One admission to an ice-skating rink costs x dollars, and renting a pair of ice skates costs y dollars. Your friend says she can determine the exact cost of one admission and one s... One admission to an ice-skating rink costs x dollars, and renting a pair of ice skates costs y dollars. Your friend says she can determine the exact cost of one admission and one skate rental. Is your friend correct?
Understand the Problem
The question is asking whether it's possible to determine the cost of one admission and one skate rental based on two receipts from an ice-skating rink. The receipts provide totals for different combinations of admissions and rentals, and the task is to evaluate if your friend is correct in claiming they can find the exact costs of admission and skate rental based on this information.
Answer
Your friend is incorrect; the exact costs cannot be determined.
Answer for screen readers
Your friend is incorrect. We cannot determine the exact cost of one admission and one skate rental.
Steps to Solve
- Set Up the Equations
Let ( x ) be the cost of one admission and ( y ) be the cost of one skate rental. Based on the receipts, we can set up the following equations from the totals:
From receipt 1 (3 admissions and 2 skate rentals for $38.00):
$$ 3x + 2y = 38 $$
From receipt 2 (15 admissions and 10 skate rentals for $190.00):
$$ 15x + 10y = 190 $$
- Simplify the Second Equation
To make calculations easier, we can simplify the second equation by dividing everything by 5:
$$ 3x + 2y = 38 $$
Now the system of equations looks like:
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( 3x + 2y = 38 )
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( 3x + 2y = 38 )
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Determine Consistency of Equations
Notice that both equations are identical:
$$ 3x + 2y = 38 $$
This means the two equations describe the same relationship, indicating they are not independent. As a result, we cannot find a unique solution for ( x ) and ( y ).
- Conclusion
Since we have only one independent equation, we cannot uniquely determine the cost of the admission and the skate rental. The relationship represents multiple possible combinations of ( x ) and ( y ) that satisfy the same equation.
Your friend is incorrect. We cannot determine the exact cost of one admission and one skate rental.
More Information
The problem illustrates a situation where two receipts lead to the same equation. This means that there are infinitely many combinations of admission and rental costs that would satisfy the given totals, hence making it impossible to find unique values for ( x ) and ( y ).
Tips
- Mistaking the similarity of the equations for having enough information to solve for unique variables.
- Not recognizing that identical equations indicate dependent relationships, leading to multiple solutions instead.
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