Question

A gym offers regular memberships for $80 per month and off-peak memberships for $60 per month. Last month, the gym sold a total of 420 memberships for a total of $31,100. How many memberships sold were regular memberships?

Answer 1

295 tickets sold at regular price and 125 sold at concessional price.

Explanation:

Number of tickets sold at regular rate = `x`,
Tickets sold at concession = `y`,

`x+y = 420`

`80x+60y=31100`

`20x+(60x+60y)=31100`

`20x+60(x+y)=31100`

`=> 20x+60*420=31100`

`20x+25200=31100`

`20x=31100 - 25200`

`20x=5900`

`x=5900/20`

`x=295`

`x+y=420`

`295+y=420 => y = 420-295 = 125`

Answer 2

Last month, `295` regular $80 memberships were sold.

Explanation:

The trick to solving problems like this is to find a way to express BOTH kinds of information in the problem:

1) Find a way to write the NUMBER of each kind of membership

2) Then find a way to express the VALUE of both kinds of memberships

1) Find a way to express the NUMBER of each kind of membership

Let `x` represent the number of `$80` memberships out of 420 total.
Then all the rest of the memberships are for `$60`

`$80` memberships . . . . . . . . . `x larr` number of `$80` memberships
`$60` memberships. . . `420 - x` `larr` number of `$60` memberships

2) Find a way to express the VALUE of each kind of membership

`80x` `larr` Value of `x` number of `$80` memberships
`60(420 - x)` `larr` Value of `(420 - x)` number of `$60` memberships

3) Together, these memberships have a value of `$31,100`

[$80 memberships] "plus" [$60 memberships] "equals" [$31,100]
[ . . . . . . . `80x` . . . . . ] . .`+` . [. . . `60(420 - x)`. . ] . . .`=` . . [. `31 100` .]

4) Write the equation and solve for x

`80x + 60(420 - x) = 31 100`
Solve for `x`, already defined as "the number of $80 memberships"

1) Divide all the terms on both sides by 10 to make the numbers smaller
`8x + 6(420-x) = 3110`

2) Divide all the terms on both sides by 2 to make the numbers even smaller
`4x + 3(420 - x) = 1555`

3) Clear the parentheses by distributing the 3
`4x + 1260 - 3x = 1555`

4) Combine like terms
`x + 1260 = 1555`

5) Subtract 1260 from both sides to isolate `x`, already defined as "the number of $80 memberships"
`x = 295` `larr` the number of $80 regular memberships

Answer:
Last month, `295` regular $80 memberships were sold

Check
`295` memberships @ `$80` ea . . . . `$23,600`
`125` memberships @ `$60` ea . . . . `$   7,500`
.................... ............................................. .................................
Total . . . . . . . . . . . . . . . . . . . . . . . . . `$31,100`
`Check!`