Math Help - word problem

1. word problem

A pair of shoes now costs $85 and the price is expected to increase 3% each year.Approximately how many years from now will the price be$100?

2. Originally Posted by gracy
A pair of shoes now costs $85 and the price is expected to increase 3% each year.Approximately how many years from now will the price be$100?
Note: A 3% increase in the price means you mulitply by 103%, or by 1.03.

This is essentially a geometric series. Let t be the number of years and P the price of the shoes in dollars. Then:
$P(t) = 85 \cdot (1.03)^t$

So, we want to know t when P = \$100.

$100 = 85 \cdot (1.03)^t$

$\frac{100}{85} = (1.03)^t$

$(1.03)^t = 1.176470588$

Now take the logarithm of both sides. I personally prefer log base e = "ln", but pick which ever base you prefer. The choice does not matter.

$ln \left ( (1.03)^t \right ) = ln(1.176470588)$

$t \cdot ln(1.03) = ln(1.176470588)$

$t = \frac{ln(1.176470588)}{ln(1.03)} = 5.498156798$

So t = 5.5 years or so.

-Dan