To find the correct proportional relationship between the number of hours Michelle works ([tex]\( w \)[/tex]) and the number of hours of break time she gets ([tex]\( b \)[/tex]), let's carefully analyze the given information:
1. For every 4 hours worked, Michelle receives a 15-minute break.
2. A 15-minute break is equivalent to [tex]\(\frac{15}{60}\)[/tex] hours (since there are 60 minutes in an hour), which simplifies to 0.25 hours.
This gives us the following proportional relationship between work hours and break hours:
- For 4 hours of work, Michelle gets 0.25 hours of break.
- We can set this up as a ratio: [tex]\( \frac{0.25 \text{ hours of break}}{4 \text{ hours of work}} \)[/tex].
Simplifying this ratio, we get:
[tex]\[ \frac{0.25}{4} = 0.0625 \text{ hours of break per hour of work} \][/tex]
Now, let's express the break time [tex]\( b \)[/tex] in terms of the work time [tex]\( w \)[/tex]:
[tex]\[ b = 0.0625w \][/tex]
If we rearrange this equation to express [tex]\( w \)[/tex] in terms of [tex]\( b \)[/tex]:
[tex]\[ w = \frac{b}{0.0625} \][/tex]
Multiplying the numerator and the denominator by 16 to simplify:
[tex]\[ w = \frac{b \times 16}{0.0625 \times 16} \][/tex]
[tex]\[ w = \frac{16b}{1} \][/tex]
[tex]\[ w = 16b \][/tex]
Thus, the equation expressing the proportional relationship between the number of hours Michelle works ([tex]\( w \)[/tex]) and the number of hours of break time she gets ([tex]\( b \)[/tex]) is:
[tex]\[ w = 16b \][/tex]
The correct answer is:
[tex]\[ \boxed{w=16b} \][/tex]
