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 involves solving for the number of adults at a concert given the total number of attendees and a difference in the number of adults and children. We will set up a system of equations to find the solution.
Answer
The number of adults is $4511$.
Answer for screen readers
The number of adults at the concert is $4511$.
Steps to Solve
- Define Variables
Let $A$ be the number of adults and $C$ be the number of children.
- Set Up the Equations
From the problem, we have two key pieces of information:
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The total number of attendees: $$ A + C = 6872 $$
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There were 2,150 more adults than children: $$ A = C + 2150 $$
- Substitute One Equation into Another
Substitute the second equation into the first equation: $$ (C + 2150) + C = 6872 $$
- Combine Like Terms
Combine the terms to simplify: $$ 2C + 2150 = 6872 $$
- Solve for Children
Isolate $C$ by subtracting 2150 from both sides: $$ 2C = 6872 - 2150 $$ $$ 2C = 4722 $$
Now, divide by 2: $$ C = \frac{4722}{2} = 2361 $$
- Calculate the Number of Adults
Now that we have $C$, use the second equation to find $A$: $$ A = C + 2150 $$ $$ A = 2361 + 2150 = 4511 $$
The number of adults at the concert is $4511$.
More Information
This problem illustrates how to set up a system of equations with two variables and solve it. Understanding how to represent relationships between quantities can help in a wide variety of problems.
Tips
- Incorrectly setting up equations: Ensure that the relationships described in the problem are accurately translated into equations.
- Forgetting to isolate variables properly: Double-check each algebraic manipulation step to avoid computational errors.
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