A group of 6 friends went ice-skating. They each paid an admission fee of $x and $2.50 for hot chocolate. Another group of 5 friends went
ice-skating at a different rink, and they paid 1.5 times as much for admission as the first group. The group of 5 friends spent $2.25 each for hot
chocolate. If they spent the same amount, choose the response that gives an equation that represents the situation and then gives the
admission price for the first group of friends. Let x represent the admission price for the first group of friends.
A) 6(x+2.50) = 2.25(1.5x + 5); admission price for the first group is $3.10
B) 6(x+1.50) = 5(2.5x + 2.25); admission price for the first group is $2.35
C) 5(x+2.50) - 6(1.5x+2.25); admission price for the first group is $2.70
D) 6(x+2.50) = 5(1.5x +2.25); admission price for the first group is $2.50
The equation that represents the situation is C) 5(x+2.50) - 6(1.5x+2.25) because both groups spent the same amount.
To find the admission price for the first group of friends, we solve the equation for x.
5(x+2.50) - 6(1.5x+2.25) = 0
5x + 12.50 - 9x - 13.50 = 0
-4x - 1 = 0
-4x = 1
x = -1/4
Since admission prices cannot be negative, the correct response is D) 6(x+2.50) = 5(1.5x +2.25) and the admission price for the first group is $2.50.
To solve this problem, let's break it down step-by-step:
Step 1: Let's identify the variables and set up the equation.
- Let x represent the admission price for the first group of friends.
- The first group of friends consists of 6 friends who paid an admission fee of $x and $2.50 for hot chocolate.
- The second group of friends consists of 5 friends who paid 1.5 times as much for admission as the first group, and $2.25 each for hot chocolate.
- Since both groups spent the same amount, we can represent this situation with an equation.
Step 2: Write the equation.
The equation representing the situation is 6(x + 2.50) = 5(1.5x + 2.25).
Step 3: Solve the equation.
Let's simplify the equation:
6x + 15 = 7.5x + 11.25
Next, subtract 7.5x from both sides:
6x - 7.5x + 15 = 7.5x - 7.5x + 11.25
-1.5x + 15 = 11.25
Then, subtract 15 from both sides:
-1.5x + 15 - 15 = 11.25 - 15
-1.5x = -3.75
Finally, divide both sides by -1.5:
-1.5x / -1.5 = -3.75 / -1.5
x = 2.5
Step 4: Determine the admission price for the first group of friends.
The admission price for the first group of friends is $2.50.
Therefore, the correct response is:
D) 6(x + 2.50) = 5(1.5x + 2.25); admission price for the first group is $2.50