Question 1: Calculate the product of \( \frac{2}{3} \) and \( \frac{3}{4} \).

Answer: \( \frac{1}{2} \)

Question 2: Divide \( \frac{5}{6} \) by \( \frac{2}{3} \).

Answer: \( \frac{5}{4} \)

Question 3: Simplify \( \frac{8}{12} \) by multiplying numerator and denominator by the same number.

Answer: \( \frac{2}{3} \)

Question 4: Calculate \( \frac{7}{9} \) ÷ \( \frac{14}{27} \).

Answer: \( \frac{3}{2} \)

Question 1: A recipe calls for \( \frac{3}{4} \) of a cup of sugar. If a student mistakenly multiplies the quantity by \( \frac{2}{3} \) instead of dividing, how much sugar will they use? Is their amount more or less than the original?

Answer: \( \frac{1}{2} \) cup, less

Question 2: Explain why multiplying \( \frac{4}{5} \) by \( \frac{3}{2} \) does not give an equivalent fraction, but instead produces a different value.

Answer: Because multiplying fractions is about scaling, not simplifying. The product \( \frac{12}{10} \) simplifies to \( \frac{6}{5} \), which is not equivalent to either original fraction.

Question 1: A car travels \( \frac{3}{4} \) of a 120-mile trip. How far has it traveled? If the driver mistakenly multiplies instead of divides when calculating remaining distance, what error do they make?

Answer: 90 miles; they will incorrectly calculate remaining distance as \( \frac{1}{4} \times 120 = 30 \) miles instead of subtracting from total.

Question 1: Construct a fraction that is not equivalent to \( \frac{2}{3} \) but results from multiplying \( \frac{2}{3} \) by \( 1.5 \).

Answer: \( \frac{3}{2} \) or any fraction equivalent to \( \frac{3}{2} \)

Question 2: If \( \frac{p}{q} \) is an equivalent fraction to \( \frac{2}{3} \), find \( p \) and \( q \) after multiplying numerator and denominator each by 4.

Answer: \( p=8, q=12 \)

Question 1: A student calculates \( \frac{3}{4} \) divided by \( \frac{2}{5} \) as \( \frac{3}{4} \times \frac{2}{5} = \frac{6}{20} \). What is the mistake? Correct it.

Answer: They should invert the second fraction; correct answer is \( \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} \).

Question 2: A student multiplies \( \frac{5}{6} \) by \( \frac{3}{4} \) but forgets to simplify the product. What is the incorrect result, and how should it be simplified?

Answer: \( \frac{15}{24} \), which simplifies to \( \frac{5}{8} \).