To evaluate the expression $\frac{1}{5} + \frac{1}{4} \div \frac{3}{4}$, we follow the proper order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction):

Step 1: Simplify the division

The division $\frac{1}{4} \div \frac{3}{4}$ is equivalent to multiplying $\frac{1}{4}$ by the reciprocal of $\frac{3}{4}$: $\frac{1}{4} \div \frac{3}{4} = \frac{1}{4} \times \frac{4}{3} = \frac{1 \cdot 4}{4 \cdot 3} = \frac{4}{12} = \frac{1}{3}.$

Step 2: Add $\frac{1}{5} + \frac{1}{3}$

To add $\frac{1}{5} + \frac{1}{3}$, we find a common denominator. The least common denominator of 5 and 3 is 15. Rewrite the fractions: $\frac{1}{5} = \frac{3}{15}, \quad \frac{1}{3} = \frac{5}{15}.$ Now add: $\frac{3}{15} + \frac{5}{15} = \frac{8}{15}.$

Final Answer:

$\frac{1}{5} + \frac{1}{4} \div \frac{3}{4} = \frac{8}{15}.$

Would you like further clarification or details?

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Related Questions:

1. What is the difference between addition and multiplication of fractions?
2. How do you determine the least common denominator of fractions?
3. Why do we multiply by the reciprocal during division of fractions?
4. Can you show examples where order of operations changes the result of an expression?
5. How can improper fractions be handled during operations?

Tip: Always simplify individual operations first when evaluating an expression to avoid mistakes.