at a school concert, 445 people paid a total of $1578 to attend. Each adult paid $6, while each student paid $2. How many students attended the concert?

Question

 this a math word problem for algebra, 

a grade 8 question and it would be much appreciated to be answered in steps so I am able to understand fully

Answer 1

First, let's break the problem down & identify variables and relationships between the variables:

 

"445 people paid a total of $1578 to attend"

total people = 445

total cost of all people = $1578

 

"Each adult paid $6, while each student paid $2."

A = number of adults: $6 each

S = number of students: $2 each

A + S = 445 (total number of people)

6A + 2S = 1578 (total cost)

 

How many students attended the concert?

Find S

 

Since we have a system of equations and need to solve for S, let's solve the first {1} for A and substitute that expression in to the second equation {2}:

 

A + S = 445 {1} (subtract S from each side)

A = 445 - S

 

6A + 2S = 1578 {2} (substitute in expression for A )

 

6(445 - S) + 2s = 1578   (distribute on left)

 

2670 - 6S + 2S = 1578   (combine like terms on left)

 

2670 - 4S = 1578   (subtract 2670 from both sides)

 

-4S = -1092  (divide both sides by -4)

 

S = 273  (this is the answer)

 

273 students attended the concert.

 

check:

445 - 273 = 172 adults

6(172) + 2(273) = 1032 + 546 = 1578

 

Answer 2

Let:

    x = number of adults

    y = number of students

 

Translate:  "445 people"     means      x+y=445       [dq1]

 

Now, the dollars paid for each group is:

     (dollars pail) = (dollars per ticket)*(number of tickets:

 

So, 

     (dollars paid for adults) + (dollars paid for students) = $1578

           $6x                        +               $2y                  = $1578

 

The math:

  To solve first for y,   use  y=445-x    from eq1.

  The problem asks "how many students," so let's solve for y using  x=445-y and substitute into eq2:

 

      $6(445-y) + $2y = $1578

      2670 - 4y = 1578              [distribute; divide by $]

        -4y = -1092                    [subtract 2670 from both sides]

          y = 273                      [divide both sides by (-4)]

  There were 273 students who paid for the concert.

 

Checking (very important):

   There were 445-273 = 172 adults.

   Is:    172*6 + 273*2 = 1578   ?

            1032 + 546 = 1578   ?

               1578 = 1578   ?yes