Question 1: Simplify the product: \( \frac{3}{4} \times \frac{2}{5} \).

Answer: \( \frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10} \)

Question 2: Calculate and simplify: \( \frac{6}{8} \times \frac{2}{3} \).

Answer: \( \frac{6 \times 2}{8 \times 3} = \frac{12}{24} = \frac{1}{2} \)

Question 3: Simplify the product: \( \frac{9}{12} \times \frac{4}{6} \).

Answer: \( \frac{9 \times 4}{12 \times 6} = \frac{36}{72} = \frac{1}{2} \)

Question 4: Calculate: \( \frac{7}{10} \times \frac{10}{7} \).

Answer: \( \frac{7 \times 10}{10 \times 7} = \frac{70}{70} = 1 \)

Question 1: A recipe requires \( \frac{2}{3} \) of a cup of sugar. If you multiply this amount by \( \frac{4}{5} \), how much sugar do you need? Simplify your answer.

Answer: \( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \)

Question 2: Explain why \( \frac{8}{12} \times \frac{3}{4} \) does not need further simplification to be correct, but it is better to simplify first.

Answer: Multiplying gives \( \frac{8 \times 3}{12 \times 4} = \frac{24}{48} = \frac{1}{2} \). Simplifying first makes calculations easier and reduces chance of errors.

Question 3: If two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) are multiplied and the result is \( \frac{3}{8} \), what are possible values for \( a, b, c, d \)? Explain your reasoning.

Answer: Possible values include \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \), so \( a=1, c=3, b=2, d=4 \).

Question 1: A painter mixes \( \frac{1}{4} \) of a litre of blue paint with \( \frac{2}{3} \) of a litre of white paint. She then uses \( \frac{1}{2} \) of this mixture for a project. How much of the original mixture does she use? Simplify.

Answer: \( \frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6} \). Half of this is \( \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} \).

Question 1: Prove that multiplying \( \frac{8}{12} \) by \( \frac{3}{4} \) simplifies to the same as multiplying \( \frac{2}{3} \) by \( \frac{1}{1} \). Show all steps.

Answer: \( \frac{8}{12} \times \frac{3}{4} = \frac{8 \times 3}{12 \times 4} = \frac{24}{48} = \frac{1}{2} \). Also, \( \frac{2}{3} \times 1 = \frac{2}{3} \), which is not equal, so the initial statement is false unless both are simplified to \( \frac{1}{2} \).

Question 2: A student mistakenly multiplies numerators and denominators but forgets to simplify the final answer. For example, they calculate \( \frac{3}{4} \times \frac{2}{5} = \frac{3 \times 2}{4 \times 5} = \frac{6}{20} \), but then state the answer as \( \frac{6}{20} \) without simplifying to \( \frac{3}{10} \). Identify and correct this mistake in their answer.

Answer: They should simplify \( \frac{6}{20} \) to \( \frac{3}{10} \) by dividing numerator and denominator by 2.