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?

2 Answers
Dec 14, 2017

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

Dec 14, 2017

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 - xlarr 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!