Solve Complex Fraction Expression: (3/5)÷(9/10) + (7/9)÷(1/3)

To solve the expression $\frac{\frac{3}{5}}{\frac{9}{10}}+\frac{\frac{7}{9}}{\frac{1}{3}}$, we need to apply the division of fractions and simplify the resulting expressions.

First, consider the expression $\frac{\frac{3}{5}}{\frac{9}{10}}$:

• When dividing by a fraction, multiply by its reciprocal. The reciprocal of $\frac{9}{10}$ is $\frac{10}{9}$.
• Therefore, $\frac{\frac{3}{5}}{\frac{9}{10}} = \frac{3}{5} \times \frac{10}{9}$.
• Multiplying the numerators and the denominators, we get $\frac{3 \times 10}{5 \times 9} = \frac{30}{45}$.
• Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 15: $\frac{30 \div 15}{45 \div 15} = \frac{2}{3}$.

Next, consider the expression $\frac{\frac{7}{9}}{\frac{1}{3}}$:

• The reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$.
• Therefore, $\frac{\frac{7}{9}}{\frac{1}{3}} = \frac{7}{9} \times \frac{3}{1}$.
• Multiplying the numerators and the denominators, we get $\frac{7 \times 3}{9 \times 1} = \frac{21}{9}$.
• Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3: $\frac{21 \div 3}{9 \div 3} = \frac{7}{3}$.

Now add the simplified fractions: $\frac{2}{3} + \frac{7}{3}$.

• The fractions have a common denominator, 3, so we can simply add the numerators: $\frac{2 + 7}{3} = \frac{9}{3}$.
• Simplify $\frac{9}{3}$ by dividing both the numerator and the denominator by 3: $\frac{9 \div 3}{3 \div 3} = 3$.

Therefore, the final solution to the expression is $3$.