Let's solve the given problem step-by-step.
### Step 1: Convert Mixed Numbers to Improper Fractions
1. [tex]\( 1 \frac{2}{3} \)[/tex]:
[tex]\[
1 \frac{2}{3} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}
\][/tex]
2. [tex]\( 2 \frac{1}{4} \)[/tex]:
[tex]\[
2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}
\][/tex]
### Step 2: Multiply [tex]\( 2 \frac{1}{4} \)[/tex] by [tex]\( \frac{4}{15} \)[/tex]
[tex]\[
2 \frac{1}{4} \times \frac{4}{15} = \frac{9}{4} \times \frac{4}{15}
\][/tex]
Multiply the numerators and the denominators:
[tex]\[
\frac{9 \times 4}{4 \times 15} = \frac{36}{60}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{36}{60} = \frac{6}{10} = \frac{3}{5}
\][/tex]
### Step 3: Add [tex]\( 1 \frac{2}{3} \)[/tex] and the result of the multiplication
Now, we need to add [tex]\( \frac{5}{3} \)[/tex] and [tex]\( \frac{3}{5} \)[/tex]. To add these fractions, find a common denominator. The least common multiple (LCM) of 3 and 5 is 15.
Convert each fraction to have a denominator of 15:
[tex]\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\][/tex]
[tex]\[
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
\][/tex]
Now, add the fractions:
[tex]\[
\frac{25}{15} + \frac{9}{15} = \frac{25 + 9}{15} = \frac{34}{15}
\][/tex]
### Step 4: Simplify the Fraction
The fraction [tex]\(\frac{34}{15}\)[/tex] is in its simplest form as the greatest common divisor (GCD) of 34 and 15 is 1.
### Final Answer
[tex]\[
1 \frac{2}{3} + 2 \frac{1}{4} \times \frac{4}{15} = \frac{34}{15}
\][/tex]
