Question

the admission fee at an amusement park is $3.25 for children and $5.40 for adults. on a certain day, 309 people entered the park, and the admission fees collected totaled $1316. how many children and how many adults were admitted? number of children: 159 number of adults: 150 incorrect! we kindly recommend reviewing the using systems of equations to investigate profits, or alternatively, you may engage in a discussion with mathgpt for a comprehensive explanation. try again attempts: 1/3

Explanation:

Step1: Define variables

Let \( x \) be the number of children and \( y \) be the number of adults.
We have two equations:

1. \( x + y = 309 \) (total number of people)
2. \( 3.25x + 5.40y = 1316 \) (total admission fees)

From the first equation, we can express \( x = 309 - y \).

Step2: Substitute into the second equation

Substitute \( x = 309 - y \) into \( 3.25x + 5.40y = 1316 \):
\( 3.25(309 - y) + 5.40y = 1316 \)

Step3: Expand and simplify

\( 3.25\times309 - 3.25y + 5.40y = 1316 \)
\( 994.25 + 2.15y = 1316 \)

Step4: Solve for \( y \)

Subtract 994.25 from both sides:
\( 2.15y = 1316 - 994.25 \)
\( 2.15y = 321.75 \)
Divide both sides by 2.15:
\( y = \frac{321.75}{2.15} = 150 \) (Wait, but let's check again. Wait, maybe miscalculation. Wait, 3.25309: 3309=927, 0.25309=77.25, so 927+77.25=1004.25? Wait, I think I made a mistake here. Let's recalculate 3.25309:

3093 = 927, 3090.25 = 77.25, so 927 + 77.25 = 1004.25. So the equation is 1004.25 - 3.25y + 5.40y = 1316. Then 1004.25 + 2.15y = 1316. Then 2.15y = 1316 - 1004.25 = 311.75. Then y = 311.75 / 2.15 = 145. Wait, that's different. Wait, maybe the initial numbers were wrong. Let's do it again.

Wait, let's start over.

Let \( x \) = number of children, \( y \) = number of adults.

Equation 1: \( x + y = 309 \) => \( x = 309 - y \)

Equation 2: \( 3.25x + 5.40y = 1316 \)

Substitute \( x = 309 - y \) into equation 2:

\( 3.25(309 - y) + 5.40y = 1316 \)

Calculate 3.25*309:

309*3 = 927

309*0.25 = 77.25

So 927 + 77.25 = 1004.25

So:

1004.25 - 3.25y + 5.40y = 1316

Combine like terms:

1004.25 + 2.15y = 1316

Subtract 1004.25 from both sides:

2.15y = 1316 - 1004.25 = 311.75

Now divide both sides by 2.15:

y = 311.75 / 2.15 = 145

Then x = 309 - 145 = 164

Wait, let's check the total fees:

1643.25 + 1455.40

1643.25: 1603.25=520, 4*3.25=13, so 520+13=533

1455.40: 1455=725, 145*0.4=58, so 725+58=783

Total: 533 + 783 = 1316. Yes, that works.

So the mistake was in the initial calculation of 3.25*309. It's 1004.25, not 994.25. So the correct number of adults is 145, and children is 164.

Answer:

Number of children: 164
Number of adults: 145