Dividing Fractions - Pre Algebra

1. A container holds $\frac{4}{5}$ liter of juice. If you pour the juice into glasses that each hold $\frac{2}{15}$ liter, how many glasses can be completely filled?
   Answer: 6 glasses When you see a word problem asking "how many can be completely filled" or "how many fit into," you're looking at a division problem. You need to divide the total amount by the size of each portion. Here, you have $\frac{4}{5}$ liter of juice total, and each glass holds $\frac{2}{15}$ liter. To find how many glasses you can fill, calculate: $\frac{4}{5} ÷ \frac{2}{15}$ To divide fractions, multiply by the reciprocal of the second fraction: $\frac{4}{5} × \frac{15}{2}$ Multiply across: $\frac{4 × 15}{5 × 2} = \frac{60}{10} = 6$ You can fill exactly 6 glasses completely. Looking at the wrong answers: Choice A (4 glasses) likely comes from incorrectly multiplying $\frac{4}{5} × \frac{2}{15} = \frac{8}{75}$, then somehow arriving at 4. Choice B (8 glasses) might result from finding a common denominator incorrectly or making computational errors during the division process. Choice D (10 glasses) could come from incorrectly dividing just the denominators (15 ÷ 5 = 3) and numerators (4 ÷ 2 = 2), then somehow getting 10, or from other calculation mistakes. Remember this key pattern: whenever you need to find "how many portions" or "how many times does X fit into Y," you're dividing the total by the portion size. For fraction division, always multiply by the reciprocal—flip the second fraction and multiply straight across.
2. James walked $\frac{5}{6}$ of a mile in $\frac{1}{4}$ of an hour. At this rate, how many miles would he walk in one full hour?
   Answer: $\frac{10}{3}$ miles per hour To find miles per hour, calculate $\frac{5}{6} \div \frac{1}{4} = \frac{5}{6} \times \frac{4}{1} = \frac{20}{6} = \frac{10}{3}$ miles per hour. Choice A results from multiplying the fractions instead of dividing. Choice C shows the unreduced form but isn't simplified. Choice D results from adding the fractions ($\frac{5}{6} + \frac{1}{4}$).
3. A machine produces $\frac{7}{12}$ of a product in $\frac{3}{8}$ of an hour. At this constant rate, what fraction of the product can be made in $\frac{1}{2}$ hour?
   Answer: $\frac{7}{9}$ of the product First find the rate: $\frac{7}{12} \div \frac{3}{8} = \frac{7}{12} \times \frac{8}{3} = \frac{56}{36} = \frac{14}{9}$ products per hour. In $\frac{1}{2}$ hour: $\frac{14}{9} \times \frac{1}{2} = \frac{14}{18} = \frac{7}{9}$ of the product. Choice A results from calculation errors. Choice B is the rate per hour, not the amount in $\frac{1}{2}$ hour. Choice D results from adding fractions incorrectly.
4. Tom runs $\frac{3}{4}$ mile in $\frac{1}{6}$ hour. Sarah runs $\frac{2}{3}$ mile in $\frac{1}{9}$ hour. How much faster is Sarah's rate than Tom's rate, in miles per hour?
   Answer: $\frac{3}{2}$ miles per hour faster Tom's rate: $\frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times 6 = \frac{18}{4} = \frac{9}{2}$ mph. Sarah's rate: $\frac{2}{3} \div \frac{1}{9} = \frac{2}{3} \times 9 = 6$ mph. Difference: $6 - \frac{9}{2} = \frac{12}{2} - \frac{9}{2} = \frac{3}{2}$ mph faster. Choice B is Tom's rate alone. Choice C results from subtracting the original fractions. Choice D results from adding the rates instead of finding the difference.
5. Elena used $\frac{5}{8}$ cup of sugar to make cookies that filled $\frac{3}{10}$ of a cookie jar. If she wants to completely fill the jar with the same type of cookies, how many cups of sugar will she need in total?
   Answer: $\frac{25}{12}$ cups When you see a problem where someone uses a certain amount of ingredients to make a portion of something, you're dealing with proportional reasoning. The key insight is finding how much ingredient is needed per unit of the final product. Elena used $\frac{5}{8}$ cup of sugar to fill $\frac{3}{10}$ of the jar. To find how much sugar fills the entire jar, you need to set up a proportion or use division. Think of it this way: if $\frac{3}{10}$ of the jar requires $\frac{5}{8}$ cup, then 1 whole jar requires $\frac{5}{8} \div \frac{3}{10}$ cups. To divide fractions, multiply by the reciprocal: $\frac{5}{8} \times \frac{10}{3} = \frac{50}{24} = \frac{25}{12}$ cups. This matches answer choice D. Answer A ($\frac{15}{80}$) likely comes from incorrectly multiplying $\frac{5}{8} \times \frac{3}{10}$ instead of dividing. This would tell you how much sugar is needed for $\frac{3}{10}$ of $\frac{3}{10}$ of the jar, which doesn't answer the question. Answer B ($\frac{31}{18}$) doesn't follow from any logical operation with these fractions and may result from arithmetic errors. Answer C ($\frac{18}{40}$) could come from flipping the wrong fraction in the division or other computational mistakes. Remember: when you have "this much ingredient makes this portion," divide the ingredient amount by the portion to find how much you need for the whole thing. Always ask yourself if your answer makes sense—you should need more sugar to fill the entire jar than Elena used for just part of it.
6. A painter uses $\frac{3}{5}$ gallon of paint to cover $\frac{2}{7}$ of a wall. How many gallons of paint are needed to cover the entire wall?
   Answer: $\frac{21}{10}$ gallons If $\frac{2}{7}$ of the wall needs $\frac{3}{5}$ gallon, then the whole wall needs $\frac{3}{5} \div \frac{2}{7} = \frac{3}{5} \times \frac{7}{2} = \frac{21}{10}$ gallons. Choice A results from multiplying the fractions. Choice C comes from adding the fractions and making calculation errors. Choice D results from inverting the wrong fraction in the division.
7. Maria has $\frac{3}{4}$ of a pizza left over from dinner. She wants to divide it equally among her friends, giving each friend $\frac{1}{8}$ of the original whole pizza. How many friends can she serve?
   Answer: 6 friends To find how many friends can be served, divide $\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1} = \frac{24}{4} = 6$. Choice A results from multiplying the numerators (3×1=3). Choice C uses the denominator of the divisor. Choice D results from multiplying both fractions incorrectly (3×4=12).
8. A rectangular garden has an area of $\frac{5}{6}$ square meters and a width of $\frac{2}{9}$ meters. What is the length of the garden?
   Answer: $\frac{15}{4}$ meters When you encounter a problem involving the area of a rectangle, remember that area equals length times width: $A = l \times w$. Since you know the area and width, you need to solve for length by rearranging this formula to $l = \frac{A}{w}$. To find the length, divide the area by the width: $l = \frac{5/6}{2/9}$. When dividing fractions, multiply by the reciprocal of the divisor. So you get: $l = \frac{5}{6} \times \frac{9}{2} = \frac{5 \times 9}{6 \times 2} = \frac{45}{12}$. Simplifying this fraction by dividing both numerator and denominator by their greatest common factor of 3: $\frac{45}{12} = \frac{15}{4}$. Therefore, answer D is correct. Looking at the wrong answers: Choice A gives $\frac{10}{54}$, which appears to come from incorrectly multiplying the fractions instead of dividing them, then making arithmetic errors. Choice B, $\frac{23}{18}$, likely results from adding the fractions rather than performing the correct division operation. Choice C, $\frac{7}{15}$, seems to stem from subtracting the width from the area, which doesn't make sense geometrically. Study tip: Always double-check fraction division problems by multiplying your answer by the divisor to see if you get back to the dividend. Here: $\frac{15}{4} \times \frac{2}{9} = \frac{30}{36} = \frac{5}{6}$ ✓. This confirms your answer matches the given area.
9. A recipe calls for $\frac{2}{3}$ cup of flour. If you want to make portions that each use $\frac{3}{8}$ cup of flour, how many complete portions can you make from the original amount?
   Answer: 1 complete portion Calculate $\frac{2}{3} \div \frac{3}{8} = \frac{2}{3} \times \frac{8}{3} = \frac{16}{9} = 1\frac{7}{9}$. Since we need complete portions only, the answer is 1. Choice B results from rounding 1.78 incorrectly. Choice C comes from adding numerators (2+3=5, then estimating). Choice D results from multiplying denominators and misinterpreting the result.

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