Sure, let's break this problem down step by step.
1. Define the Costs:
- Let [tex]\( M_1 \)[/tex] represent the total cost for the first gym membership.
- Let [tex]\( M_2 \)[/tex] represent the total cost for the second gym membership.
2. Write Out the Expressions:
- For the first gym membership:
- Monthly cost: [tex]$10
- Sign-up fee: $[/tex]45
- After [tex]\( m \)[/tex] months, the total cost will be:
[tex]\[
M_1 = 10m + 45
\][/tex]
- For the second gym membership:
- Monthly cost: $25
- No sign-up fee
- After [tex]\( m \)[/tex] months, the total cost will be:
[tex]\[
M_2 = 25m
\][/tex]
3. Set Up the Equation to Find When the Costs Are Equal:
We need to find the month [tex]\( m \)[/tex] when [tex]\( M_1 = M_2 \)[/tex]. So, we set the two expressions equal to each other:
[tex]\[
10m + 45 = 25m
\][/tex]
4. Solve the Equation:
- First, move all terms involving [tex]\( m \)[/tex] to one side of the equation and constants to the other:
[tex]\[
10m + 45 = 25m
\][/tex]
Subtract [tex]\( 10m \)[/tex] from both sides:
[tex]\[
45 = 25m - 10m
\][/tex]
Simplify the equation:
[tex]\[
45 = 15m
\][/tex]
- Next, solve for [tex]\( m \)[/tex] by dividing both sides by 15:
[tex]\[
m = \frac{45}{15}
\][/tex]
[tex]\[
m = 3
\][/tex]
Therefore, the total cost of both gym memberships will be the same after 3 months.
