I drove 280 miles, made up of both highway and city driving. I used 15 gallons of gas that cost $3.15 per gallon. If my car gets 24mpg highway and 16mpg city, how much did it cost me for the highway portion of the trip, and how much for the city portion of the trip? Explain work.
If my car gets 24mpg highway and 16mpg city, how much did it cost me for the highway portion of the trip, and how much for the city portion of the trip?
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let x= # of gallons used driving on the highway
let y= # of gallons used driving in the city
if x+y=15, then y=15-x
substitute y=15-x in the second equation
x=5 gallons used on the highway
5+y=15 and y=10
y=10 gallons used in the city
5x$3.15=$15.75 for the highway cost
10x$3.15=$31.50 for the city cost
check: 15x$3.15=47.25 and $15.75+$31.50=$47.25
By the way, if you want to determine your average gas mileage and you drive the same number of miles in the city as on the highway, you can use a special case of what is called the harmonic mean., For example, if you get 10 mpg in the city and 15 mpg on the highway and you drive 30 mile in the city and 30 miles on the highway, you use 3 gallons of gas in the city and 2 gallons of gas on the highway. Therefore you use 5 gallons altogether(3+2) and you drive 60 miles altogether(30+30).
Therefore 60/5=12 mpg. Using the special case of the harmonic mean, we have (2xrate#1xrate#2)/(rate#1+rate#2)=(2x10x15)/(10+15)=
Using your problem, we have (2x24x16)/(24+16)=768/40=19.2 mpg.
This is your average miles per gallon if you drive the same distance in the city as on the highway. Another example:20 mpg city and 30 mpg highway-(2x20x30)/(20+30)=1200/50=24 mpg average
Notice the average is always closer to the city mpg !
24 m/g highway
16 m/g city
Let's cancel our units...
Gallons is a denominator and numerator...
$47.25[(280 m)/(15 g)]=[(24 m/g)(x)]+[(16 m/g)($47.25-x)]
First, let cancel our units...
On both sides, we have miles/gallon...
$47.25[(280 m)/(15 g)]=[(24 m/g)(x)]+[(16 m/g)($47.25-x)]
The only unit remaining is $, which is what we are solving for...
Let's combine like units...
Let's subtract $756.00 from both sides...
Let's divide both sides by 8...
$15.75 spent on highway
$47.25-$15.75=$31.50 spent on city
Let's check our work...
[($15.75)(24 m/g)]+[($31.50)(16 m/g)]=($3.15/g)(280 m)
Our units align in that each side is $-m/g...
x = miles of hwy driving
280-x = miles of city driving
Gallons for hwy + Gallons for city = Total gallons for the trip
(x miles)/(24mpg) + (280-x)/16 = 15
2x/48 + 3(280-x)/48 = 15
(2x + 840 - 3x)/48 = 15
(-x + 840)/48 = 15
-x + 840 = 48 * 15
-x + 840 = 720
x = 120 miles hwy
x/24 = 120/24 = 5 gallons for the hwy
(280-x)/16 = (280 - 120) / 16 = 160/16 = 10 gallons city
5 gallons hwy * (3.15) = $15.75 cost of hwy
10 gallons city * (3.15) = $31.50 cost of city
15 gallons * (3.15) = 47.25
x = no. of gallons used on highway
y = no. of gallons used in city
according to your question,
x+y = 15 ................................................................equation 1
your 2nd equation is
24x+16y = 280 .......................................................equation 2
Multiply Equation 1 by 16, u will get
16x+16y = 240 ........................................................equation 3
Now subtraction equation 3 from 2, u will get
8x = 40
therefore x= 5
putting the value of x in equation 1
5+y = 15, y=10
Check: putting the value of x and y in equation 2
5*3.15 = 15.75 cost on highway
10*3.15 = 31.5 cost in the city
Total cost = 15.75+31.5 = 47.25
Let H represent the number of gallons of gas used for Highway travel.
Let C represent the number of gallons of gas used for City travel.
15 total gallons of gas are used, so one equation, I'll call the "gallons" equation is:
H gal + C gal = 15 gal or more simply: H + C = 15
We have 2 unknowns, but only one equation so far. So the second equation comes from the MPG information (I will write this equation including the units, so you can see how it "works").
This one I'll call the "miles" equation:
24 mi / gal * H gal + 16 mi / gal * C gal = 280 mi
Notice that for the left side of this equation, in both terms the "gallon" units will "cancel," since one of them is in the denominator (from the MPG part -- miles per gallon, or mi / gal), and the other is in the numerator (from the unknowns H and C, which represent "gallons").
After the "gallon" units cancel, the left side of the equation is only expressed in terms of "miles," which matches the "miles" on the right side of the equation.
Here is the updated "miles" equation:
24 mi * H + 16 mi * C = 280 mi or more simply: 24 H + 16 C = 280
When we combine these 2 equations (the "gallons" equation with the "miles" equation, we get the following system of equations:
H + C = 15 ("gallon" equation)
24 H + 16 C = 280 ("miles" equation)
You can now solve this system in any way you choose, depending on what your teacher requires.
I suspect that the solution method (substitution, elimination, graphing, Cramer's rule, or matrix methods) is not what was giving you trouble; it was setting up the system of equations in the first place.
If you are familiar with matrix methods on the graphing calculator, then that is probably the fastest way to go. You simply enter this system of equations as a 2x3 augmented matrix, and then find the solution using the "rref" (reduced-row echelon form) method.
(feel free to message me if you would like the detailed, step-by-step instructions)
To solve using the substitution method (probably the fastest "by-hand" way)
Solve "gallons" equation for either variable (I'll choose H):
H = 15 - C
Now, substitute (15 - C) for H in the second, "miles" equation of 24 H + 16 C = 280, like so:
24 ( 15 - C ) + 16 C = 280 Now, you have an equation only in terms of one variable, C.
So, simplify and solve this equation:
360 - 24 C + 16 C = 280
- 8 C = 280 - 360
- 8 C = - 80
So C = 10 gallons. But the question is not asking how many gallons were used for City travel; it is asking how much it COST for the City travel. All you do is multiply the cost per gallon from the given information:
Cost of City driving = 10 gal * $3.15 / gal = $31.50 total cost for City driving.
(Note that this problem did NOT ask about the Highway driving, but this is easy enough to get.)
You already know that C = 10, so substitute this back into the "gallons" equation of H + C = 15:
H + 10 = 15 so H = 5 gallons,
and cost of Highway driving = 5 gal * $3.15 / gal = $15.75 total cost for Highway driving.
Total cost of gas = $3.15 per gallon * 15 gallons = $47.25.
distance of highway travel is unknown; call it x.
distance of city travel is 280-x.
distance = rate * time.
x = 24 *time1
280-x = 16 * time 2
Without knowing how long you traveled on each type of road, you cannot solve this question.
in other words, time 1 + time 2 has not been specified, and without it, I cannot solve.
Have a suspicion that there is way to solve this, but I only spent a minute or so looking at the question. At the least, this gives you a framework to start you off.