Let's answer this step-by-step.
1. Determine the least common denominator (LCD):
To perform operations with fractions, we often convert them to have a common denominator. The least common denominator for the fractions [tex]\(\frac{1}{8}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex] is the smallest number that both denominators can divide without leaving a remainder.
The denominators are 8 and 3. The least common multiple (LCM) of 8 and 3 is 24.
Thus, the least common denominator is 24.
2. Convert the fractions to have the same denominator (LCD):
- Convert [tex]\(\frac{1}{8}\)[/tex] to have a denominator of 24:
[tex]\[
\frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}
\][/tex]
- Convert [tex]\(\frac{2}{3}\)[/tex] to have a denominator of 24:
[tex]\[
\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}
\][/tex]
3. Subtract the dad's part from the sisters' part:
- The fraction the dad ate is [tex]\(\frac{3}{24}\)[/tex].
- The fraction the sisters ate is [tex]\(\frac{16}{24}\)[/tex].
To find out how much less the sisters ate than the dad, subtract the dad's fraction from the sisters' fraction:
[tex]\[
\frac{16}{24} - \frac{3}{24} = \frac{16 - 3}{24} = \frac{13}{24}
\][/tex]
4. Express the difference as a decimal:
Convert the fraction [tex]\(\frac{13}{24}\)[/tex] to a decimal by dividing the numerator by the denominator:
[tex]\[
\frac{13}{24} \approx 0.5416666666666666
\][/tex]
Therefore, Jose's sisters ate [tex]\(\frac{13}{24}\)[/tex] or approximately 0.5417 less of the cake than their dad.
