Let's break down the expression step-by-step:

$\frac{5}{6} + \frac{7}{9} \times \frac{4}{3} - \frac{1}{2}$

Step 1: Multiplication First (Order of Operations)

We need to handle the multiplication part first:

$\frac{7}{9} \times \frac{4}{3} = \frac{7 \times 4}{9 \times 3} = \frac{28}{27}$

Step 2: Now we have the expression:

$\frac{5}{6} + \frac{28}{27} - \frac{1}{2}$

Step 3: Find a Common Denominator

The denominators are 6, 27, and 2. The least common denominator (LCD) of 6, 27, and 2 is 54.

Convert each fraction:

• For $\frac{5}{6}$, multiply the numerator and denominator by 9: $\frac{5}{6} = \frac{5 \times 9}{6 \times 9} = \frac{45}{54}$

• For $\frac{28}{27}$, multiply the numerator and denominator by 2: $\frac{28}{27} = \frac{28 \times 2}{27 \times 2} = \frac{56}{54}$

• For $\frac{1}{2}$, multiply the numerator and denominator by 27: $\frac{1}{2} = \frac{1 \times 27}{2 \times 27} = \frac{27}{54}$

Step 4: Combine the Fractions

Now the expression becomes:

$\frac{45}{54} + \frac{56}{54} - \frac{27}{54}$

Combine the numerators:

$\frac{45 + 56 - 27}{54} = \frac{74}{54}$

Step 5: Simplify the Fraction

The greatest common divisor (GCD) of 74 and 54 is 2, so divide both the numerator and denominator by 2:

$\frac{74}{54} = \frac{74 \div 2}{54 \div 2} = \frac{37}{27}$

Final Answer:

$\frac{37}{27}$

Would you like a more detailed explanation of any of the steps?

---

Here are some follow-up questions to further explore:

1. How do you find the least common denominator (LCD)?
2. What is the greatest common divisor (GCD), and how do you calculate it?
3. What are other strategies to simplify fractions?
4. How does the order of operations apply to expressions with fractions?
5. How do you handle subtraction of fractions with different denominators?

Tip: Always simplify fractions by finding the GCD to ensure you have the most reduced form!