Question 1: Solve for x: \( \frac{2}{3}x = 8 \).

Answer: 12

Question 2: Solve the equation: \( \frac{5}{4}x = 15 \).

Answer: 12

Question 3: If \( \frac{3}{5}x = 9 \), find the value of x.

Answer: 15

Question 4: Simplify and solve: \( \frac{1}{2}x = 7 \).

Answer: 14

Question 5: Solve for x: \( \frac{4}{7}x = 16 \).

Answer: 28

Question 6: Construct a step-by-step solution to solve: \( \frac{3}{4}x = 9 \).

Answer: x = 12

Question 7: A recipe calls for \( \frac{3}{4} \) cup of sugar for 12 cookies. How much sugar is needed for 24 cookies? (Use multiply through)

Answer: 1.5 cups

Question 8: A student solves the equation \( \frac{2}{3}x = 10 \) but makes a mistake. The student’s work shows: \( \frac{2}{3}x \times \frac{3}{2} = 10 \times \frac{3}{2} \), resulting in \( x = 15 \). Identify and correct the mistake.

Answer: The mistake is multiplying both sides by \( \frac{3}{2} \) instead of dividing. Correct method: multiply both sides by \( \frac{3}{2} \) or better, multiply both sides by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). So, x = \( 10 \times \frac{3}{2} = 15 \).

Question 9: Solve for x: \( \frac{7}{8}x = 21 \).

Answer: 24

Question 10: A box contains water with volume \( \frac{2}{5} \) liters. If the total volume is increased by multiplying both sides by \( \frac{5}{2} \), what is the new volume?

Answer: 2 liters

Question 11: Challenge: Solve for x in the equation \( \frac{4}{9}x - \frac{1}{3}x = 5 \).

Answer: x = 15

Question 12: Identify the common mistake in solving: \( \frac{3}{4}x = 12 \) by dividing both sides by \( \frac{3}{4} \).

Answer: Dividing both sides by \( \frac{3}{4} \) is equivalent to multiplying by its reciprocal, \( \frac{4}{3} \). The mistake is thinking division is the only way; but division by a fraction is the same as multiplication by its reciprocal. Proper solution: multiply both sides by \( \frac{4}{3} \).