Divide Fractions by Fractions
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6th Grade Math › Divide Fractions by Fractions
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.
$\tfrac{1}{2}$ test tube; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{1}{2}\right)=\tfrac{1}{3}$
$\tfrac{9}{8}$ test tubes; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{9}{8}\right)=\tfrac{3}{4}$
$\tfrac{6}{12}$ test tube; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{6}{12}\right)=\tfrac{1}{3}$
$\tfrac{8}{9}$ test tube; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{8}{9}\right)=\tfrac{16}{27}$
Explanation
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 student walks $\tfrac{1}{2}$ mile. Each lap around a short track is $\tfrac{3}{4}$ mile. What fraction of a lap did the student complete? Compute $\left(\tfrac{1}{2}\right)\div\left(\tfrac{3}{4}\right)$ and verify using multiplication.
$\tfrac{3}{2}$ laps; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{3}{2}\right)=\tfrac{9}{8}$
$\tfrac{3}{8}$ of a lap; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{3}{8}\right)=\tfrac{9}{32}$
$\tfrac{4}{6}$ laps; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{4}{6}\right)=\tfrac{1}{2}$
$\tfrac{2}{3}$ of a lap; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{2}{3}\right)=\tfrac{1}{2}$
Explanation
This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like fractions of a lap, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (1/2)÷(3/4)=(1/2)×(4/3)=4/6=2/3 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((3/4)×(2/3)=6/12=1/2 confirms quotient correct). Context: 'what fraction of a 3/4-mile lap is 1/2 mile' divides: 1/2÷3/4=2/3 of a lap. Multiplication-division relationship: (1/2)÷(3/4)=2/3 because (3/4) of (2/3) equals (1/2) (3/4×2/3=1/2, division is inverse of multiplication). The correct choice is A, which uses the reciprocal method to get 2/3 and verifies correctly. Common errors include incorrect reciprocal use, leading to 3/2 as in B.
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 $q$?
$\tfrac{8}{9}$
$\tfrac{9}{8}$
$\tfrac{4}{9}$
$\tfrac{1}{2}$
Explanation
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 ribbon is $\tfrac{3}{4}$ meter long. You cut pieces that are each $\tfrac{1}{3}$ meter long. How many pieces can you cut? Compute $\left(\tfrac{3}{4}\right)\div\left(\tfrac{1}{3}\right)$ and verify by multiplication.
$\tfrac{4}{9}$ piece; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{4}{9}\right)=\tfrac{4}{27}$
$\tfrac{1}{4}$ piece; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{1}{4}\right)=\tfrac{1}{12}$
$\tfrac{3}{12}$ pieces; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{3}{12}\right)=\tfrac{1}{12}$
$\tfrac{9}{4}$ pieces; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{9}{4}\right)=\tfrac{3}{4}$
Explanation
This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like cutting ribbons into pieces, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (3/4)÷(1/3)=(3/4)×(3/1)=9/4 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((1/3)×(9/4)=9/12=3/4 confirms quotient correct). Context: 'how many 1/3-meter pieces in 3/4 meter ribbon?' divides: 3/4÷1/3=9/4 (more than 2 full pieces). Multiplication-division relationship: (3/4)÷(1/3)=9/4 because (1/3) of (9/4) equals (3/4) (1/3×9/4=3/4, division is inverse of multiplication). The correct choice is A, which uses the reciprocal method to get 9/4 and verifies correctly. Common errors include using the wrong reciprocal or arithmetic mistakes, like getting 4/9 as in B.
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 $\left(\tfrac{1}{2}\right)\div 3$.
$\tfrac{1}{3}$ liter
$\tfrac{3}{2}$ liter
$\tfrac{2}{3}$ liter
$\tfrac{1}{6}$ liter
Explanation
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 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 $\tfrac{1}{2}$.
$\tfrac{3}{2}$ bottles; and $\left(\tfrac{1}{2}\right)\times\left(\tfrac{3}{2}\right)=\tfrac{3}{4}$
$\tfrac{2}{3}$ bottle; and $\left(\tfrac{1}{2}\right)\times\left(\tfrac{2}{3}\right)=\tfrac{1}{3}$
$\tfrac{3}{8}$ bottle; and $\left(\tfrac{1}{2}\right)\times\left(\tfrac{3}{8}\right)=\tfrac{3}{16}$
$\tfrac{1}{4}$ bottle; and $\left(\tfrac{1}{2}\right)\times\left(\tfrac{1}{4}\right)=\tfrac{1}{8}$
Explanation
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 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.)
$\tfrac{2}{3}$
$\tfrac{9}{8}$
$\tfrac{1}{2}$
$\tfrac{8}{9}$
Explanation
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 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.
$\tfrac{1}{2}$ serving (because $\tfrac{2}{3}\times\tfrac{3}{4}=\tfrac{1}{2}$)
$\tfrac{2}{3}$ serving (from an arithmetic slip: $\tfrac{2}{3}\times\tfrac{4}{3}=\tfrac{6}{9}$)
$\tfrac{3}{8}$ serving (dividing numerators and denominators separately)
$\tfrac{8}{9}$ of a serving (less than 1 full $\tfrac{3}{4}$-cup serving)
Explanation
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.
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.
$\tfrac{1}{2}$ room; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{1}{2}\right)=\tfrac{1}{6}$
$2$ rooms; and $\left(\tfrac{1}{3}\right)\times 2=\tfrac{2}{3}$
$\tfrac{2}{9}$ room; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{2}{9}\right)=\tfrac{2}{27}$
$\tfrac{3}{2}$ rooms; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{3}{2}\right)=\tfrac{1}{2}$
Explanation
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.
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)?
$\left(\tfrac{8}{9}\right)\times\left(\tfrac{2}{3}\right)=\tfrac{3}{4}$
$\left(\tfrac{2}{3}\right)\times\left(\tfrac{8}{9}\right)=\tfrac{3}{4}$
$\left(\tfrac{3}{4}\right)\times\left(\tfrac{8}{9}\right)=\tfrac{2}{3}$
$\left(\tfrac{2}{3}\right)\times\left(\tfrac{3}{4}\right)=\tfrac{8}{9}$
Explanation
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.