Question 1: Find the least common multiple (LCM) of 4 and 6. Show your working.

Answer: 12

Question 2: Calculate the LCM of 8 and 12 and then use it to write an equivalent fraction to \(rac{3}{8}\) with denominator equal to the LCM.

Answer: {"lcm":"24","new_fraction":"\\(\\frac{9}{24}\\)"}

Question 3: What is the LCM of 5 and 7? Use this to add \(\frac{2}{5}\) and \(\frac{3}{7}\).

Answer: {"lcm":"35","sum":"\\(\\frac{14}{35} + \\frac{15}{35} = \\frac{29}{35}\\)"}

Question 4: Find the LCM of 9 and 15. Then, convert \(\frac{2}{9}\) and \(\frac{1}{15}\) to equivalent fractions with the same denominator.

Answer: {"lcm":"45","fractions":["\\(\\frac{10}{45}\\)","\\(\\frac{3}{45}\\)"]}

Question 5: A recipe requires \(\frac{1}{4}\) cup of sugar and \(\frac{1}{6}\) cup of honey. Find the total amount of sweetener needed by adding these fractions. Show your working.

Answer: \(\frac{5}{12}\) cup

Question 6: A car travels 60 km every 4 hours, and a cyclist travels 40 km every 2 hours. Find the LCM of 4 and 2 to determine after how many hours both will have traveled a multiple of their respective distances.

Answer: LCM of 4 and 2 is 4 hours; both will have traveled 60 km and 40 km respectively after 4 hours.

Question 7: Calculate the LCM of 18 and 24. Then, add the fractions \(\frac{3}{18}\) and \(\frac{2}{24}\) by converting them to have this common denominator.

Answer: {"lcm":"72","sum":"\\(\\frac{12}{72} + \\frac{6}{72} = \\frac{18}{72} = \\frac{1}{4}\\)"}

Question 8: A printer uses \(\frac{1}{3}\) of a cartridge every week, while a scanner uses \(\frac{1}{4}\) of a cartridge every week. Find the LCM of their weekly usage to determine after how many weeks they will both need a new cartridge at the same time.

Answer: LCM of 3 and 4 is 12; after 12 weeks, both will need a new cartridge.

Question 9: Find the LCM of 14 and 35. Show how to convert \(\frac{3}{14}\) and \(\frac{2}{35}\) to equivalent fractions with the same denominator.

Answer: {"lcm":"70","fractions":["\\(\\frac{15}{70}\\)","\\(\\frac{4}{70}\\)"]}

Question 10: Identify the common mistake made when finding the LCM of 6 and 8. Explain how to correct it.

Answer: Common mistake: Using the greater of the two numbers instead of the LCM. Correction: Find all multiples of both numbers and identify the smallest common multiple.