Mental Arithmetic

If it were me doing this in my head... first, I'd drop the trialing 0's, remembering to multiply by 10 at the end - so 13mpg and 5 mpg. Next, some simple multiplication - 13x5 = 65. So, lets just say the distance traveled was 65 miles each way. That means you used 5 gallons going, and 13 gallons returning - a total of 18 gallons of gas, over a total of 130 miles. 130/18 = 7 R4. 4/18 =2/9=0.2222. So we have 7.2222... now remember those trailing 0's we took off at the beginning? Multiply by 10 and you've got 72.222 as the answer.

All told, you do a single multiplication, a single addition, and a single division. Removing the trailing 0's at the beginning and adding them back on at the end just makes that easier (who wants to keep track of numbers like 6500 and 130 when you don't have to?)

 

I get as far as (2 * 130 * 50) / (130 + 50) and then do the arithmetic in my head. Others might have a different algebra vs arithmetic balance point.

 

how do you know you got 130 if the gauge only goes to 99.9?

 

Wouldn't you just add 130 to 50, then divide by 2? :huh:

130+50= 180
180 / 2 = 90

Unless you went different distances for each way (different routes).
Or, unless your 130 was on the entire tank, in which case, I'm not playing.

 

Not quite, although that is a very common mistake. Since the distance is the same each way, what you really care about is how much gas is used each (gallons per mile) way. 130 MPG is actually 130 miles / 1 gallon - to find out how much gas is used, you would formally do it like this:

X miles * (1 / 130 MPG) = X/130 gallons
X miles * (1 / 50 MPG) = X/50 gallons.

Lets assume X=1 mile, just to get rid of the variable.

For the total amount of gas used, you have to add them... but first get common denominators:

50/(130*50) = 50/6500
130/(50*130) = 130/6500
50/6500 + 130/6500 = 180/6500 gallons

Now, we need this back in MPG, right? So take our total distance traveled (we said X=1, so each direction was 1 mile - total distance traveled was 2 miles) and divide by the amount of gas used:

2 / (180/6500) = 2*6500/180 = 72.22222


The math becomes odd because we use Miles Per Gallon, and not Gallons Per Mile. It's the same kind of problem you see most often with speed - a guy drove 20 miles to work going 45MPH, and 20 miles home going 60MPH. What was his average speed? Everyone hears that and knows immediately it's a "trick question", since the answer isn't (45+60)/2. But most people only know it's a trick question because they had to solve it back in school... not necessarily because they understand the logic behind it.

 

Can one of you quote the math rule that 'splains why you can't average mpg like that? I instinctively know you can't average two rates like that but it's been so long since I took Algebra that I don't remember the rule.

 

The rule of common denominators. You're dealing with Miles / Gallon... your miles stay the same in both directions, but your gallons change. It helps to understand if you assign some arbitrary distance - lets say 5 miles:

130mpg going there, 50mpg coming back, 5 miles traveled in each direction.

MPG = total miles traveled / total gas used

We know the total miles traveled is 10 for this example... but how much gas was used?

5 miles / 130 mpg + 5 miles / 50 mpg

Since we now have our mpg's on the bottom, we have to use common denominators to do the addition:
250 miles / 6500 mpg + 650 miles / 6500mpg = 900 miles / 6500 mpg = 9/65

So, we can plug that in now: 10 / (9/65) = 650/9 = 72.222

Taking that whole process into a single equation:

Code:

                   2 * distance per leg
------------------------------------------------
    distance per leg           distance per leg
    ---------------      +    ---------------
    mileage for leg 1          mileage for leg 2

= (This is the common denominator step!)
                                          2 * distance per leg
-----------------------------------------------------------------------------------
    distance per leg * mileage for leg 2           distance per leg * mileage for leg 1
    ---------------------------------      +    ----------------------------------
    mileage for leg 1 * mileage for leg 2          mileage for leg 1 * mileage for leg 2 

=
                                          2 * distance per leg
-----------------------------------------------------------------------------------
    distance per leg * mileage for leg 2 + distance per leg * mileage for leg 1
    -------------------------------------------------------------------
                          mileage for leg 1 * mileage for leg 2

=

                   2 * distance per leg * mileage for leg 1 * mileage for leg 2
-----------------------------------------------------------------------------------
      distance per leg * mileage for leg 2 + distance per leg * mileage for leg 1

=

   2 * mileage for leg 1 * mileage for leg 2
--------------------------------------------
     mileage for leg 2 + mileage for leg 1

The key is simply to understand that you need to break it down into miles traveled and gas used, then recombine them to make MPG.

At least, back in school that's the process my teacher would have wanted to see in order to get full credit for "showing my work"... even though I'd write down the answer first!