Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. A caterer is making cookie trays for upcoming holiday parties. This mor... Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. A caterer is making cookie trays for upcoming holiday parties. This morning, she made 5 small trays and 1 large tray, which contain a total of 116 cookies. In the afternoon, she made 5 small trays and 5 large trays, which contain a total of 320 cookies. How many cookies do the different sized trays contain? The small tray contains ____ cookies and the large one contains ____ cookies.
Understand the Problem
The question is asking us to formulate and solve a system of equations based on the number of cookies in small and large trays as described in the scenario. We'll create two equations representing the morning and afternoon situations, then solve for the number of cookies in each type of tray.
Answer
The small tray contains $13$ cookies and the large one contains $51$ cookies.
Answer for screen readers
The small tray contains ( 13 ) cookies and the large one contains ( 51 ) cookies.
Steps to Solve
- Define Variables
Let ( x ) be the number of cookies in a small tray and ( y ) be the number of cookies in a large tray.
- Write the First Equation
From the morning situation, we know that 5 small trays and 1 large tray contain a total of 116 cookies. This can be expressed as: $$ 5x + y = 116 $$
- Write the Second Equation
From the afternoon situation, we know that 5 small trays and 5 large trays contain a total of 320 cookies. This can be expressed as: $$ 5x + 5y = 320 $$
- Simplify the Second Equation
We can simplify the second equation by dividing everything by 5: $$ x + y = 64 $$
- Setup the System of Equations
Now, we have the following system of equations:
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( 5x + y = 116 )
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( x + y = 64 )
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Substitute to Solve for ( y )
From the second equation, we can express ( y ) in terms of ( x ): $$ y = 64 - x $$
- Substitute and Solve for ( x )
Substituting this expression for ( y ) into the first equation: $$ 5x + (64 - x) = 116 $$
Now simplify: $$ 5x + 64 - x = 116 $$ $$ 4x + 64 = 116 $$ $$ 4x = 116 - 64 $$ $$ 4x = 52 $$ $$ x = 13 $$
- Find ( y )
Now substitute ( x = 13 ) back into the equation ( y = 64 - x ): $$ y = 64 - 13 = 51 $$
The small tray contains ( 13 ) cookies and the large one contains ( 51 ) cookies.
More Information
The caterer's cookie situation can be modeled using a system of equations, which is a common approach in algebra to solve for unknowns based on given relationships.
Tips
- Forgetting to simplify the second equation can lead to a more complex system than necessary.
- Mixing up the variables or incorrectly interpreting the scenario can cause errors in the equations.
- Not checking the solutions against the original problem statement to ensure they fit all conditions.
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