How many Students and how many adults bought tickets?

Question

The spring concert at a certain high school sold 170 tickets. Students were charge $3 each and adults $6 each. The income from the sale of tickets was $804. How many students and how many adults bought tickets?

Answer 1

Hello Jordan!

This doesn't look to me like an elementary math problem, since it requires the use of variables to solve.

We can create an equation using the information given. Let's create variables for the unknown numbers.

S = the number of student tickets sold at the concert

A = the number of adult tickets sold at the concert

We know that there were 170 tickets sold in total. Therefore, the number of student tickets sold plus the number of adult tickets sold equals 170.

S + A = 170

This also means that:

170 = S + A

170 - S = A

S = 170 - A

Now we can use only one variable, the letter A, to represent both the number of adult tickets sold and the number of student tickets sold.

S = 170 - A

A = A

We are told that each student ticket cost $3. Therefore we can represent the total income from student tickets by the expression $3 x S. Since S = 170 - A, we can use this expression instead: $3 x (170 - A).

We are told that each adult ticket cost $6. Therefore we can represent the total income from the adult tickets by the expression $6 x A.

We also know that the total income from the concert tickets was $804. If we add the income from all the adult tickets to the income from all the student tickets, we will get $804. This can be represented by the following equation.

$3 x (170 - A) + $6 x A = $804

(The income from student tickets plus the income from the adult tickets equals the total income from the concert.)

This can also be written this way:

3(170 - A) + 6A = 804

(We can ignore the dollar signs for now.)

Now we can multiply the items inside the parenthesis by 3.

3 x 170 -3A + 6A = 804

Multiply.

3 x 170 = 510

Combine like terms.

-3A + 6A = 3A

Now we have:

510 + 3A = 804

Subtract 510 from both sides of the equation.

510 + 3A = 804

-510 -510

3A = 294

Now we can just divide both sides by 3 and we will find the value of "A".

3A = 294

divided by 3 divided by 3

A = 98

The number of adult tickets sold was 98.

We can find the number of student tickets sold by substituting 98 for "A".

S = 170 - 98 = 72

Therefore 72 students bought tickets, and 98 adults bought tickets!

We can check our answer by plugging these numbers in.

$3 x 72 = $216

$6 x 98 = $588

$216 + $588 = $804

Answer 2

You typically solve this sort of problem by setting up a system of equations

Suppose

x = the # of adult tickets

y= the # of student tickets

We are told that 170 tickets were sold.

So we express that as the equation:

x + y = 170

We are also told that student tickets are $3 each and adult tickets are $6 each. And that the total income was $804 dollars

We write that as the equation"

6x + 3y = 804

So we solve the system

x + y = 170

6x + 3y = 804

We could use the addition method or substitution method to solve. Lets use the substitution method. Using the first equation we can solve for y and get that y = 170 - x.

Then we substitute this into the second equation to get:

6x + 3(170-x) = 804

6x + 510 - 3x = 804 combine like terms:

3x + 510 = 804 subtract 510 from both sides

3x = 804- 510 = 294 divide by 3

x = 294/3 = 98

If x= 98 then we can plug that value into our substitution equation 7= 170 - x = 170 - 98 = 72

So 98 adult tickets were sold, and 72 student tickets.