Divide Fractions by Fractions - 6th Grade Math

Three friends share $\tfrac{1}{2}$ pound of trail mix equally. How much does each friend get? Write the division expression and compute $\left(\tfrac{1}{2}\right)\div 3$ . Verify by multiplying your answer by 3.
Answer: $\tfrac{1}{6}$ lb each; and $3\times\tfrac{1}{6}=\tfrac{1}{2}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like sharing among friends, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (1/2)÷3=(1/2)×(1/3)=1/6 (treat 3 as 3/1, flip to reciprocal, multiply). Verification: divisor×quotient=dividend (3×(1/6)=3/6=1/2 confirms quotient correct). Context: 'how much trail mix each of 3 friends gets from 1/2 lb' divides: 1/2÷3=1/6 lb each. Multiplication-division relationship: (1/2)÷3=1/6 because 3 times (1/6) equals (1/2) (3×1/6=1/2, division is inverse of multiplication). The correct choice is A, which uses the reciprocal method to get 1/6 and verifies correctly. Common errors include treating division incorrectly, like getting 3/2 as in B.
A recipe uses $\tfrac{2}{3}$ cup of yogurt. Each smoothie serving needs $\tfrac{3}{4}$ cup of yogurt. How many servings can you make from $\tfrac{2}{3}$ cup? Compute $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)$ and interpret the result (is it more or less than 1 full serving?). Then verify your answer by multiplying: $\left(\tfrac{3}{4}\right)\times(\text{quotient})=\tfrac{2}{3}$ .
Answer: $\tfrac{8}{9}$ of a serving (less than 1); and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{8}{9}\right)=\tfrac{2}{3}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like servings, sharing, or rectangle area÷length=width, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (2/3)÷(3/4)=(2/3)×(4/3)=8/9 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((3/4)×(8/9)=24/36=2/3 confirms quotient correct). Context: 'how many 3/4-cup servings in 2/3 cup yogurt?' divides: 2/3÷3/4=8/9 (less than 1 full serving, 8/9 of a 3/4-cup portion). Multiplication-division relationship: (2/3)÷(3/4)=8/9 because (3/4) of (8/9) equals (2/3) (3/4×8/9=24/36=2/3, division is inverse of multiplication). The correct choice is B, which uses the reciprocal method to get 8/9 and verifies correctly. Common errors include multiplying instead of dividing ((2/3)×(3/4)=1/2 as in A), or flipping the wrong fraction leading to incorrect quotients like in D.
A ribbon is $\tfrac{3}{4}$ yard long. Each bracelet needs $\tfrac{1}{2}$ yard of ribbon. The division expression is $\left(\tfrac{3}{4}\right)\div\left(\tfrac{1}{2}\right)$ . How many bracelets can be made from the ribbon? (Simplify.)
Answer: $\tfrac{3}{2}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method, interpreting in contexts like how many segments of one length fit into another, and understanding via visual models and the multiplication-division relationship. To divide (3/4)÷(1/2), flip the divisor to its reciprocal and multiply: (3/4) × (2/1) = 6/4 = 3/2, so 3/2 bracelets can be made. Verification: multiply divisor by quotient to get dividend, (1/2) × (3/2) = 3/4, which confirms. For example, a ribbon of 3/4 yard divided by 1/2 yard per bracelet: (3/4) × 2 = 3/2, meaning 1 full and half another. The correct division using the reciprocal gives simplified 3/2. Common mistakes include subtracting fractions or not simplifying 6/4. Method: write division, flip divisor, multiply, reduce fraction, verify.
A recipe uses $\tfrac{2}{3}$ cup of yogurt. One serving size is $\tfrac{3}{4}$ cup. How many $\tfrac{3}{4}$ -cup servings can you make from $\tfrac{2}{3}$ cup? Compute $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)$ and choose the best interpretation.
Answer: $\tfrac{8}{9}$ of a serving (less than 1 full $\tfrac{3}{4}$ -cup serving) This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like servings from a given amount of yogurt. To divide (2/3)÷(3/4), flip the divisor to its reciprocal 4/3 and multiply: (2/3)×(4/3)=(2×4)/(3×3)=8/9. Verification: multiply the divisor by the quotient, (3/4)×(8/9)=(3×8)/(4×9)=24/36=2/3, which matches the dividend. In context, this means you can make 8/9 of a 3/4-cup serving from 2/3 cup of yogurt, which is less than one full serving. A common error is multiplying instead of dividing, like (2/3)×(3/4)=1/2 as in choice A, or flipping the wrong fraction leading to incorrect results like 2/3 in choice C. To compute: write the division, flip the divisor, multiply numerators and denominators, simplify if needed, and verify by multiplying back. Visual models, like a bar representing 2/3 cup divided into 3/4-cup segments, show it fits 8/9 of such a segment.
On a number line, $\tfrac{2}{3}$ is a point to the right of 0. If you take steps of size $\tfrac{3}{4}$ , how many steps reach $\tfrac{2}{3}$ ? This is $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)$ . Which value is correct?
Answer: $\tfrac{8}{9}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreted on a number line as steps to reach a point. Compute (2/3)÷(3/4)=(2/3)×(4/3)=8/9 steps. Verification: (3/4)×(8/9)=24/36=2/3, reaching the point. This is less than 1 step since 2/3 < 3/4. Mistakes include inverting the operation to get 9/8 as in choice B, or multiplying to get 1/2. Method: division to multiplication by reciprocal, compute 8/9, verify. Visual: number line with steps of 3/4 up to 2/3 fits 8/9 of a step.
A science club has $\tfrac{1}{2}$ liter of solution and pours it equally into 3 identical containers. How much solution goes in each container? Compute $\left(\tfrac{1}{2}\right)\div 3$ .
Answer: $\tfrac{1}{6}$ liter This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), here dividing by a whole number interpreted as sharing equally among containers. Compute (1/2)÷3=(1/2)×(1/3)=(1×1)/(2×3)=1/6 liter per container. Verification: multiply one share by 3, (1/6)×3=3/6=1/2 liter, matching the total. Context: sharing 1/2 liter equally into 3 parts is like finding how many 1/3 portions fit into 1/2, but directly it's division by 3. Errors include flipping incorrectly to get 3/2 as in choice A, or confusing with multiplication to get 2/3. Steps: rewrite division as multiplication by reciprocal, compute numerator and denominator, simplify, verify by multiplying back. Visual: a bar of 1/2 liter divided into 3 equal parts shows each as 1/6.
Compute $\left(\tfrac{3}{4}\right)\div\left(\tfrac{2}{3}\right)$ and choose the simplified result.
Answer: $\tfrac{9}{8}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), focusing on computing and simplifying the result. Compute (3/4)÷(2/3)=(3/4)×(3/2)=(3×3)/(4×2)=9/8. Verification: (2/3)×(9/8)=(2×9)/(3×8)=18/24=3/4, correct. This is greater than 1 since the dividend is larger than the divisor. Errors include flipping the wrong fraction to get 8/9 as in choice B, or multiplying directly to get 6/12=1/2. Steps: write division, reciprocal of divisor, multiply fractions, simplify 9/8 (already simplest), verify. Common example: (1/2)÷(1/4)=2 shows how many quarters in a half.
A ribbon is $\tfrac{3}{4}$ yard long. Each bow needs $\tfrac{1}{2}$ yard of ribbon. How many bows can be made? Compute $\left(\tfrac{3}{4}\right)\div\left(\tfrac{1}{2}\right)$ .
Answer: $\tfrac{3}{2}$ This question tests dividing fractions $(a/b) \div (c/d)$ using the reciprocal method (multiply by $d/c$ ), in context of how many items can be made from a given length. Compute $(3/4) \div (1/2) = (3/4) \times (2/1) = (3 \times 2)/(4 \times 1) = 6/4 = 3/2$ bows. Verification: $(1/2) \times (3/2) = 3/4$ yard, matching the total. This means 1 full bow and half another from $3/4$ yard. Errors include multiplying fractions to get 3/8 as in choice A, or confusing reciprocals to get 1/2. Steps: flip divisor, multiply, simplify $6/4$ to $3/2$ , verify. Context like ribbons shows practical division application.
Which value makes this verification true? If $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)=q$ , then $\left(\tfrac{3}{4}\right)\times q=\tfrac{2}{3}$ . What is $q$ ?
Answer: $\tfrac{8}{9}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), verifying through the multiplication-division relationship. To find q where (3/4)×q=2/3, q=(2/3)÷(3/4)=(2/3)×(4/3)=8/9. Verification: (3/4)×(8/9)=24/36=2/3, correct. This confirms division as the inverse of multiplication. Common errors: flipping incorrectly to 9/8 as in choice D, or multiplying to 1/2. Method: solve for q using reciprocal, compute, verify by plugging back. Examples like (1/2)÷(1/4)=2 verify similarly.
A recipe needs $\tfrac{2}{3}$ cup of yogurt. You have a scoop that holds $\tfrac{3}{4}$ cup. The division expression is $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)$ . How many full scoops of size $\tfrac{3}{4}$ cup fit into $\tfrac{2}{3}$ cup of yogurt? (Choose the simplified value.)
Answer: $\tfrac{8}{9}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method, interpreting in contexts like servings or how many scoops of one size fit into another amount, and understanding via visual models and the multiplication-division relationship. To divide (2/3)÷(3/4), flip the divisor to its reciprocal and multiply: (2/3) × (4/3) = 8/9, meaning 8/9 of a full 3/4-cup scoop fits into 2/3 cup of yogurt. Verification: multiply divisor by quotient to get dividend, (3/4) × (8/9) = 24/36 = 2/3, which confirms the result. For example, imagine a bar representing 2/3 cup; dividing it into 3/4-cup segments shows it fits 8/9 of one segment. The correct method is to use the reciprocal for division, yielding 8/9 as the simplified value. A common mistake is multiplying without flipping, like (2/3) × (3/4) = 1/2, or flipping the wrong fraction. Remember the steps: write the division, flip the divisor, multiply, simplify, and verify with multiplication.
A road segment is $\tfrac{1}{2}$ mile long. A student walks at a pace of $\tfrac{1}{4}$ mile per minute. The time (in minutes) is $\left(\tfrac{1}{2}\right)\div\left(\tfrac{1}{4}\right)$ . How many minutes does it take? (Simplify.)
Answer: $2$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method, interpreting in contexts like time taken to cover distance at a given speed, and understanding via visual models and the multiplication-division relationship. To divide (1/2)÷(1/4), flip the divisor to its reciprocal and multiply: (1/2) × (4/1) = 4/2 = 2, so it takes 2 minutes. Verification: multiply divisor by quotient to get dividend, (1/4) × 2 = 1/2, which confirms. For example, distance 1/2 mile at 1/4 mile per minute: how many quarters in a half is 2. The correct division using the reciprocal gives 2. A common mistake is adding fractions or using wrong reciprocal. Method: write division, flip divisor, multiply, simplify, verify with multiplication.
A rectangular garden has area $\tfrac{1}{2}$ square mile and length $\tfrac{3}{4}$ mile. The width is found by $\left(\tfrac{1}{2}\right)\div\left(\tfrac{3}{4}\right)$ . What is the width (in miles), simplified?
Answer: $\tfrac{2}{3}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method, interpreting in contexts like finding the width of a rectangle given area and length, and understanding via visual models and the multiplication-division relationship. To divide (1/2)÷(3/4), flip the divisor to its reciprocal and multiply: (1/2) × (4/3) = 4/6 = 2/3, so the width is 2/3 mile. Verification: multiply divisor by quotient to get dividend, (3/4) × (2/3) = 6/12 = 1/2, which matches the area. For example, a rectangle with area 1/2 square mile and length 3/4 mile has width (1/2) × (4/3) = 2/3 mile. The correct division using the reciprocal gives the simplified 2/3. A common error is dividing numerators and denominators separately, like 1÷3 / 2÷4 = 1/3 / 1/2 = 2/3 incorrectly, or arithmetic mistakes. Steps: write division, flip divisor, multiply, simplify 4/6 to 2/3, verify with multiplication.
Use the multiplication-division relationship to check a quotient: If $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)=\tfrac{8}{9},$ which multiplication correctly verifies the result (divisor $\times$ quotient $=$ dividend)?
Answer: $\left(\tfrac{3}{4}\right)\times\left(\tfrac{8}{9}\right)=\tfrac{2}{3}$ This question tests dividing fractions $\frac{a}{b} \div \frac{c}{d}$ using the reciprocal method, emphasizing the multiplication-division relationship for verification, and understanding via contexts and visual models. The division $\frac{2}{3} \div \frac{3}{4} = \frac{8}{9}$ is verified by multiplying divisor by quotient to equal dividend: $\frac{3}{4} \times \frac{8}{9} = \frac{24}{36} = \frac{2}{3}$ . Other options like $\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$ do not verify correctly. For example, the relationship shows division is the inverse of multiplication, so if quotient is correct, divisor $\times$ quotient $=$ dividend. The correct verification is option B. A common error is confusing which to multiply, like quotient $\times$ dividend. Remember: to check, always multiply divisor by proposed quotient and see if it equals dividend.
Number-line reasoning: On a number line, $\tfrac{3}{4}$ is one step size. How many $\tfrac{3}{4}$ -steps are in $\tfrac{1}{3}$ ? Compute $\left(\tfrac{1}{3}\right)\div\left(\tfrac{3}{4}\right)$ and verify by multiplication.
Answer: $\tfrac{4}{9}$ step; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{4}{9}\right)=\tfrac{1}{3}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in number-line contexts like step sizes, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (1/3)÷(3/4)=(1/3)×(4/3)=4/9 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((3/4)×(4/9)=12/36=1/3 confirms quotient correct). Context: 'how many 3/4-steps in 1/3' divides: 1/3÷3/4=4/9 step (less than 1). Multiplication-division relationship: (1/3)÷(3/4)=4/9 because (3/4) of (4/9) equals (1/3) (3/4×4/9=1/3, division is inverse of multiplication). The correct choice is A, which uses the reciprocal method to get 4/9 and verifies correctly. Common errors include flipping the wrong fraction, leading to 9/4 as in B.
A painter has $\tfrac{2}{3}$ gallon of paint. One room needs $\tfrac{1}{3}$ gallon. How many rooms can be painted? Compute $\left(\tfrac{2}{3}\right)\div\left(\tfrac{1}{3}\right)$ and verify by multiplication.
Answer: $2$ rooms; and $\left(\tfrac{1}{3}\right)\times 2=\tfrac{2}{3}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like painting rooms, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (2/3)÷(1/3)=(2/3)×(3/1)=2 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((1/3)×2=2/3 confirms quotient correct). Context: 'how many rooms needing 1/3 gallon from 2/3 gallon' divides: 2/3÷1/3=2 rooms. Multiplication-division relationship: (2/3)÷(1/3)=2 because (1/3) of 2 equals (2/3) (1/3×2=2/3, division is inverse of multiplication). The correct choice is B, which uses the reciprocal method to get 2 and verifies correctly. Common errors include wrong division, like getting 3/2 as in C.
Compute and interpret: A container holds $\tfrac{3}{4}$ liter of juice. Each small bottle holds $\tfrac{1}{2}$ liter. How many small bottles can be filled? Compute $\left(\tfrac{3}{4}\right)\div\left(\tfrac{1}{2}\right)$ and verify by multiplying the quotient by $\tfrac{1}{2}$ .
Answer: $\tfrac{3}{2}$ bottles; and $\left(\tfrac{1}{2}\right)\times\left(\tfrac{3}{2}\right)=\tfrac{3}{4}$ This question tests dividing fractions $(a/b) \div (c/d)$ using the reciprocal method (multiply by $d/c$ ), interpreting in contexts like filling bottles, and understanding via visual models and the multiplication-division relationship. Dividing fractions: $(3/4) \div (1/2) = (3/4) \times (2/1) = 6/4 = 3/2$ (flip divisor to reciprocal, multiply). Verification: divisor $\times$ quotient $=$ dividend $((1/2) \times (3/2) = 3/4$ confirms quotient correct). Context: 'how many $1/2$ -liter bottles from $3/4$ liter' divides: $(3/4) \div (1/2) = 3/2$ (1.5 bottles). Multiplication-division relationship: $(3/4) \div (1/2) = 3/2$ because $(1/2)$ of $(3/2)$ equals $(3/4)$ ( $(1/2) \times (3/2) = 3/4$ , division is inverse of multiplication). The correct choice is C, which uses the reciprocal method to get $3/2$ and verifies correctly. Common errors include multiplying instead of dividing, like getting $2/3$ as in B.
A science class has $\tfrac{3}{4}$ cup of saltwater. Each test tube needs $\tfrac{2}{3}$ cup. How many test tubes can be filled? Compute $\left(\tfrac{3}{4}\right)\div\left(\tfrac{2}{3}\right)$ and verify by multiplication.
Answer: $\tfrac{9}{8}$ test tubes; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{9}{8}\right)=\tfrac{3}{4}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like filling test tubes, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (3/4)÷(2/3)=(3/4)×(3/2)=9/8 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((2/3)×(9/8)=18/24=3/4 confirms quotient correct). Context: 'how many 2/3-cup test tubes from 3/4 cup' divides: 3/4÷2/3=9/8 (more than 1). Multiplication-division relationship: (3/4)÷(2/3)=9/8 because (2/3) of (9/8) equals (3/4) (2/3×9/8=3/4, division is inverse of multiplication). The correct choice is B, which uses the reciprocal method to get 9/8 and verifies correctly. Common errors include incorrect multiplication, like getting 8/9 as in A.
A rectangle has area $\tfrac{1}{2}$ square mile and length $\tfrac{3}{4}$ mile. What is the width? Compute $\left(\tfrac{1}{2}\right)\div\left(\tfrac{3}{4}\right)$ .
Answer: $\tfrac{2}{3}$ mile This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like finding the width of a rectangle given area and length. To find the width, compute (1/2)÷(3/4)=(1/2)×(4/3)=(1×4)/(2×3)=4/6=2/3 mile. Verification: multiply length by width, (3/4)×(2/3)=(3×2)/(4×3)=6/12=1/2 square mile, matching the area. This uses the area formula where area = length × width, so width = area ÷ length, the inverse of multiplication. Common errors include multiplying instead of dividing to get 3/8 as in choice B, or arithmetic slips like not simplifying 4/6 correctly. Method: write the division, flip the divisor, multiply, simplify the fraction, and verify. Visual aids like drawing a rectangle with area 1/2 and length 3/4 help see the width as 2/3.
Maria has $\frac{3}{4}$ of a yard of fabric. She wants to cut it into pieces that are each $\frac{1}{8}$ of a yard long to make bookmarks. After cutting all possible pieces, she realizes she made an error and each bookmark actually needs $\frac{1}{6}$ of a yard of fabric. How many complete bookmarks can she make with her original $\frac{3}{4}$ yard of fabric?
Answer: 4 bookmarks because $\frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = 4.5$ To find how many $\frac{1}{6}$ -yard bookmarks can be made from $\frac{3}{4}$ yard, we calculate $\frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = 4.5$ . Since we can only make complete bookmarks, the answer is 4. Choice A incorrectly assumes the previously cut pieces can be reused. Choice C uses the wrong division (by $\frac{1}{8}$ instead of $\frac{1}{6}$ ). Choice D incorrectly rounds up when we need complete bookmarks.
Use a visual model idea: Imagine a bar showing $\tfrac{2}{3}$ of a whole. Each group is $\tfrac{1}{4}$ of a whole. How many $\tfrac{1}{4}$ -size groups fit into $\tfrac{2}{3}$ ? Compute $\left(\tfrac{2}{3}\right)\div\left(\tfrac{1}{4}\right)$ and verify by multiplication.
Answer: $\tfrac{8}{3}$ groups; and $\left(\tfrac{1}{4}\right)\times\left(\tfrac{8}{3}\right)=\tfrac{2}{3}$ This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in visual model contexts like fitting groups into a bar, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (2/3)÷(1/4)=(2/3)×(4/1)=8/3 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((1/4)×(8/3)=8/12=2/3 confirms quotient correct). Context: 'how many 1/4-size groups in 2/3' divides: 2/3÷1/4=8/3 (more than 2 groups). Multiplication-division relationship: (2/3)÷(1/4)=8/3 because (1/4) of (8/3) equals (2/3) (1/4×8/3=2/3, division is inverse of multiplication). The correct choice is B, which uses the reciprocal method to get 8/3 and verifies correctly. Common errors include wrong reciprocal or simplification, like getting 1/6 as in C.

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