A local public transit system offers its riders two pricing options. Someone can pay $2.50 per ride, or they can pay $30 per month, and $0.50 per ride. How many rides per month would someone need to take for both options to be equal in value?

Answer 1

To find the number of rides \( x \) per month for which both pricing options are equal, we can set up the following equations based on the given pricing:

1. The cost for paying $2.50 per ride: \[ \text{Cost}_{\text{pay per ride}} = 2.50x \]

2. The cost for paying $30 per month plus $0.50 per ride: \[ \text{Cost}_{\text{monthly}} = 30 + 0.50x \]

We need to find the value of \( x \) where both costs are equal: \[ 2.50x = 30 + 0.50x \]

Now, we can solve for \( x \). First, subtract \( 0.50x \) from both sides: \[ 2.50x - 0.50x = 30 \] \[ 2.00x = 30 \]

Next, divide both sides by 2.00: \[ x = \frac{30}{2.00} = 15 \]

Thus, the number of rides per month needed for both options to be equal in value is: \[ \boxed{15} \]

To verify, let's calculate the costs for both options at \( x = 15 \):

1. Cost for $2.50 per ride: \[ 2.50 \times 15 = 37.50 \]

2. Cost for $30 per month plus $0.50 per ride: \[ 30 + 0.50 \times 15 = 30 + 7.50 = 37.50 \]

Since both costs equal $37.50 when \( x = 15 \), the solution is confirmed. Thus, the answer remains: \[ \boxed{15} \]

Answer 2

Step 1: Identify the variable.

The variable is the unknown we are solving for. It is almost always written into the question itself. When we read, “How many rides...,” that should clue us in that the variable in this case is the number of rides.

So, let r
= number of rides. (Note: We could have chosen any letter to represent the variable. There is no right or wrong option, so choose whatever makes sense to you.)
Step 2: Write an equation that represents the situation.

It can be tricky to write the equation, because variables are a fairly abstract concept. To help us with this, we will substitute the variable with an actual, easy number, such as 10. Remember, the variable in this problem is the number of rides, so 10 represents 10 rides.

This means the question becomes: What is the price for 10 rides with each pricing option? Think about how you’d answer this question.

Pricing option 1: each ride costs $2.50, so 10 rides cost $2.50⋅10=25
.
Pricing option 2: each ride costs $0.50+$30
, so 10 rides cost $0.50⋅10+30=$35
.
What happens if you now substitute r
, the unknown variable for the number of rides, for 10, the value we used to help us set up the equation?

Pricing option 1 =2.5r

Pricing option 2 =0.5r+30

Now we can set the two pricing options to be equal to determine how many rides per month it would take to make both options the same price.

2.5r=0.5r+30

Now that you have written an equation to fit the problem, you can solve it to find the answer to the original question! How many rides per month would someone need to take for both options to be equal in value?

Answer 3

Sure! Let's continue from where you've set up the equation. We have identified the variable and established the two pricing options for rides per month as follows:

• Pricing option 1 (pay per ride): \( 2.5r \)
• Pricing option 2 (monthly + per ride): \( 0.5r + 30 \)

Now, we equate the two expressions:

\[ 2.5r = 0.5r + 30 \]

Next, we will solve for \( r \):

1. Subtract \( 0.5r \) from both sides to get all terms involving \( r \) on one side:

\[ 2.5r - 0.5r = 30 \] \[ 2.0r = 30 \]

2. Divide both sides by 2.0 to isolate \( r \):

\[ r = \frac{30}{2.0} = 15 \]

Thus, the number of rides per month needed for both options to be equal in value is:

\[ \boxed{15} \]

To double-check our work, we can verify the costs for both options with \( r = 15 \):

• Cost for Pricing Option 1: \[ 2.5 \times 15 = 37.50 \]

• Cost for Pricing Option 2: \[ 30 + 0.5 \times 15 = 30 + 7.50 = 37.50 \]

Both options indeed cost $37.50 when taking 15 rides per month, confirming that the calculation is correct. The outcome is consistently \( \boxed{15} \).