There was a total of 6,872 adults and children at a concert. There were 2,150 more adults than children. How many adults were there at the concert?
Understand the Problem
The question is asking for the number of adults at a concert where the total attendance was 6,872, and there were 2,150 more adults than children. We will need to set up a system of equations to solve for the number of adults and children.
Answer
The number of adults at the concert is \( 4511 \).
Answer for screen readers
The number of adults at the concert is ( 4511 ).
Steps to Solve
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Define Variables Let ( a ) be the number of adults and ( c ) be the number of children.
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Set Up Equations From the problem, we have two equations:
- The total attendance equation: $$ a + c = 6872 $$
- The relationship between adults and children: $$ a = c + 2150 $$
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Substitute the Second Equation into the First Replace ( a ) in the first equation with the expression from the second equation: $$ (c + 2150) + c = 6872 $$
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Combine Like Terms Combine the terms on the left side: $$ 2c + 2150 = 6872 $$
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Isolate the Variable ( c ) Subtract 2150 from both sides: $$ 2c = 6872 - 2150 $$ $$ 2c = 4722 $$
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Solve for ( c ) Divide both sides by 2: $$ c = \frac{4722}{2} $$ $$ c = 2361 $$
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Calculate ( a ) Use the value of ( c ) to find ( a ): $$ a = c + 2150 $$ $$ a = 2361 + 2150 $$ $$ a = 4511 $$
The number of adults at the concert is ( 4511 ).
More Information
This problem illustrates how to set up and solve a system of equations based on a word problem. The relationships defined by the total attendance and the difference between adults and children can be represented mathematically to find a solution.
Tips
- Misunderstanding the relationships: Ensure to clearly define the relationships between adults and children.
- Arithmetic errors in calculations: Double-check addition and subtraction steps.
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