To multiply the fraction \(\frac{2}{3}\) by the mixed number \(1 \frac{1}{4}\), you first need to convert the mixed number into an improper fraction.
The mixed number \(1 \frac{1}{4}\) can be converted as follows:
\[
1 \frac{1}{4} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}
\]
Now, you can multiply \(\frac{2}{3}\) by \(\frac{5}{4}\):
\[
\frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12}
\]
Next, you can simplify \(\frac{10}{12}\) by dividing both the numerator and denominator by their greatest common divisor, which is 2:
\[
\frac{10 \div 2}{12 \div 2} = \frac{5}{6}
\]
So, \(\frac{2}{3} \times 1 \frac{1}{4} = \frac{5}{6}\).