Multiplying Fractions - Pre Algebra

1. A rectangular piece of fabric is $\frac{7}{8}$ yard long and $\frac{2}{3}$ yard wide. Sarah cuts off $\frac{3}{7}$ of the total area for her project. How much area does she cut off?
   Answer: $\frac{1}{4}$ square yard First find the total area: $\frac{7}{8} \times \frac{2}{3} = \frac{14}{24} = \frac{7}{12}$ square yard. Then find $\frac{3}{7}$ of this area: $\frac{3}{7} \times \frac{7}{12} = \frac{21}{84} = \frac{1}{4}$ square yard. Choice B is an unsimplified intermediate calculation. Choice C comes from incorrectly multiplying $\frac{7}{8}$ by $\frac{3}{7}$. Choice D results from multiplying $\frac{2}{3}$ by $\frac{3}{7}$ directly.
2. A rectangular garden plot has dimensions $\frac{5}{6}$ meters by $\frac{4}{9}$ meters. If tomatoes are planted in $\frac{3}{5}$ of the total area, what area is planted with tomatoes?
   Answer: $\frac{2}{9}$ square meters First find the total area: $\frac{5}{6} \times \frac{4}{9} = \frac{20}{54} = \frac{10}{27}$ square meters. Then find $\frac{3}{5}$ of this area: $\frac{3}{5} \times \frac{10}{27} = \frac{30}{135} = \frac{2}{9}$ square meters. Choice A represents an incorrect multiplication without simplification. Choice C comes from incorrectly multiplying $\frac{4}{9} \times \frac{3}{5}$ directly. Choice D is the unsimplified form of the correct answer.
3. A water tank loses $\frac{1}{6}$ of its water each day due to evaporation. If the tank starts with $\frac{4}{5}$ of its capacity filled, what fraction of the tank's total capacity remains after 2 days?
   Answer: $\frac{5}{9}$ of capacity Each day, $\frac{5}{6}$ of the water remains (since $\frac{1}{6}$ evaporates). After day 1: $\frac{4}{5} \times \frac{5}{6} = \frac{20}{30} = \frac{2}{3}$ of capacity. After day 2: $\frac{2}{3} \times \frac{5}{6} = \frac{10}{18} = \frac{5}{9}$ of capacity. Choice A represents the amount after only 1 day. Choice B is the unsimplified form of $\frac{2}{3}$. Choice D comes from incorrectly subtracting $\frac{1}{6}$ twice from $\frac{4}{5}$.
4. A machine produces widgets at a rate of $\frac{7}{12}$ widgets per minute. If the machine operates at $\frac{4}{7}$ of its normal capacity for $\frac{3}{2}$ hours, how many widgets will it produce?
   Answer: $30$ widgets First, find the reduced rate: $\frac{4}{7} \times \frac{7}{12} = \frac{1}{3}$ widgets per minute. Convert time to minutes: $\frac{3}{2} \times 60 = 90$ minutes. Total widgets: $\frac{1}{3} \times 90 = 30$ widgets. Choice A comes from multiplying the fractions without converting time units. Choice C results from incorrect fraction multiplication. Choice D represents an intermediate calculation error.
5. A bakery uses $\frac{3}{5}$ cup of sugar for each batch of cookies. If they want to make $\frac{4}{9}$ of their usual daily production, and their usual daily production is $7\frac{1}{2}$ batches, how much sugar will they need?
   Answer: $2$ cups of sugar First find how many batches they'll make: $\frac{4}{9} \times 7\frac{1}{2} = \frac{4}{9} \times \frac{15}{2} = \frac{60}{18} = \frac{10}{3}$ batches. Then find the sugar needed: $\frac{10}{3} \times \frac{3}{5} = \frac{30}{15} = 2$ cups. Choice B represents the number of batches, not cups of sugar. Choice C comes from incorrectly calculating $\frac{3}{5} \times \frac{15}{9}$. Choice D results from multiplying $\frac{4}{9} \times \frac{3}{5}$ without considering the batch count.
6. A pizza is cut into equal slices. Alex eats $\frac{1}{4}$ of the pizza, and then Ben eats $\frac{2}{3}$ of what Alex ate. What fraction of the entire pizza did Ben eat?
   Answer: $\frac{1}{6}$ of the pizza When you see a fraction word problem with phrases like "of what someone ate," you're dealing with multiplication of fractions. The key is identifying what each fraction refers to. Let's work through this step by step. Alex eats $\frac{1}{4}$ of the entire pizza. Then Ben eats $\frac{2}{3}$ of what Alex ate - not $\frac{2}{3}$ of the whole pizza, but $\frac{2}{3}$ of Alex's portion. To find Ben's portion, multiply: $\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6}$ So Ben ate $\frac{1}{6}$ of the entire pizza. Looking at the wrong answers: Choice A ($\frac{2}{3}$) is the trap of thinking Ben ate $\frac{2}{3}$ of the whole pizza instead of $\frac{2}{3}$ of Alex's portion. Choice B ($\frac{11}{12}$) comes from incorrectly adding $\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}$, which would be the total amount eaten by both people, not just Ben's portion. Choice C ($\frac{2}{12}$) is the unreduced form of the correct answer - while mathematically equivalent to $\frac{1}{6}$, it's not in simplest form. Remember: when you see "of what someone ate" or similar phrasing, you're multiplying fractions. Always reduce your final answer to simplest form, and double-check that you're answering the right question - what Ben ate, not what both people ate combined.
7. Jason reads $\frac{2}{5}$ of a book on Monday and $\frac{3}{8}$ of the remaining pages on Tuesday. What fraction of the entire book did Jason read on Tuesday?
   Answer: $\frac{9}{40}$ of the book After Monday, Jason has $1 - \frac{2}{5} = \frac{3}{5}$ of the book remaining. On Tuesday, he reads $\frac{3}{8}$ of this remaining portion: $\frac{3}{8} \times \frac{3}{5} = \frac{9}{40}$ of the entire book. Choice A incorrectly assumes Tuesday's reading is $\frac{3}{8}$ of the whole book. Choice B comes from multiplying $\frac{2}{5} \times \frac{3}{8}$. Choice D represents the total read over both days ($\frac{2}{5} + \frac{9}{40}$).

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