The cost of a gym membership is a onetime joining fee of $140, plus a monthly fee of $40. Joe wrote a $500 check to pay his gym membership for a certain number of months, including the onetime fee. How many months of membership did he pay for?

The cost of membership at gym A can be written as:
c
a
=
$
10
m
+
$
15
The cost of membership at gym B can be written as:
c
b
=
$
4
m
+
$
40
Where
c
g
y
m
is the cost after
m
months.
To find when the cost is equal, equate the right side of both equations and solve for
m
:
$
10
m
+
$
15
=
$
4
m
+
$
40
Next, subtract
$
15
and
$
4
m
from each side of the equation to isolate the
m
term while keeping the equation balanced:
−
$
4
m
+
$
10
m
+
$
15
−
$
15
=
−
$
4
m
+
$
4
m
+
$
40
−
$
15
(
−
$
4
+
$
10
)
m
+
0
=
0
+
$
25
$
6
m
=
$
25
Now, divide each side of the equation by
$
6
to solve for
m
while keeping the equation balanced:
$
6
m
$
6
=
$
25
$
6
$
6
m
$
6
=
$
25
$
6
m
=
25
6
m
=
4.17
At 4 months the cost of the two gym memberships will be about the same.