Introduction
Dividing Fractions by Fractions is an important Grade 6 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with dividing fractions by fractions.
What Is Dividing Fractions by Fractions?
Dividing Fractions by Fractions means using equal parts, number lines, and clear fraction language to describe parts of a whole.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Dividing Fractions by Fractions
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Identify the whole before naming a fraction.
- Make sure each part is equal in size.
- Use a number line or model to show where the fraction belongs.
- Explain whether two fractions have the same size or different sizes.
Visual Models
For Dividing Fractions by Fractions, a useful visual model is a quick drawing, array, table, number line, graph, or labeled diagram that shows what each number means.
Worked Examples
Example 1
Question: What is \(\frac{3}{4}\div\frac{1}{2}\)?
- A. \(\frac{3}{8}\)
- B. \(\frac{2}{3}\)
- C. \(1\frac{1}{2}\)
- D. \(2\)
- Friendly reminder: when you divide by a fraction, you can multiply by its reciprocal (“flip” the divisor and multiply). \(\frac{3}{4}\div\frac{1}{2}=\frac{3}{4}\times\frac{2}{1}=\frac{6}{4}\), which simplifies to \(\frac{3}{2}\), or \(1\frac{1}{2}\).
Answer: \(1\frac{1}{2}\)
Example 2
Question: Evaluate: \(\frac{5}{6}\div\frac{1}{3}\)
- A. \(\frac{5}{18}\)
- B. \(\frac{5}{6}\)
- C. \(\frac{5}{2}\)
- D. \(\frac{15}{30}\)
- Rewrite the division as multiplication by flipping \(\frac{1}{3}\): \(\frac{5}{6}\times\frac{3}{1}=\frac{15}{6}\).
- Simplify \(\frac{15}{6}\) to \(\frac{5}{2}\) (or mixed form \(2\frac{1}{2}\)).
Answer: \(\frac{5}{2}\) or \(2\frac{1}{2}\)
Example 3
Question: What is \(\frac{2}{3}\div\frac{4}{5}\)?
- A. \(\frac{8}{15}\)
- B. \(\frac{5}{6}\)
- C. \(\frac{6}{7}\)
- D. \(\frac{10}{15}\)
- Invert \(\frac{4}{5}\) to \(\frac{5}{4}\), then multiply: \(\frac{2}{3}\times\frac{5}{4}=\frac{10}{12}\).
- That reduces nicely to \(\frac{5}{6}\).
Answer: \(\frac{5}{6}\)
Real-World Word Problems
Problem 1
Question: A recipe calls for \(\frac{3}{4}\) cup of sugar. If you want to make exactly half of the recipe (so that every ingredient is halved), how much sugar do you need? Which expression represents this situation?
- A. \(\frac{3}{4}\div\frac{1}{2}\)
- B. \(\frac{3}{4}\times\frac{1}{2}\)
- C. \(\frac{3}{4}+\frac{1}{2}\)
- D. \(\frac{1}{2}\div\frac{3}{4}\)
Why it works: "Half the recipe" means multiply each ingredient by \(\frac{1}{2}\), not divide by \(\frac{1}{2}\). So you want \(\frac{3}{4}\times\frac{1}{2}=\frac{3}{8}\) cup of sugar.
Answer: \(\frac{3}{4}\times\frac{1}{2}=\frac{3}{8}\) cup
Problem 2
Question: A ribbon is \(\frac{5}{6}\) yard long. You need pieces that are \(\frac{1}{6}\) yard long. How many pieces can you cut?
- A. \(1\)
- B. \(4\)
- C. \(5\)
- D. \(6\)
Why it works: \(\frac{5}{6}\div\frac{1}{6}=\frac{5}{6}\times 6=5\).
Answer: \(5\) pieces
Common Mistakes
- Counting unequal parts as if they were equal.
- Forgetting that the denominator tells how many equal parts make the whole.
- Comparing fractions without first checking the size of the whole.
- Placing a fraction on a number line without counting equal intervals.
Strategy Tips
- Draw the whole first, then divide it into equal parts.
- Use number lines when the question asks about order or location.
- Say the fraction out loud to connect numerator and denominator meanings.
- Check whether the answer should be closer to 0, 1/2, or 1.
Practice Questions
Question 1
Compute: \(\frac{7}{8}\div\frac{1}{4}\)
- A. \(\frac{7}{32}\)
- B. \(\frac{7}{16}\)
- C. \(3\frac{1}{2}\)
- D. \(\frac{11}{8}\)
Question 2
What is \(\frac{4}{5}\div\frac{2}{3}\)?
- A. \(\frac{8}{15}\)
- B. \(\frac{2}{15}\)
- C. \(\frac{10}{12}\)
- D. \(\frac{6}{5}\)
Question 3
How many \(\frac{1}{4}\)-cup servings are in \(2\) cups?
- A. \(6\)
- B. \(8\)
- C. \(\frac{1}{2}\)
- D. \(4\)
Question 4
Evaluate: \(\frac{1}{2}\div\frac{1}{8}\)
- A. \(\frac{1}{16}\)
- B. \(4\)
- C. \(\frac{1}{4}\)
- D. \(16\)
Question 5
A trail is \(3\frac{1}{2}\) miles long. How many \(\frac{3}{4}\)-mile lengths fit along the trail?
- A. \(2\frac{5}{8}\)
- B. \(4\)
- C. \(4\frac{2}{3}\)
- D. \(5\)
Question 6
What is \(\frac{9}{10}\div\frac{3}{5}\)?
- A. \(\frac{27}{50}\)
- B. \(\frac{27}{15}\)
- C. \(\frac{10}{15}\)
- D. \(\frac{3}{2}\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(3\frac{1}{2}\)
\(\frac{7}{8}\times\frac{4}{1}=\frac{28}{8}=\frac{7}{2}=3\frac{1}{2}\).
Question 2
Answer: \(\frac{6}{5}\) or \(1\frac{1}{5}\)
\(\frac{4}{5}\times\frac{3}{2}=\frac{12}{10}=\frac{6}{5}\).
Question 3
Answer: \(8\)
You are counting how many \(\frac{1}{4}\)-cup servings fit inside \(2\) cups---that’s a division problem: \(2\div\frac{1}{4}\). Rewrite as \(2\times 4\), which equals \(8\) servings.
Question 4
Answer: \(4\)
Dividing by \(\frac{1}{8}\) is the same as multiplying by \(8\). So \(\frac{1}{2}\div\frac{1}{8}=\frac{1}{2}\times 8\), which simplifies to \(4\).
Question 5
Answer: \(4\frac{2}{3}\) lengths
Change \(3\frac{1}{2}\) to \(\frac{7}{2}\), then divide by multiplying by \(\frac{4}{3}\): \(\frac{7}{2}\times\frac{4}{3}=\frac{28}{6}\), which simplifies to \(\frac{14}{3}\) or \(4\frac{2}{3}\) lengths along the trail.
Question 6
Answer: \(\frac{3}{2}\) or \(1\frac{1}{2}\)
\(\frac{9}{10}\times\frac{5}{3}=\frac{45}{30}=\frac{3}{2}\).
Connection to Standards
This lesson supports Grade 6 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Dividing Fractions by Fractions becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Equal parts first, fraction name second.
