Divide Fractions by Fractions
Help Questions
6th Grade Math › Divide Fractions by Fractions
A rectangular garden plot has an area of $$\frac{5}{6}$$ square meters and a length of $$\frac{5}{4}$$ meters. If the garden is divided into smaller rectangular sections, each with a width equal to the original garden's width, and each section has a length of $$\frac{1}{3}$$ meter, how many complete sections can be formed?
2 sections because the original width is $$\frac{2}{3}$$ meter and $$\frac{5}{4} \div \frac{1}{3} = \frac{15}{4}$$
2 sections because the original width is $$\frac{2}{3}$$ meter and $$\frac{2}{3} \div \frac{1}{3} = 2$$
3 sections because the original width is $$\frac{2}{3}$$ meter and $$\frac{5}{4} \div \frac{1}{3} = 3.75$$
4 sections because the original length $$\frac{5}{4}$$ divided by new length $$\frac{1}{3}$$ gives $$\frac{15}{4} = 3.75$$
Explanation
First find the original width: $$\text{width} = \frac{\text{area}}{\text{length}} = \frac{5/6}{5/4} = \frac{5}{6} \times \frac{4}{5} = \frac{2}{3}$$ meter. The number of sections is determined by how many $$\frac{1}{3}$$-meter lengths fit in the original $$\frac{5}{4}$$-meter length: $$\frac{5}{4} \div \frac{1}{3} = \frac{5}{4} \times 3 = \frac{15}{4} = 3.75$$, so 3 complete sections. Choice A miscalculates the division. Choice B incorrectly divides width by new length. Choice D rounds 3.75 up instead of down for complete sections.
A recipe for trail mix calls for $$\frac{5}{8}$$ cup of nuts. Sarah wants to make multiple smaller portions, where each portion contains $$\frac{3}{4}$$ of the nuts called for in the original recipe. If she has $$2\frac{1}{8}$$ cups of nuts available, and after making as many complete smaller portions as possible she uses the remaining nuts to make one final portion at $$\frac{1}{2}$$ the original recipe size, will she have enough nuts?
No, because 4 smaller portions use $$1\frac{7}{8}$$ cups, but the final portion needs $$\frac{5}{16}$$ cup
No, because she can make 4 smaller portions using $$2\frac{1}{16}$$ cups total
Yes, because 4 smaller portions use $$1\frac{7}{8}$$ cups, leaving $$\frac{1}{4}$$ cup for the final portion
Yes, because she can make exactly 4 smaller portions with $$\frac{1}{8}$$ cup remaining for other uses
Explanation
Each smaller portion needs $$\frac{3}{4} \times \frac{5}{8} = \frac{15}{32}$$ cup. Number of portions from $$2\frac{1}{8} = \frac{17}{8}$$ cups: $$\frac{17}{8} \div \frac{15}{32} = \frac{17}{8} \times \frac{32}{15} = \frac{544}{120} = \frac{68}{15} \approx 4.53$$, so 4 complete portions. These use $$4 \times \frac{15}{32} = \frac{60}{32} = \frac{15}{8} = 1\frac{7}{8}$$ cups. Remaining: $$2\frac{1}{8} - 1\frac{7}{8} = \frac{17}{8} - \frac{15}{8} = \frac{2}{8} = \frac{1}{4}$$ cup. The final portion at $$\frac{1}{2}$$ original size needs $$\frac{1}{2} \times \frac{5}{8} = \frac{5}{16}$$ cup. Since $$\frac{1}{4} = \frac{4}{16} < \frac{5}{16}$$, she doesn't have enough. Choice A has wrong calculation. Choice B incorrectly concludes she has enough. Choice D ignores the final portion requirement.
A baker uses $$\frac{7}{8}$$ cup of sugar to make a batch of cookies. She wants to make smaller batches using only $$\frac{2}{3}$$ of the sugar amount each time. If she has exactly 2$$\frac{1}{4}$$ cups of sugar available, how many of these smaller batches can she make?
3 batches because each small batch uses $$\frac{7}{12}$$ cup and $$2\frac{1}{4} \div \frac{7}{12} = \frac{27}{7}$$
3 batches because each small batch uses $$\frac{7}{12}$$ cup and $$\frac{9}{4} \div \frac{7}{12} = \frac{27}{7} \approx 3.86$$
4 batches because $$2\frac{1}{4} \div \frac{2}{3} = \frac{9}{4} \times \frac{3}{2} = \frac{27}{8} = 3.375$$
4 batches because $$\frac{7}{8} \times \frac{2}{3} = \frac{7}{12}$$ and $$\frac{9}{4} \div \frac{7}{12} = \frac{27}{7} = 3.86$$
Explanation
When you encounter word problems involving fractions and division, break them down step by step to identify what you're actually calculating.
First, you need to find how much sugar each smaller batch requires. Since each small batch uses $$\frac{2}{3}$$ of the original $$\frac{7}{8}$$ cup, multiply: $$\frac{7}{8} \times \frac{2}{3} = \frac{14}{24} = \frac{7}{12}$$ cup per small batch.
Next, determine how many small batches you can make with $$2\frac{1}{4}$$ cups total. Convert the mixed number to an improper fraction: $$2\frac{1}{4} = \frac{9}{4}$$. Now divide the total sugar by the amount per batch: $$\frac{9}{4} \div \frac{7}{12}$$. To divide fractions, multiply by the reciprocal: $$\frac{9}{4} \times \frac{12}{7} = \frac{108}{28} = \frac{27}{7} \approx 3.86$$.
Since you can only make whole batches, the answer is 3 batches.
Answer A is correct because it shows the complete calculation with the proper approximation. Answer B makes the same calculation but fails to convert $$\frac{27}{7}$$ to a decimal to show it's approximately 3.86, making the "3 batches" conclusion less clear. Answer C incorrectly divides the total sugar by $$\frac{2}{3}$$ instead of by the actual amount per batch ($$\frac{7}{12}$$). Answer D shows correct intermediate steps but incorrectly concludes that 3.86 rounds to 4 batches.
Remember: in division problems involving "how many groups," always round down to the nearest whole number since you can't make partial batches.
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?
5 bookmarks because $$\frac{3}{4} \div \frac{1}{6} = \frac{18}{4} = 4.5$$ rounds up to 5
6 bookmarks because $$\frac{3}{4} \div \frac{1}{8} = 6$$ pieces were already cut
4 bookmarks because she can reuse all the pieces she already cut
4 bookmarks because $$\frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = 4.5$$
Explanation
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.
During a science experiment, students found that $$\frac{3}{5}$$ of a solution evaporated. The remaining solution was then divided equally among 4 test tubes, with each test tube containing $$\frac{1}{10}$$ liter of solution. What was the original amount of solution before any evaporation?
1 liter because $$4 \times \frac{1}{10} = \frac{2}{5}$$ and $$\frac{2}{5} \div \frac{2}{5} = 1$$
$$\frac{2}{3}$$ liter because the remaining fraction is $$\frac{2}{5}$$ and $$\frac{2}{5} \div \frac{3}{5} = \frac{6}{10}$$
$$\frac{2}{3}$$ liter because $$4 \times \frac{1}{10} = \frac{2}{5}$$ and $$\frac{2}{5} \div \frac{3}{5} = \frac{2}{3}$$
1 liter because $$4 \times \frac{1}{10} = \frac{2}{5}$$ remaining, so original $$= \frac{2}{5} \div \frac{2}{5} = 1$$
Explanation
If $$\frac{3}{5}$$ evaporated, then $$1 - \frac{3}{5} = \frac{2}{5}$$ remained. The remaining amount is $$4 \times \frac{1}{10} = \frac{4}{10} = \frac{2}{5}$$ liter. Since $$\frac{2}{5}$$ of the original equals $$\frac{2}{5}$$ liter, the original amount was $$\frac{2}{5} \div \frac{2}{5} = 1$$ liter. Choice A incorrectly divides by the evaporated fraction instead of the remaining fraction. Choice B makes a calculation error. Choice C shows the correct work but arrives at the same answer through correct reasoning.
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.
$\tfrac{6}{12}$ test tube; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{6}{12}\right)=\tfrac{1}{3}$
$\tfrac{1}{2}$ test tube; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{1}{2}\right)=\tfrac{1}{3}$
$\tfrac{9}{8}$ test tubes; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{9}{8}\right)=\tfrac{3}{4}$
$\tfrac{8}{9}$ 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 $\left(\tfrac{3}{4}\right)\times\left(\tfrac{3}{8}\right)=\tfrac{9}{32}$
$\tfrac{2}{3}$ of a lap; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{2}{3}\right)=\tfrac{1}{2}$
$\tfrac{4}{6}$ laps; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{4}{6}\right)=\tfrac{1}{2}$
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 $q$?
$\tfrac{1}{2}$
$\tfrac{4}{9}$
$\tfrac{8}{9}$
$\tfrac{9}{8}$
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 $\left(\tfrac{1}{3}\right)\times\left(\tfrac{4}{9}\right)=\tfrac{4}{27}$
$\tfrac{9}{4}$ pieces; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{9}{4}\right)=\tfrac{3}{4}$
$\tfrac{3}{12}$ pieces; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{3}{12}\right)=\tfrac{1}{12}$
$\tfrac{1}{4}$ piece; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{1}{4}\right)=\tfrac{1}{12}$
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 $\left(\tfrac{1}{2}\right)\div 3$.
$\tfrac{3}{2}$ liter
$\tfrac{1}{3}$ liter
$\tfrac{1}{6}$ liter
$\tfrac{2}{3}$ 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.