Let's determine the total number of guests at Joy's wedding step-by-step using the given information.
1. Determine the fractional parts:
- The fraction of guests who chose beef is [tex]\( \frac{1}{3} \)[/tex].
- The fraction of guests who chose chicken is [tex]\( \frac{5}{12} \)[/tex].
2. Sum of the fractions for beef and chicken:
We need to add the fractions of guests who chose beef and chicken to see what fraction of the total these two groups account for.
[tex]\[
\frac{1}{3} + \frac{5}{12}
\][/tex]
To add these fractions, we need a common denominator. The common denominator of 3 and 12 is 12.
[tex]\[
\frac{1}{3} = \frac{4}{12}
\][/tex]
Now, add the fractions:
[tex]\[
\frac{4}{12} + \frac{5}{12} = \frac{9}{12} = \frac{3}{4}
\][/tex]
Therefore, the combined fraction of guests who chose beef or chicken is [tex]\( \frac{3}{4} \)[/tex].
3. Determine the fraction of guests who chose vegetarian:
Since the total fraction of all guests must sum to 1, the fraction of guests who chose vegetarian is:
[tex]\[
1 - \frac{3}{4} = \frac{1}{4}
\][/tex]
4. Calculate the total number of guests:
We know that 69 guests chose vegetarian meals. This represents [tex]\( \frac{1}{4} \)[/tex] of the total number of guests. Let [tex]\( x \)[/tex] be the total number of guests. Using the fractional representation:
[tex]\[
\frac{1}{4} x = 69
\][/tex]
Solving for [tex]\( x \)[/tex] (the total number of guests):
[tex]\[
x = 69 \times 4 = 276
\][/tex]
So, the total number of guests at the wedding is [tex]\( \boxed{276} \)[/tex].
