Let's determine the difference between the fraction of counters that Alicia has at the start of the game and the fraction she has at the end.
### Step 1: Initial Fraction of Counters
At the start of the game, the ratio of Alicia's counters (A) to William's counters (W) to Isabella's counters (I) is given as [tex]\(2:7:3\)[/tex].
To find the fraction of the total number of counters that Alicia has initially (Fraction_initial_alicia):
- The total number of parts in the initial ratio is [tex]\(2 + 7 + 3 = 12\)[/tex].
- Alicia's part of the counters is [tex]\(2\)[/tex].
Thus, the initial fraction of counters that Alicia has is:
[tex]\[
\frac{Alicia's\ part}{Total\ parts} = \frac{2}{12} = \frac{1}{6}
\][/tex]
### Step 2: Final Fraction of Counters
At the end of the game, the ratio of Alicia's counters to William's counters to Isabella's counters changes to [tex]\(3:8:4\)[/tex].
To find the fraction of the total number of counters that Alicia has at the end (Fraction_final_alicia):
- The total number of parts in the final ratio is [tex]\(3 + 8 + 4 = 15\)[/tex].
- Alicia's part of the counters is [tex]\(3\)[/tex].
Thus, the final fraction of counters that Alicia has is:
[tex]\[
\frac{Alicia's\ part}{Total\ parts} = \frac{3}{15} = \frac{1}{5}
\][/tex]
### Step 3: Calculating the Difference in Fractions
Now, we need to find the difference between the fraction of counters that Alicia has at the end and the fraction she has at the start.
The initial fraction is [tex]\(\frac{1}{6}\)[/tex] and the final fraction is [tex]\(\frac{1}{5}\)[/tex].
To find the difference, we subtract the initial fraction from the final fraction:
[tex]\[
\frac{1}{5} - \frac{1}{6}
\][/tex]
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 6 is 30. Rewriting both fractions with this common denominator:
[tex]\[
\frac{1}{5} = \frac{6}{30}
\][/tex]
[tex]\[
\frac{1}{6} = \frac{5}{30}
\][/tex]
Now, subtract the fractions:
[tex]\[
\frac{6}{30} - \frac{5}{30} = \frac{1}{30}
\][/tex]
Therefore, the difference between the fraction of counters that Alicia has at the start of the game and the fraction she has at the end is:
[tex]\[
\boxed{\frac{1}{30}}
\][/tex]
