SOLUTION: I have been struggling with a word problem. The problem is "Best rentals charges a daily fee plus a milage fee for renting its cars. Mateo was charged $111.00 for 3 days and 300

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Question 1196509: I have been struggling with a word problem. The problem is "Best rentals charges a daily fee plus a milage fee for renting its cars. Mateo was charged $111.00 for 3 days and 300 miles, while Dara was charged $207.00 for 5 days and 600 miles. What does Best Rentals charge per day and per mile?"
I have tried to come up with an equation for this, but I've only managed to confuse myself. I have tried putting the time in as "x" and miles in as "y" and putting them equal to the total rental charge, then solving for "x" and "y". Obviously I'm not solving anything with that arrangement. I have tried making a "tic tac toe" table with the given data, should I be adding all my data up and using that to make a bigger equation. I am just confused with how to set this up. Any help is greatly appreciated, thank you.
Found 4 solutions by math_tutor2020, ikleyn, josgarithmetic, MathTherapy:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

I appreciate you showing your work and thought process, rather than just copy/pasting the problem only.

x = cost per day
y = cost per mile
Both costs are in dollars

Let's list the given facts
[1] Mateo was charged $111.00 for 3 days and 300 miles
[2] Dara was charged $207.00 for 5 days and 600 miles

Fact [1] gives us the equation
3x+300y = 111
since
3x = 3 times x = cost for the 3 days
300y = cost for the 300 miles
3x+300y = total cost

Fact [2] yields the equation 5x+600y = 207 through similar reasoning.

Here's our system of equations
3x+300y = 111
5x+600y = 207

Let's double the first equation to go from 3x+300y=111 to 6x+600y = 222

We arrive at this equivalent system of equations
6x+600y = 222
5x+600y = 207

Notice the 600y terms matching up. This allows us to cancel out the y terms if we subtract straight down
The x terms combine to 6x-5x = 1x = x
The right hand sides combine to 222-207 = 15

Therefore, x = 15

Then we can determine y
3x+300y = 111
3*15+300y = 111
45+300y = 111
300y = 111-45
300y = 66
y = 66/300
y = 0.22
You could use other equations previously mentioned involving x and y.


We found that: x = 15 and y = 0.22
Therefore, it costs $15 per day and $0.22 per mile (aka 22 cents per mile).

---------------------------------------
Check:

Mateo:
1 day = 15 dollars
3 days = 15*3 = 45 dollars so far
1 mile = 0.22 dollars
300 miles = 300*0.22 = 66 dollars added on
45+66 = 111 dollars total for Mateo

Or we could say
3x+300y = 111
3*15+300*0.22 = 111
45+66 = 111
111 = 111
which confirms Mateo's equation

Let's follow a similar idea for Dara
5x+600y = 207
5*15+600*0.22 = 207
75+132 = 207
207 = 207
Her equation is confirmed as well.

Both equations are true for (x,y) = (15, 0.22)

---------------------------------------

Answers:
$15 per day
$0.22 per mile (aka 22 cents per mile)
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.
Best rentals charges a daily fee plus a milage fee for renting its cars.
Mateo was charged $111.00 for 3 days and 300 miles, while Dara was charged $207.00 for 5 days and 600 miles.
What does Best Rentals charge per day and per mile?
~~~~~~~~~~~~~~~~~~~~ 

x = charge per day

y = charge per mile.


The problem is described by the system of 2 equations in 2 unknowns

    3x + 300y = 111   (1)    (Mateo case)

    5x + 600y = 207   (2)    (Dara case)


To solve, multiply equation (1) by 2; keep equation (2) as is.  You will get

    6x + 600y = 222   (3)    

    5x + 600y = 207   (4)  


Now subtract equation (4) from equation (3).  You will get

     x        = 222 - 207 = 15     (the terms with y will cancel each other).


Thus charge per day is 15 dollars.                   <<<---===   ANSWER


Now from equation (1)

    3*15 + 300*y = 111,


which gives

    300y = 111 - 45 = 66,   y = 66/300 = 22/100 = 0.22.


So, charge per 1 mile is $0.22, or 22 cents.         <<<---===   ANSWER

Solved.



Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
If choose r for daily rate and d for distance rate, then:


You could try to find a strategy to solve the system. Maybe first simplify the "Mateo" equation, dividing both sides by 3.



and then use five-times-Mateo's equation:


and then follow with,....

---------------- 22 cents per mile.
-




-------------------- dollars per day, daily rate

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

I have been struggling with a word problem. The problem is "Best rentals charges a daily fee plus a milage fee for renting its cars. Mateo was charged $111.00 for 3 days and 300 miles, while Dara was charged $207.00 for 5 days and 600 miles. What does Best Rentals charge per day and per mile?"
I have tried to come up with an equation for this, but I've only managed to confuse myself. I have tried putting the time in as "x" and miles in as "y" and putting them equal to the total rental charge, then solving for "x" and "y". Obviously I'm not solving anything with that arrangement. I have tried making a "tic tac toe" table with the given data, should I be adding all my data up and using that to make a bigger equation. I am just confused with how to set this up. Any help is greatly appreciated, thank you. 
Let daily and mileage fees be d, and m, respectively
We then get: 3d + 300m = 111 ---- eq (i)
       Also, 5d + 600m = 207 ---- eq (ii)
             6d + 600m = 222 ---- Multiplying eq (i) by 2 ------ eq (iii)
Subtracting eq (ii) from eq (iii), we get the daily fee, or  

3(15) + 300m = 111 -----Substituting 15 for d in eq (i)
   15 + 100m = 37 ----- Dividing each expression by GCF, 3 
        100m = 22
Mileage fee, or

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