) = nt * ln( (1+r/n))

Now, just divide both sides by n*ln(1+r/n) to have t isolated

t = ln(A/P) / (n * ln(1 + r/n))

All we have to do is plug in our values now. NOTE: quarterly compounded interest takes place 4 times a year, so n = 4. Our rate, r, is 6.2/100 = 0.062 (remember what was discussed above about r as the percentage rate being represented in decimal form).

t = ln(2280/760) / (4 * ln( 1 + 0.062 / 4))

t = 17.85 years

Hope this helps! :)

Answer 3