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If 65 people attend a concert and tickets for adults cost
$3 while tickets for children cost
$2.5 and total receipts for the concert was
$180, how many of each went to the concert?

If $65$ people attend a concert and tickets for adults cost $\$ 3$ while tickets for children cost $\$ 2.5$ and total receipts for the concert was $\$ 180$ , how many of each went to the concert?

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Identify quotients of rational numbers: word problems

Full solution

Q. If

65

people attend a concert and tickets for adults cost

\$ 3

while tickets for children cost

\$ 2.5

and total receipts for the concert was

\$ 180

, how many of each went to the concert?

Denote Variables: Let's denote the number of adult tickets sold as $A$ and the number of children tickets sold as $C$ . We are given two pieces of information that will lead to two equations: $\newline$ $1$ . The total number of people who attended the concert is $65$ . $\newline$ $2$ . The total amount of money collected from ticket sales is $\$180$ . $\newline$ From the first piece of information, we can write the equation: $\newline$ $A + C = 65$
Write Equations: From the second piece of information, we can write the equation based on the cost of the tickets: $3A + 2.5C = 180$
Solve System: Now we have a system of two equations with two variables: $\newline$ $1$ . $A + C = 65$ $\newline$ $2$ . $3A + 2.5C = 180$ $\newline$ We can solve this system using substitution or elimination. Let's use substitution. We can express $A$ in terms of $C$ from the first equation: $\newline$ $A = 65 - C$
Substitute A: Substitute $A = 65 - C$ into the second equation: $\newline$ $3(65 - C) + 2.5C = 180$ $\newline$ Now, distribute the $3$ : $\newline$ $195 - 3C + 2.5C = 180$
Combine Like Terms: Combine like terms: $\newline$ $195 - 0.5C = 180$ $\newline$ Now, subtract $195$ from both sides: $\newline$ $-0.5C = 180 - 195$ $\newline$ $-0.5C = -15$
Divide and Solve for C: Divide both sides by $-0.5$ to solve for $C$ : $C = \frac{-15}{-0.5}$ $C = 30$ So, there were $30$ children's tickets sold.
Substitute C: Now, substitute $C = 30$ back into the first equation to find $A$ : $A + 30 = 65$ $A = 65 - 30$ $A = 35$ So, there were $35$ adult tickets sold.

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