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Question

Blakes 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 Blakes

Answer 1

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:

1. y = the cost of the plan after talking for x minutes
2. m = 0.40 (slope, it affects x)
3. 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:

1. y = the cost of the plan after talking for x minutes
2. m = 0.25 (slope, it affects x)
3. 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:

1. 0.40x + 20 = 0.25x + 45
2. 0.40x - 0.25x = 45 - 20
3. 0.15x = 25
4. 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:

1. 0.40x + 20 ?? 0.25x + 45
2. 0.40•167 + 20 ?? 0.25•167 + 45
3. 86.80 > 86.75 ✔

And when x = 166:

1. 0.40x + 20 ?? 0.25x + 45
2. 0.40•166 + 20 ?? 0.25•166
3. 86.40<86.50✔✔

😎

Answer 2

Let x be the number of minutes

Blake: 20 + 0.4x

Steve: 40 + 0.25x

We want to find when Steve's plan becomes better than Blake so we need to write an inequality

Blakes Plan > Steve's Plan

Blakes Plan becomes more expensive :( than Steve's

20 + 0.4x > 45 + 0.25x

Multiply by 100 on both sides

2000+40x > 4500 +25x

Subtract 2000 each side

40x>25x+2500

Simplify

15x>2500

x> 2500/15

x>500/3

Check our answer

Blake at x =45+ 0.4(500/3)

Blake at x = 86.667

Steve at x = 20+0.25(500.3)

Steve at x = 86.667

Black greater than x at 500/3

Example : 501/3

Blake: 86.8

Steve: 86.75

86.8>86.75 :)

Answer 3

Both of these plans are linear expressions, that is they increase at a constant rate. For the purposes of this question, let's have B stand for Blake's plan, S for Steven's plan, and m for the number of minutes each could spend talking.

To calculate Blake's cell phone bill we can note it as B = 0.40m + 20 This is because Blake's company charges him $20 before he makes any calls and $0.40 for each minute.

Likewise, Steven's phone bill could be found by saying S = 0.25m + 45 with Steven's company charging $45 before making any calls and $0.25 each minute.

To find where Steven's plan is better than Blake's, you will want to find when Steven pays less than Blake. For this we will need yet another equation. Well, it's really just a new way to organize equations we already made.

We are finding where S < B

This means that 0.25m + 45 < 0.40m + 20

Now all that's left is to solve for m. Tt do this, get all variables with m on one side and other variables on the other then get m by itself. Remember, if you divide by a negative number, the < or > sign will be reversed.

Please feel free to contact me if you have any further questions on this or need additional help with other problems.