Suzanne O. answered • 10/25/19
International Experience and Multiple State Certifications
Word problems always look hard, but they get easier as you learn how to break them down into smaller steps and translate the words into mathspeak.
The problem: Blake's cell phone company charges him $20 plus $0.40 a minute. Steven’s cell phone company charges him $45 and $0.25 a minute. For how many minutes would Steven and Blake have to talk on their cell phones so that Steven’s plan is better than Blake's?
As soon as I saw this problem, I saw two lines (y = mx + b) on a graph.
The first line would represent Blake's cell phone plan, where:
- y = the cost of the plan after talking for x minutes
- m = 0.40 (slope, it affects x)
- b = 20 (y-intercept, it is a constant)
So Blake's y = 0.40x + 20.
The second line would represent Steven's cell phone plan, where:
- y = the cost of the plan after talking for x minutes
- m = 0.25 (slope, it affects x)
- b = 45 (y-intercept, it is a constant)
And Steven's y = 0.25x + 45.
The reason I saw lines is because the value of the y's will change depending on how many minutes you use.
Before anyone makes a call, Blake's plan is less expensive (smaller y-intercept). However, because Steven's plan costs less per minute to use, we expect to see Steven's plan become a better deal the more they talk.
This means that we would see a point where the two lines cross on the graph. That crossing point is where the two plans are of equal value.
After that, Steven's plan with the smaller slope (m) will grow less quickly. So it will be less expensive than Blake's.
Now we just have to find that point where the two graphs intersect.
To do this, set the two lines equal to each other, and solve for x:
- 0.40x + 20 = 0.25x + 45
- 0.40x - 0.25x = 45 - 20
- 0.15x = 25
- x = 166.67
This means that at any usage over 166.67 minutes and Steven's plan costs less than Blake's. At any usage under 166.67 minutes, and Blake's is the better deal.
To test the answer, just pick a number for x, substitute it for x in the two equations, solve and see which way the inequality goes.
When x = 167:
- 0.40x + 20 ?? 0.25x + 45
- 0.40•167 + 20 ?? 0.25•167 + 45
- 86.80 > 86.75 ✔
And when x = 166:
- 0.40x + 20 ?? 0.25x + 45
- 0.40•166 + 20 ?? 0.25•166
- 86.40<86.50✔✔
😎