One cellular phone carrier charges $26.50 a month and $0.15 a minute for local calls. Another carrier charges $14.50 a month and $0.25 per minute for local calls. For how many minutes is the cost of the plans the same?
The answer given “120 minutes” is incorrect.
This question is determined not only by math but also by grammar. The question was not “at how many minutes…” or “at what time…”. The question was “for how many minutes…”
“For” indicates duration. In order to ask the moment at which the charges would equalize, the question would need to be worded using “at,” indicating a specific time when this was to occur. Since the author instead used the word “for,” the meaning instead is about duration.
For example: at what time…? At how many miles…? At what distance…?
As opposed to: for how much time…? For how many miles…? For what distance…?
So, rather than “120 minutes,” which is the moment at which the two phone companies are charging the same rate, the correct answer is “1 minute,” which is the length of time at which the two phone companies are charging the same rate. For the first 119 they charged differently, and from 121 on they will again be charging differently.
To find the number of minutes for which the cost of the plans is the same, we can set up an equation.
Let's use 'x' to represent the number of minutes.
For the first carrier:
Monthly cost = $26.50 + ($0.15 * x)
For the second carrier:
Monthly cost = $14.50 + ($0.25 * x)
Setting up the equation:
$26.50 + ($0.15 * x) = $14.50 + ($0.25 * x)
To solve for 'x', let's simplify the equation:
$26.50 - $14.50 = ($0.25 * x) - ($0.15 * x)
$12.00 = $0.10 * x
Now, divide both sides of the equation by $0.10:
$12.00 / $0.10 = x
120 = x
Therefore, the cost of the plans will be the same for 120 minutes.
To find the number of minutes for which the cost of the two plans is the same, we need to set up an equation.
Let's assume the number of minutes is 'm'.
For the first carrier, the monthly cost is $26.50, and each local call costs $0.15 per minute.
So, the cost of the first carrier's plan would be $26.50 + ($0.15 × m) = $26.50 + $0.15m.
For the second carrier, the monthly cost is $14.50, and each local call costs $0.25 per minute.
So, the cost of the second carrier's plan would be $14.50 + ($0.25 × m) = $14.50 + $0.25m.
To find the point where the costs are equal, we can set up the equation:
$26.50 + $0.15m = $14.50 + $0.25m.
Now, let's solve this equation to determine the number of minutes that make the costs equal:
$26.50 - $14.50 = $0.25m - $0.15m.
Simplifying further:
$12 = $0.10m.
Next, we divide both sides of the equation by $0.10 to isolate 'm':
m = $12 ÷ $0.10.
Calculating this division:
m = 120 minutes.
So, the cost of the two plans will be the same after 120 minutes.
when is .15t + 26.50 = .25t + 14.50
multiply by 100 to clean up the numbers
15t + 2650 = 25t + 1450
-10t = -1200
t = 120 minutes
at 120 minutes the charge would be the same