Question 1: Solve for x: \( \frac{1}{2}x + \frac{1}{4} = \frac{3}{4} \).

Answer: x = 1

Question 2: Solve the equation: \( \frac{3}{5}x - \frac{2}{5} = 1 \).

Answer: x = 5

Question 3: Find x: \( \frac{2}{3}x + \frac{1}{3} = \frac{5}{3} \).

Answer: x = 2

Question 4: Construct an equation with fractions where multiplying through clears fractions, and solve for x: \( \frac{2}{7}x - \frac{1}{7} = \frac{3}{7} \).

Answer: x = 2

Question 5: A recipe calls for \( \frac{1}{2} \) cup of sugar per serving. Write an equation to find the total sugar for x servings, and solve for x if the total sugar is 2 cups.

Answer: x = 4

Question 6: Explain how multiplying through helps eliminate fractions in an equation.

Answer: Multiplying through by the common denominator clears the fractions, converting the equation into whole numbers for easier solving.

Question 7: Solve for x: \( \frac{4}{9}x + \frac{2}{3} = \frac{7}{9} \).

Answer: x = 1

Question 8: A student has the equation \( \frac{5}{8}x - \frac{1}{4} = \frac{3}{8} \). Use multiply through to solve for x.

Answer: x = 2

Question 9: A technician needs to solve: \( \frac{2}{3}x + \frac{1}{2} = 1 \). Show your process and find x.

Answer: x = \frac{5}{4}

Question 10: Identify and correct the mistake: Solve \( \frac{3}{4}x + \frac{1}{4} = 2 \) by multiplying through. Wrong step: multiplying both sides by \( \frac{4}{3} \) gives \( x + \frac{1}{3} = \frac{8}{3} \).

Answer: Correct approach: multiply both sides by 4 (the denominator's common multiple) to clear fractions: \( 4 \times \left( \frac{3}{4}x + \frac{1}{4} \right) = 4 \times 2 \) which simplifies to \( 3x + 1 = 8 \). Then solve for x: \( 3x = 7 \), so x = \( \frac{7}{3} \). The mistake was multiplying by \( \frac{4}{3} \) instead of 4.