We are asked to solve the multiplication problems involving fractions. We need to evaluate the three expressions: (1) $\frac{3}{7} \times \frac{2}{5}$ (3) $\frac{4}{15} \times \frac{5}{12}$ (5) $1\frac{1}{9} \times \frac{5}{7}$

1. Problem Description

We are asked to solve the multiplication problems involving fractions. We need to evaluate the three expressions:

(1) $\frac{3}{7} \times \frac{2}{5}$

(3) $\frac{4}{15} \times \frac{5}{12}$

(5) $1\frac{1}{9} \times \frac{5}{7}$

2. Solution Steps

(1) $\frac{3}{7} \times \frac{2}{5}$

To multiply fractions, multiply the numerators together and the denominators together.

$\frac{3}{7} \times \frac{2}{5} = \frac{3 \times 2}{7 \times 5} = \frac{6}{35}$

(3) $\frac{4}{15} \times \frac{5}{12}$

We can multiply directly and then simplify:

$\frac{4}{15} \times \frac{5}{12} = \frac{4 \times 5}{15 \times 12} = \frac{20}{180}$

We can simplify by dividing both the numerator and the denominator by

2

0. $\frac{20}{180} = \frac{20 \div 20}{180 \div 20} = \frac{1}{9}$

Alternatively, we can simplify first by canceling common factors:

$\frac{4}{15} \times \frac{5}{12} = \frac{4 \times 5}{15 \times 12} = \frac{4}{12} \times \frac{5}{15} = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$.

(5) $1\frac{1}{9} \times \frac{5}{7}$

First, convert the mixed number to an improper fraction: $1\frac{1}{9} = \frac{1 \times 9 + 1}{9} = \frac{10}{9}$.

Then, multiply the fractions:

$\frac{10}{9} \times \frac{5}{7} = \frac{10 \times 5}{9 \times 7} = \frac{50}{63}$

3. Final Answer

(1) $\frac{6}{35}$

(3) $\frac{1}{9}$

(5) $\frac{50}{63}$