I'm stuck on two problems. Can anyone help?
1. At an annual interest rate of five percent, how long would it take for your savings to double?
What grade level is it? If it's Algebra II or above, the easiest way to solve the first question is using logs:
Start out using the formula for calculating interest: A = P(1 + i) ^n
where
A = the amount you end up with
P = the principle (what you start with)
i = annual interest rate
n = number of years interest accrues
Let your savings equal one dollar, and therefore you want to learn how long it takes for your savings to reach two dollars:
2 = 1 (1.05)^n
Take the log of both sides:
log 2 = log (1.05)^n; remember that an exponent in a log can be moved to the front:
log 2 = n log (1.05) ; rearrange so variable is on the left
n log (1.05) = log 2 ; divide both sides by log (1.05)
n = log 2/log(1.05) ; use calculator to solve log(2) and log(1.05)
n = 14.2
If it's Algebra I, I would solve it using a table:
Start with 1 dollar.
After 1 year, you have 1 * 1.05 = $1.05
After 2 years, you have 1.05 * 1.05 = $1.1025
After 3 years, you have 1.1025 * 1.05 = $1.1576
and so on, until you reach $2 (which should be at 14 years)
3. In the mid-1990s, selected automobiles had an average cost of $12,000. The average cost of those same motor vehicles is now $20,000. What was the rate of increase for this item between the two time periods?
Let the price in 1995 = $12,000
Let the price in 2010 = $20,000
The number of years passed = 15 years
Using the same formula as above,
A = 20,000
P = 12,000
n = 15
Therefore, 20,000 = 12,000 (1+i)^15
20,000/12,000 = (1+i)^15
1.6667 = (1 + i)^15; take the log of both sides:
log 1.6667 = log ((1 + i)^15) ; move the exponent to the front of the log
log 1.6667 = 15 log (1+i); flip so i is on the left side
15 log (1+i) = log 1.6667 ; divide both sides by 15
log (1+i) = log 1.6667/15 ; use calculator to find log 1.6667
log (1+i) = .2218/15
log (1+i)= .01479; make each side an exponent of 10
10^log(1+i) = 10^.01479 ; remember that 10^log(A) = (A)
1+i = 1.0346
i = 0.0346 or 3.46%
This could also be solved using trial and error, where we are trying to get A of 20,000:
12,000 (1+0.01)^15 = 13932
12,000 (1 + 0.02)^15 = 16150
12,000 (1 + 0.03)^15 = 18696 (just under)
12,000 (1 + 0.04)^15 = 21611 (just over)
Try in between 0.03 and 0.04:
12,000 (1 + 0.035)^15 = 20104
12,000 (1 + 0.0347)^15 = 20017
12,000 (1+0.0346)^15 = 19988
Therefore, interest rate is about 0.0346 or 3.46%
