Question

During one month of cell phone use, Noah used 200 anytime minutes and 400 text messages, and paid $80.00. The next month, he used 150 anytime minutes and 350 text messages, and paid $67.50. Which statement is true?

Each text message costs 5 cents more than each anytime minute.
Each anytime minute costs 10 cents more than each text message.
A text message and an anytime minute each cost 25 cents.
Each text message costs double the amount of an anytime minute.

Answer 1

To solve this problem, let's denote the cost of each anytime minute as \( m \) (in dollars) and the cost of each text message as \( t \) (in dollars).

From the information provided, we can set up the following equations based on Noah's usage and total payments:

1. **First month**:
\[
200m + 400t = 80
\]

2. **Second month**:
\[
150m + 350t = 67.50
\]

Now, let's simplify these equations.

### First month equation:
\[
200m + 400t = 80 \quad \Rightarrow \quad 2m + 4t = 0.8 \quad \Rightarrow \quad m + 2t = 0.4 \quad \text{(Equation 1)}
\]

### Second month equation:
\[
150m + 350t = 67.50 \quad \Rightarrow \quad 15m + 35t = 6.75 \quad \Rightarrow \quad 3m + 7t = 1.35 \quad \text{(Equation 2)}
\]

Now we can solve these two equations (Equation 1 and Equation 2):

**From Equation 1**:
\[
m = 0.4 - 2t
\]

**Substituting \( m \) into Equation 2**:
\[
3(0.4 - 2t) + 7t = 1.35
\]
\[
1.2 - 6t + 7t = 1.35
\]
\[
1.2 + t = 1.35
\]
\[
t = 1.35 - 1.2 = 0.15
\]

Now substitute \( t = 0.15 \) back into Equation 1 to find \( m \):
\[
m + 2(0.15) = 0.4
\]
\[
m + 0.3 = 0.4
\]
\[
m = 0.1
\]

So we have:
- The cost of each anytime minute \( m = 0.10 \) dollars (or 10 cents).
- The cost of each text message \( t = 0.15 \) dollars (or 15 cents).

Now let's evaluate the given options:

1. **Each text message costs 5 cents more than each anytime minute.**
- True: \( 0.15 - 0.10 = 0.05 \) (5 cents)

2. **Each anytime minute costs 10 cents more than each text message.**
- False: \( 0.10 \) is not more than \( 0.15 \).

3. **A text message and an anytime minute each cost 25 cents.**
- False: They cost 10 cents and 15 cents respectively.

4. **Each text message costs double the amount of an anytime minute.**
- False: \( 0.15 \) is not double \( 0.10 \).

The only true statement is:
**Each text message costs 5 cents more than each anytime minute.**