Fraction Multiplication/Division - ISEE Middle Level Mathematics Achievement

A farmer plants corn on $\frac{3}{5}$ of his 90-acre farm. He then sells $\frac{1}{3}$ of the cornfield to a neighbor. How many acres of corn does the farmer have left?
Answer: 36 acres This problem involves working with fractions in multiple steps, which is common on fraction word problems. You need to carefully track what fraction refers to what quantity at each stage. Start by finding how many acres are planted with corn: $\frac{3}{5} \times 90 = \frac{3 \times 90}{5} = \frac{270}{5} = 54$ acres of corn initially. Next, the farmer sells $\frac{1}{3}$ of his cornfield to a neighbor. This means he sells $\frac{1}{3}$ of the 54 acres of corn: $\frac{1}{3} \times 54 = 18$ acres sold. Therefore, the farmer has left: $54 - 18 = 36$ acres of corn remaining. Looking at the wrong answers: Choice A (18 acres) represents the amount of cornfield sold, not what remains. Choice B (24 acres) might result from incorrectly calculating $\frac{2}{3}$ of 36 instead of $\frac{2}{3}$ of 54. Choice C (54 acres) is the original amount of corn planted, ignoring the sale entirely. The key strategy for multi-step fraction problems is to work through them systematically: identify what fraction applies to which quantity, calculate each step completely before moving to the next, and always check that your final answer makes sense in context. Here, 36 acres should be less than the original 54 acres but more than half of it, which it is.
A gardener has a bag of fertilizer that weighs $10\frac{1}{2}$ pounds. He wants to spread the fertilizer evenly over 7 sections of his garden. How many pounds of fertilizer should he use for each section?
Answer: $1\frac{1}{2}$ pounds To find the amount for each section, divide the total weight by the number of sections. First, convert the mixed number to an improper fraction: $10\frac{1}{2} = \frac{21}{2}$ . Then, divide by 7: $\frac{21}{2} \div 7 = \frac{21}{2} \times \frac{1}{7} = \frac{21}{14}$ . Simplify the fraction by dividing the numerator and denominator by 7: $\frac{3}{2}$ , which is equal to $1\frac{1}{2}$ pounds.
Liam and Chloe painted a fence that is 72 feet long. Liam painted $\frac{3}{8}$ of the fence. Chloe painted $\frac{1}{3}$ of the remaining portion of the fence. How many more feet of fence did Liam paint than Chloe?
Answer: 12 feet First, calculate the length Liam painted: $72 \times \frac{3}{8} = 9 \times 3 = 27$ feet. Next, find the length of the remaining portion: $72 - 27 = 45$ feet. Then, calculate the length Chloe painted, which is $\frac{1}{3}$ of the remaining part: $45 \times \frac{1}{3} = 15$ feet. Finally, find the difference between the lengths they painted: $27 - 15 = 12$ feet. Liam painted 12 more feet than Chloe.
For a student's final grade, the average of three tests counts for $\frac{2}{3}$ of the grade, and a final exam counts for the remaining $\frac{1}{3}$ . The student's test scores are 80, 90, and 100. If the student scores an 84 on the final exam, what is the student's final grade?
Answer: 88 First, find the average of the three test scores: $(80 + 90 + 100) \div 3 = 270 \div 3 = 90$ . This average counts for $\frac{2}{3}$ of the final grade, so its contribution is $90 \times \frac{2}{3} = 60$ points. The final exam score is 84, and it counts for $\frac{1}{3}$ of the grade, so its contribution is $84 \times \frac{1}{3} = 28$ points. The final grade is the sum of these two parts: $60 + 28 = 88$ .
A bookshelf holds 120 books. One-third of the books are fiction. Of the remaining books, $\frac{3}{4}$ are non-fiction. The rest are reference books. How many reference books are on the bookshelf?
Answer: 20 First, find the number of fiction books: $120 \times \frac{1}{3} = 40$ . The number of remaining books is $120 - 40 = 80$ . Of these 80 books, $\frac{3}{4}$ are non-fiction: $80 \times \frac{3}{4} = 60$ . The rest of the 80 books are reference books, so subtract the non-fiction books from the remainder: $80 - 60 = 20$ . There are 20 reference books.
A baker has $13\frac{1}{2}$ cups of sugar. A recipe for one batch of muffins requires $\frac{3}{4}$ cup of sugar. How many full batches of muffins can the baker make?
Answer: 18 To find the number of batches, divide the total amount of sugar by the amount needed per batch. First, convert the mixed number to an improper fraction: $13\frac{1}{2} = \frac{27}{2}$ . Now, divide: $\frac{27}{2} \div \frac{3}{4} = \frac{27}{2} \times \frac{4}{3}$ . Simplify by cross-cancellation: $27 \div 3 = 9$ and $4 \div 2 = 2$ . The calculation becomes $\frac{9}{1} \times \frac{2}{1} = 18$ . The baker can make exactly 18 full batches.
A piece of wood is 6 feet long. A carpenter needs to cut it into smaller pieces that are each $4\frac{1}{2}$ inches long. How many full pieces can be cut from the piece of wood? (1 foot = 12 inches)
Answer: 16 First, convert the total length of the wood into inches. Since 1 foot = 12 inches, 6 feet is $6 \times 12 = 72$ inches. Next, convert the length of the smaller pieces to an improper fraction: $4\frac{1}{2} = \frac{9}{2}$ inches. To find the number of pieces, divide the total length by the length of one piece: $72 \div \frac{9}{2} = 72 \times \frac{2}{9}$ . Simplify: $\frac{72}{9} \times 2 = 8 \times 2 = 16$ . The carpenter can cut 16 full pieces.
A roll of ribbon contains 16 yards. A crafter needs to cut pieces that are each $\frac{3}{4}$ of a yard long. After cutting as many full pieces as possible, what length of ribbon, in yards, will be left over?
Answer: $\frac{1}{4}$ yard First, find how many pieces can be cut by dividing the total length by the length of one piece: $16 \div \frac{3}{4} = 16 \times \frac{4}{3} = \frac{64}{3} = 21\frac{1}{3}$ . This means 21 full pieces can be cut. The fractional part, $\frac{1}{3}$ , represents $\frac{1}{3}$ of a piece, not $\frac{1}{3}$ of a yard. To find the leftover length, calculate the total length used for the 21 pieces: $21 \times \frac{3}{4} = \frac{63}{4} = 15\frac{3}{4}$ yards. Subtract the used length from the original length: $16 - 15\frac{3}{4} = \frac{1}{4}$ yard.
A bicycle originally priced at $320 is on sale for $\frac{1}{4}$ off the original price. A customer also has a coupon for an additional $\frac{1}{5}$ off the sale price. What is the final price the customer pays for the bicycle?
Answer: $192 First, calculate the sale price. A discount of $\frac{1}{4}$ means the price is $1 - \frac{1}{4} = \frac{3}{4}$ of the original. Sale price = $\$320 \times \frac{3}{4} = \$240$ . The coupon gives an additional $\frac{1}{5}$ off this sale price, so the customer pays $1 - \frac{1}{5} = \frac{4}{5}$ of the sale price. Final price = $\$240 \times \frac{4}{5} = \$192$ .
A hiking trail is $12\frac{1}{2}$ kilometers long. A hiker has completed $\frac{2}{5}$ of the trail. How many kilometers has the hiker traveled?
Answer: 5 km To find the distance traveled, multiply the total length of the trail by the fraction completed. First, convert the mixed number to an improper fraction: $12\frac{1}{2} = \frac{25}{2}$ . Now multiply: $\frac{25}{2} \times \frac{2}{5}$ . You can simplify by canceling the 2s and dividing 25 by 5: $\frac{25}{2} \times \frac{2}{5} = \frac{5}{1} \times \frac{1}{1} = 5$ . The hiker has traveled 5 kilometers.
A rectangular painting is $3\frac{1}{2}$ feet long and $2\frac{1}{4}$ feet wide. An artist covers $\frac{2}{3}$ of the painting with a protective glaze. What is the area of the glazed portion of the painting in square feet?
Answer: $5\frac{1}{4}$ square feet First, calculate the total area of the painting by multiplying its length and width. Convert mixed numbers to improper fractions: $3\frac{1}{2} = \frac{7}{2}$ and $2\frac{1}{4} = \frac{9}{4}$ . Area = $\frac{7}{2} \times \frac{9}{4} = \frac{63}{8}$ square feet. Next, find $\frac{2}{3}$ of this area: $\frac{63}{8} \times \frac{2}{3} = \frac{126}{24}$ . Simplify by dividing the numerator and denominator by their greatest common divisor, 6: $\frac{21}{4}$ . Convert this to a mixed number: $5\frac{1}{4}$ square feet.
A company's annual budget is $240,000. One-fifth of the budget is for marketing. Of the marketing budget, $\frac{1}{2}$ is for online advertising. Of the online advertising budget, $\frac{3}{4}$ is for social media campaigns. How much money is budgeted for social media campaigns?
Answer: $18,000 This problem requires multiplying the total budget by a series of fractions. The amount for social media is $240,000 \times \frac{1}{5} \times \frac{1}{2} \times \frac{3}{4}$ . First step: Marketing budget = $240,000 \times \frac{1}{5} = \$48,000$ . Second step: Online advertising = $\$48,000 \times \frac{1}{2} = \$24,000$ . Third step: Social media = $\$24,000 \times \frac{3}{4} = \$18,000$ .
A recipe for one dozen cookies requires $\frac{3}{4}$ cup of flour. A baker wants to make $2\frac{1}{2}$ dozen cookies. How many cups of flour will the baker need?
Answer: $1\frac{7}{8}$ cups To find the total amount of flour needed, multiply the amount per dozen by the number of dozens. First, convert the mixed number to an improper fraction: $2\frac{1}{2} = \frac{5}{2}$ . Then, multiply: $\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$ . Convert the result back to a mixed number: $\frac{15}{8} = 1\frac{7}{8}$ cups.
A container held $2\frac{1}{2}$ gallons of paint. A painter used $\frac{1}{5}$ of the paint for a wall. Then, he used $\frac{3}{4}$ of a gallon for a door. How much paint was left in the container?
Answer: $1\frac{1}{4}$ gallons First, find the amount of paint used for the wall. Convert the starting amount to an improper fraction: $2\frac{1}{2} = \frac{5}{2}$ . Paint for wall = $\frac{5}{2} \times \frac{1}{5} = \frac{1}{2}$ gallon. The amount remaining after painting the wall is $\frac{5}{2} - \frac{1}{2} = \frac{4}{2} = 2$ gallons. Then, he used $\frac{3}{4}$ of a gallon for a door. The final amount left is $2 - \frac{3}{4} = 1\frac{1}{4}$ gallons.
A water tank is currently filled with 45 gallons of water, which is $\frac{3}{5}$ of its total capacity. What is the total capacity of the tank in gallons?
Answer: 75 gallons Let C be the total capacity. The problem states that $\frac{3}{5}$ of the total capacity is 45 gallons, so $\frac{3}{5}C = 45$ . To find C, divide 45 by $\frac{3}{5}$ : $C = 45 \div \frac{3}{5} = 45 \times \frac{5}{3} = \frac{225}{3} = 75$ . The total capacity is 75 gallons.
A running track is $\frac{1}{4}$ of a mile long. A runner completed several laps, running a total of $3\frac{1}{2}$ miles. How many laps did the runner complete?
Answer: 14 laps This is a division problem involving mixed numbers and fractions. When you need to find how many times one quantity fits into another, you divide the total by the individual amount. To find the number of laps, you need to divide the total distance by the length of one lap: $3\frac{1}{2} \div \frac{1}{4}$ First, convert the mixed number to an improper fraction: $3\frac{1}{2} = \frac{7}{2}$ Now divide: $\frac{7}{2} \div \frac{1}{4}$ Remember that dividing by a fraction is the same as multiplying by its reciprocal: $\frac{7}{2} \times \frac{4}{1} = \frac{28}{2} = 14$ The runner completed 14 laps, making D correct. Looking at the wrong answers: A) $\frac{7}{8}$ laps results from incorrectly multiplying $3\frac{1}{2} \times \frac{1}{4}$ instead of dividing—this gives you a fraction of a lap, which doesn't make sense when someone ran multiple miles. B) 12 laps comes from converting $3\frac{1}{2}$ incorrectly to $\frac{6}{2}$ and then calculating $\frac{6}{2} \times 4 = 12$ . C) 16 laps results from converting $3\frac{1}{2}$ to 4 (rounding up) and then multiplying by 4. When solving "how many times" problems, always ask yourself whether your answer makes logical sense. If someone ran $3\frac{1}{2}$ miles on a $\frac{1}{4}$ -mile track, they must have completed more than 3 laps but fewer than 20, which helps you verify that 14 is reasonable.
A car travels $52\frac{1}{2}$ miles in $\frac{3}{4}$ of an hour. What is the car's average speed in miles per hour?
Answer: 70 miles per hour Average speed is calculated by dividing the total distance by the total time. First, convert the distance to an improper fraction: $52\frac{1}{2} = \frac{105}{2}$ miles. Now, divide the distance by the time: $\frac{105}{2} \div \frac{3}{4} = \frac{105}{2} \times \frac{4}{3}$ . Simplify before multiplying: $\frac{105 \div 3}{2 \div 2} \times \frac{4 \div 2}{3 \div 3} = \frac{35}{1} \times \frac{2}{1} = 70$ . The car's average speed is 70 miles per hour.
In a school band, $\frac{3}{5}$ of the members play a brass instrument. Of those who play a brass instrument, $\frac{1}{3}$ play the trumpet. If there are 12 trumpet players, how many members are in the band in total?
Answer: 60 First, determine the fraction of the entire band that plays the trumpet. This is $\frac{3}{5} \times \frac{1}{3} = \frac{3}{15} = \frac{1}{5}$ . So, $\frac{1}{5}$ of the total band members are trumpet players. If T is the total number of members, then $\frac{1}{5}T = 12$ . To find T, multiply 12 by 5: $12 \times 5 = 60$ . There are 60 members in the band.
Maria has $\frac{4}{5}$ of a liter of juice. She pours it equally into 6 small glasses. What fraction of a liter of juice is in each glass?
Answer: $\frac{2}{15}$ To find the amount of juice in each glass, divide the total amount of juice by the number of glasses. This is calculated as $\frac{4}{5} \div 6$ . To divide by a whole number, multiply by its reciprocal: $\frac{4}{5} \times \frac{1}{6} = \frac{4}{30}$ . This fraction can be simplified by dividing both the numerator and the denominator by 2, which gives $\frac{2}{15}$ . Each glass contains $\frac{2}{15}$ of a liter.
After a party, $\frac{1}{4}$ of a pizza was left over. The next day, David ate $\frac{2}{3}$ of the leftover pizza. The piece David ate weighed 6 ounces. What was the original weight of the whole pizza in ounces?
Answer: 36 ounces This is a two-step problem working backwards. Let L be the weight of the leftover pizza. David ate $\frac{2}{3}$ of L, which weighed 6 ounces. So, $\frac{2}{3}L = 6$ . To find L, calculate $6 \div \frac{2}{3} = 6 \times \frac{3}{2} = 9$ ounces. So, the leftover pizza weighed 9 ounces. This leftover amount was $\frac{1}{4}$ of the original pizza's weight, P. So, $\frac{1}{4}P = 9$ . To find P, calculate $9 \div \frac{1}{4} = 9 \times 4 = 36$ ounces.

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