To find the ratio of Jerome's share to Kevin's share to Seth's share of the sandwich, let's first summarize how much of the sandwich each person ate:
- Jerome ate [tex]\(\frac{1}{2}\)[/tex] of the sandwich.
- Kevin ate [tex]\(\frac{1}{3}\)[/tex] of the sandwich.
- Seth ate the rest of the sandwich.
To figure out Seth's share, we need to calculate what remains of the sandwich after accounting for Jerome's and Kevin's shares.
1. Add Jerome's and Kevin's shares together:
[tex]\[
\frac{1}{2} + \frac{1}{3}
\][/tex]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 2 and 3 is 6.
So, convert each fraction:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
[tex]\[
\frac{1}{3} = \frac{2}{6}
\][/tex]
2. Now we can add the fractions:
[tex]\[
\frac{3}{6} + \frac{2}{6} = \frac{5}{6}
\][/tex]
This means Jerome and Kevin together ate [tex]\(\frac{5}{6}\)[/tex] of the sandwich.
3. Seth's share is the remaining part of the sandwich:
[tex]\[
1 - \frac{5}{6}
\][/tex]
Again, we convert the 1 to a fraction with the same denominator:
[tex]\[
1 = \frac{6}{6}
\][/tex]
So,
[tex]\[
\frac{6}{6} - \frac{5}{6} = \frac{1}{6}
\][/tex]
Hence, Seth ate [tex]\(\frac{1}{6}\)[/tex] of the sandwich.
Next, we will determine the ratio of their shares. We need a common baseline to compare the fractions directly. Since [tex]\(\frac{1}{2}\)[/tex], [tex]\(\frac{1}{3}\)[/tex], and [tex]\(\frac{1}{6}\)[/tex] have the denominators 2, 3, and 6 respectively, we use the least common multiple (LCM) of these denominators, which is 6.
Let's express each share with the common base of 6:
- Jerome's share:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
- Kevin's share:
[tex]\[
\frac{1}{3} = \frac{2}{6}
\][/tex]
- Seth's share:
[tex]\[
\frac{1}{6} = \frac{1}{6}
\][/tex]
Thus, the ratio of Jerome's share to Kevin's share to Seth's share is:
[tex]\[
\frac{3}{6} : \frac{2}{6} : \frac{1}{6}
\][/tex]
To express this ratio in its simplest form (removing the common denominator 6), we get:
[tex]\[
3 : 2 : 1
\][/tex]
Therefore, the ratio of Jerome's share to Kevin's share to Seth's share is [tex]\(3 : 2 : 1\)[/tex].
