Let's solve the problem step-by-step to find the fraction of the original pie that was not eaten.
1. Fraction of the pie eaten by Lara:
Lara ate [tex]\(\frac{1}{4}\)[/tex] of the pie.
2. Fraction of the pie eaten by Ryan:
Ryan ate [tex]\(\frac{3}{10}\)[/tex] of the pie.
3. Total fraction of the pie eaten by Lara and Ryan:
The fractions of the pie eaten by Lara and Ryan need to be added together:
[tex]\[
\frac{1}{4} + \frac{3}{10}
\][/tex]
To add these fractions, we need a common denominator. The least common multiple of 4 and 10 is 20. So, we convert each fraction:
[tex]\[
\frac{1}{4} = \frac{5}{20}
\][/tex]
[tex]\[
\frac{3}{10} = \frac{6}{20}
\][/tex]
Now add the fractions:
[tex]\[
\frac{5}{20} + \frac{6}{20} = \frac{11}{20}
\][/tex]
4. Fraction of the pie remaining after Lara and Ryan ate:
The total fraction of the pie they did not eat is:
[tex]\[
1 - \frac{11}{20}
\][/tex]
Convert 1 to a fraction with the same denominator:
[tex]\[
1 = \frac{20}{20}
\][/tex]
Subtract the fractions:
[tex]\[
\frac{20}{20} - \frac{11}{20} = \frac{9}{20}
\][/tex]
5. Fraction of the remaining pie eaten by Cassie:
Cassie ate [tex]\(\frac{2}{3}\)[/tex] of the remaining [tex]\(\frac{9}{20}\)[/tex] of the pie. So we multiply:
[tex]\[
\frac{2}{3} \times \frac{9}{20}
\][/tex]
Multiply the numerators and the denominators:
[tex]\[
\frac{2 \times 9}{3 \times 20} = \frac{18}{60}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{18}{60} = \frac{3}{10}
\][/tex]
6. Fraction of the original pie eaten by all three:
We have to add together the fraction of the pie eaten by Lara, Ryan, and Cassie:
[tex]\[
\frac{11}{20} + \frac{3}{10}
\][/tex]
Convert [tex]\(\frac{3}{10}\)[/tex] to a fraction with the same denominator (20):
[tex]\[
\frac{3}{10} = \frac{6}{20}
\][/tex]
Now add the fractions:
[tex]\[
\frac{11}{20} + \frac{6}{20} = \frac{17}{20}
\][/tex]
7. Fraction of the original pie that was not eaten:
The fraction of the pie that was not eaten is:
[tex]\[
1 - \frac{17}{20}
\][/tex]
Convert 1 to a fraction with the same denominator:
[tex]\[
1 = \frac{20}{20}
\][/tex]
Subtract the fractions:
[tex]\[
\frac{20}{20} - \frac{17}{20} = \frac{3}{20}
\][/tex]
Therefore, the fraction of the original pie that was not eaten is [tex]\(\boxed{\frac{3}{20}}\)[/tex].
