I was helping my Algebra I brother with his math homework tonight. One of the problems he had to do went something like this:
There are two rental car companies. One charges $17 per day of rental and $0.13 per mile driven. Another charges $19 per day of rental and $0.07 per mile driven. How many miles must you drive for the costs to be equal?
I looked at this and thought "Well there can't be any one solution, as the number of miles necessary will fluctuate based on the number of days the car is rented." Every other problem in the packet had a single solution, so I was curious about this.
So the first thing I did was to create two equations to model the situation, and set them equal to each other:
17x + 0.13y = 19x + 0.07y
Simplifying, this becomes:
0.13y = 2x + 0.07y
Plugging in values of 1 and 2 for x, the number of necessary miles came about to be 33.3... and 66.6... respectively (I could also have extended the solution of the equation to model this, but just plugging in values was simpler). So the number of miles necessary to create equal cost is 33.3...x, where x is number of days of rental.
Another blank asked for total costs of car rental when this number of miles was driven (so it would be the same for both car companies). By setting 33.3... for y and 1 for x in both of the original equations, the total cost came out to be 21.3...x, where x is number of days of rental.
I'm fairly positive that this is correct, but I'm wary, as that was the only problem in the entire packet which required a general, rather than a particular, solution. I made a note of this on my brother's packet, so that the teacher would know. I'm wondering if this is correct, however. I can't help but think that I missed something obvious when the solution to this one problem is so drastically different than the solutions to the others.
If anyone could help confirm or deny this, I'd appreciate it. Thanks.
There are two rental car companies. One charges $17 per day of rental and $0.13 per mile driven. Another charges $19 per day of rental and $0.07 per mile driven. How many miles must you drive for the costs to be equal?
I looked at this and thought "Well there can't be any one solution, as the number of miles necessary will fluctuate based on the number of days the car is rented." Every other problem in the packet had a single solution, so I was curious about this.
So the first thing I did was to create two equations to model the situation, and set them equal to each other:
17x + 0.13y = 19x + 0.07y
Simplifying, this becomes:
0.13y = 2x + 0.07y
Plugging in values of 1 and 2 for x, the number of necessary miles came about to be 33.3... and 66.6... respectively (I could also have extended the solution of the equation to model this, but just plugging in values was simpler). So the number of miles necessary to create equal cost is 33.3...x, where x is number of days of rental.
Another blank asked for total costs of car rental when this number of miles was driven (so it would be the same for both car companies). By setting 33.3... for y and 1 for x in both of the original equations, the total cost came out to be 21.3...x, where x is number of days of rental.
I'm fairly positive that this is correct, but I'm wary, as that was the only problem in the entire packet which required a general, rather than a particular, solution. I made a note of this on my brother's packet, so that the teacher would know. I'm wondering if this is correct, however. I can't help but think that I missed something obvious when the solution to this one problem is so drastically different than the solutions to the others.
If anyone could help confirm or deny this, I'd appreciate it. Thanks.
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