To determine how many heads of lettuce, \( x \), Timothy would need to feed 275 guests based on the ratio of heads of lettuce to guests, we can set up a proportion.
The original proportion is:
\[ \frac{37 \text{ heads of lettuce}}{100 \text{ guests}} = \frac{x \text{ heads of lettuce}}{275 \text{ guests}} \]
This can be rearranged to match one of the given options. Let's analyze the options:
A. \( \frac{37}{100} = \frac{x}{175} \): This option does not correspond correctly to what we're looking for because the denominator on the right side should correspond to 275 guests.
B. \( \frac{x}{100} = \frac{37}{275} \): This does not match either since it has the wrong sides of the ratio.
C. \( \frac{100}{275} = \frac{x}{37} \): This also doesn’t match correctly since it switches the positions.
D. \( \frac{100}{37} = \frac{x}{275} \): This is the correct setup. When rearranging the initial proportion, we want to say that the ratio of guests to heads of lettuce must be equal, which matches this setup.
Thus, the correct answer is:
D. \( \frac{100}{37} = \frac{x}{275} \)