Question 1: Simplify the expression \( \frac{3x}{4} + \frac{5x}{6} \) and express it as a single fraction.

Answer: \( \frac{19x}{12} \)

Question 2: Rearrange the formula \( \frac{a}{b} = c \) to make \( a \) the subject.

Answer: a = b c

Question 3: Calculate the value of \( x \) in the equation \( \frac{2x + 3}{5} = \frac{7}{10} \).

Answer: x = 2

Question 4: Add \( \frac{1}{3} \) and \( \frac{2}{5} \) and express your answer as a simplified fraction.

Answer: \( \frac{11}{15} \)

Question 1: The formula for the area \( A \) of a rectangle is given by \( A = \frac{l \times w}{2} \). If the length \( l \) is expressed as \( \frac{3x}{4} \) and the width \( w \) as \( \frac{2x}{3} \), find an expression for \( A \) in terms of \( x \).

Answer: A = \frac{x^2}{4}

Question 2: A student solves the equation \( \frac{4x}{5} - \frac{2}{3} = \frac{7}{15} \). They mistakenly multiply through by 15 and get \( 12x - 10 = 7 \). Identify and correct the mistake, then solve for \( x \).

Answer: The mistake is multiplying through by 15; correct is multiplying by 15. Solving gives x = 1

Question 3: If \( \frac{a}{b} = \frac{c}{d} \), derive an expression for \( a \) in terms of \( b, c, d \).

Answer: a = \frac{b c}{d}

Question 4: A recipe requires \( \frac{2}{3} \) of a cup of sugar. If you want to make half the recipe, how much sugar do you need?

Answer: \( \frac{1}{3} \)

Question 1: A car travels \( \frac{3}{4} \) of a mile in \( \frac{1}{8} \) of an hour. Calculate its speed in miles per hour.

Answer: 6 mph

Question 2: A box contains 3/5 kg of almonds. If you use 2/3 of the almonds for baking, how much almonds do you use?

Answer: \( \frac{2}{5} \) kg

Question 3: A survey shows that \( \frac{2}{7} \) of students prefer tea over coffee. If there are 210 students, how many prefer tea?

Answer: 60

Question 1: Solve for \( x \) in the equation \( \frac{5x - 2}{3} = \frac{x + 4}{6} \).

Answer: x = 5

Question 2: Construct an algebraic expression that represents the total cost of buying \( n \) items, each costing \( \frac{a}{b} \) dollars, with a fixed shipping fee of \( \frac{c}{d} \).

Answer: Total cost = n \times \frac{a}{b} + \frac{c}{d}

Question 1: Express \( \frac{9}{12} \) as a simplified fraction.

Answer: \( \frac{3}{4} \)

Question 2: Rewrite the equation \( \frac{x}{5} = \frac{2}{3} \) to solve for \( x \).

Answer: x = \frac{10}{3}

Question 1: A student solves \( \frac{2x}{3} = 4 \) by multiplying both sides by 3 and gets \( 2x = 12 \), then divides by 2 to get \( x = 6 \). Is this correct? If not, what is the correct value of \( x \)?

Answer: No, the correct value is \( x = 6 \). The work is correct.