Question 1: Multiply through the equation: \( \frac{1}{2}x + 3 = 5 \).

Answer: x = 4

Question 2: Solve for x: \( \frac{2}{3}x - 4 = 2 \) by multiplying through.

Answer: x = 9

Question 3: Simplify and solve: \( \frac{3}{4}x + 1 = \frac{1}{2}x + 3 \).

Answer: x=4

Question 1: A recipe requires \( \frac{2}{3} \) of a cup of sugar. If you want to make a batch that is twice as large, how much sugar do you need? Show your working.

Answer: 1 \text{ cup}

Question 2: Solve for x: \( 2x - \frac{1}{3}x = 4 \). Explain each step.

Answer: x=6

Question 1: A painter is mixing paint. If \( \frac{3}{4} \) of a liter is needed for one wall, how much paint is needed for 5 walls? Show your work.

Answer: 3.75 liters

Question 1: Solve the equation: \( \frac{5}{6}x + \frac{1}{3}x = 7 \) by multiplying through. Provide detailed steps.

Answer: x=6

Question 2: Construct an algebraic equation involving fractions where multiplying through simplifies the problem, then solve it.

Answer: Sample: \( \frac{1}{2}x + 2 = \frac{3}{4}x - 1 \), solution: \( x=8 \).

Question 1: Identify and correct the mistake in the following solution: \( \frac{1}{3}x + 2 = 4 \), student multiplies both sides by 3 and gets \( x + 6 = 12 \).

Answer: Incorrect; should multiply the entire equation correctly, but maintaining balance requires careful steps.

Question 2: Explain why multiplying through an equation with fractions must be done carefully.

Answer: To avoid distributing incorrectly or losing fractions, ensuring each term is multiplied by the same non-zero number.