Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I purchased the Prius for 20k and goes 45 mpg. The other car costs 15k and goes 25mpg. What is the formula to calculate how long do I need to keep my car in order to see the saving.

Number of miles traveled: 800miles per month.

Gas price: $4 per gallon

Or how much money would I save each year I keep the Prius?


[ADDED] don't worry about other factors such as car resale value

share|cite|improve this question
What is the estimated price for a gallon of fuel? – t.b. May 9 '11 at 21:15
How are you calculating depreciation for each vehicle? – user1736 May 9 '11 at 21:16

If you drive the same number of miles $d$ in a year, you would use $d/45$ gallons of gas in the Prius, or $d/25$ in the other car. If gas costs $c$ dollars per gallon, the savings would be $s = cd/25 - cd/45$ dollars per year. It would then take $5000/s$ years to save enough on gas to make up for the difference in initial cost. This rather crude calculation doesn't take into account the interest that you could have earned if you invested the 5000 dollars rather than putting it into the Prius, or possible differences in costs of other things (e.g. insurance or maintenance).

share|cite|improve this answer
Substituting values from the question, d = 12*800 = 9600, c = 4, so s ≈ 682.67 and the answer 5000/s ≈ 7.32 (years). – ShreevatsaR May 9 '11 at 21:51
Caution: if you're so thrilled with your Prius that you drive it 1.8 times as much as you would have driven the old gas-guzzler, that will nullify any savings. – Robert Israel May 10 '11 at 17:13

First off, go ahead and calculate the number of gallons used per month, like so:

Prius: $\displaystyle \frac{800}{45} \approx 17.78$ gallons per month.

Other: $\displaystyle \frac{800}{25} = 32.00$ gallons per month.

Next, assuming gas is four dollars a gallon, calculate the cost per month...

Prius: $17.78*4 \approx 71.11$ dollars per month.
Other: $32.00*4 \approx 128.00$ dollars per month.

Now we want to know how a car costs after $t$ months, taking into account the initial cost.

$C_{\text{Prius}}(t) = 20000 + 71.11*t$
$C_{\text{Other}}(t) = 15000 + 128.00*t$

Next, we want to know what the difference is.

$\begin{eqnarray*} C_{\text{difference}}(t) &=& C_{\text{Prius}}(t) - C_{\text{Other}}(t) \\ &\approx& 5000 - 56.88*t \end{eqnarray*}$

In order to make any savings just from gas alone, we want to know what the minimum value of $t$ is such that $C_{\text{difference}}(t) \leq 0$. In other words...

$5000 - 56.88*t \leq 0$

Solving this equation through any number of methods yields $t \approx 87.891$ months.

In other words, keep your Prius for at least 88 months (7 years 4 months) and you have savings from the reduced gas consumption. :D

EDIT: General formula for when you don't know the price of gas ($g=4$ in my example):

$\displaystyle t \geq \frac{5000}{14.22g}$ months

share|cite|improve this answer
The other car uses $800/25 = 32$ gallons per month. – t.b. May 9 '11 at 21:50
facepalm Editing... – El'endia Starman May 9 '11 at 21:52

Starman has a good model; but you left out two details. How often do you need to replace the battery and what's it cost? Also does it cost you extra to charge the battery from your home?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.