Let's solve the problem step-by-step.
Given:
- The initial fee for Big Time Movers is [tex]\( \$24.50 \)[/tex].
- The hourly rate for their moving services is [tex]\( \$12.75 \)[/tex].
- On holidays, the charge is 2.5 times the regular total amount.
- They earned [tex]\( \$188.75 \)[/tex] on New Year's Day.
We need to find the number of hours they worked, denoted as [tex]\( x \)[/tex].
We start with the equation:
[tex]\[ 2.5 \times (12.75x + 24.50) = 188.75 \][/tex]
First, we'll isolate the expression inside the parentheses by dividing both sides by 2.5:
[tex]\[ 12.75x + 24.50 = \frac{188.75}{2.5} \][/tex]
Calculate the division on the right-hand side:
[tex]\[ 12.75x + 24.50 = 75.50 \][/tex]
Next, isolate the term with [tex]\( x \)[/tex] by subtracting [tex]\( 24.50 \)[/tex] from both sides:
[tex]\[ 12.75x = 75.50 - 24.50 \][/tex]
Perform the subtraction:
[tex]\[ 12.75x = 51.00 \][/tex]
Now, solve for [tex]\( x \)[/tex] by dividing both sides by the hourly rate [tex]\( 12.75 \)[/tex]:
[tex]\[ x = \frac{51.00}{12.75} \][/tex]
Finally, calculate the division to find [tex]\( x \)[/tex]:
[tex]\[ x = 4.0 \][/tex]
Therefore, Big Time Movers worked [tex]\(\boxed{4.0}\)[/tex] hours on New Year's Day.
