$\frac{\frac{2}{3} \div \frac{4}{5}}{\frac{1}{4} + \frac{3}{5} - \frac{1}{3}}$

To solve this expression, we will simplify the numerator and the denominator separately before performing the final division.

1. Simplify the Numerator

The numerator is: $\frac{2}{3} \div \frac{4}{5}$

When dividing fractions, we multiply by the reciprocal of the divisor:

Simplifying $\frac{10}{12}$ by dividing both numerator and denominator by 2 gives: Numerator = $\frac{5}{6}$

2. Simplify the Denominator

The denominator is: $\frac{1}{4} + \frac{3}{5} - \frac{1}{3}$

To add and subtract these, we find the Least Common Multiple (LCM) of 4, 5, and 3, which is 60.

• $\frac{1}{4} = \frac{15}{60}$
• $\frac{3}{5} = \frac{36}{60}$
• $\frac{1}{3} = \frac{20}{60}$

Now, combine them:

Denominator = $\frac{31}{60}$

3. Final Division

Now, divide the simplified numerator by the simplified denominator:

Again, multiply by the reciprocal:

We can simplify before multiplying: 6 goes into 60 ten times:

Conclusion

Comparing our result to the given options, we find that it matches option D.

Exam Tip: When dealing with complex fractions, always handle the numerator and denominator as two separate problems first. This prevents "fraction fatigue" and reduces the likelihood of making a silly calculation error in the middle of the process.