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Fraction Operations

A comprehensive sequence covering fraction multiplication with cross-cancelling techniques and fraction division using reciprocals.

Ttsthibeault

Cross Cancelling Fractions

A collection of worksheets and practice materials focused on multiplying signed fractions and mixed numbers using cross-cancelling techniques.

Dividing Fractions

Instructional materials introducing fraction division through reciprocal multiplication and the Keep Change Flip strategy.

Fraction Operations

A comprehensive sequence covering fraction multiplication with cross-cancelling techniques and fraction division using reciprocals.

Ttsthibeault

Cross Cancelling Fractions Worksheet

Multiplying Fractions with Cross Cancelling

Name:

Solve each problem. Answer as an improper fraction (if necessary).

\(\displaystyle {-\frac{1}{6} \times \frac{4}{5} =}\)

\(\displaystyle {\frac{1}{2} \times \left(-\frac{2}{3}\right) =}\)

\(\displaystyle {-\frac{1}{3} \times \left(-\frac{3}{5}\right) =}\)

\(\displaystyle {\frac{4}{5} \times \frac{7}{8} =}\)

\(\displaystyle {-\frac{4}{7} \times \frac{1}{8} =}\)

\(\displaystyle {\frac{1}{3} \times \left(-\frac{3}{4}\right) =}\)

\(\displaystyle {-\frac{3}{7} \times \left(-\frac{7}{9}\right) =}\)

\(\displaystyle {-\frac{2}{5} \times \frac{5}{6} =}\)

\(\displaystyle {\frac{2}{3} \times \frac{3}{4} =}\)

\(\displaystyle {-\frac{5}{54} \times \left(-\frac{9}{40}\right) =}\)

\(\displaystyle {\frac{6}{27} \times \left(-\frac{3}{42}\right) =}\)

\(\displaystyle {-\frac{9}{30} \times \frac{3}{27} =}\)

\(\displaystyle {-\frac{5}{2} \times \frac{10}{7} =}\)

\(\displaystyle {\frac{10}{3} \times \left(-\frac{13}{4}\right) =}\)

\(\displaystyle {-\frac{17}{8} \times \left(-\frac{20}{7}\right) =}\)

\(\displaystyle {-\frac{11}{8} \times 1\frac{5}{7} =}\)

\(\displaystyle {\frac{7}{8} \times \left(-\frac{10}{7}\right) =}\)

\(\displaystyle {-\frac{10}{9} \times \left(-1\frac{1}{4}\right) =}\)

Answers

Mixed Fraction Cancellation Worksheet

Multiplying Fractions with Cross Cancelling

Name:

Solve each problem. Answer as a simplified proper or improper fraction.

\(\displaystyle {-\frac{2}{3} \times \frac{3}{8} =}\)

\(\displaystyle {\frac{5}{6} \times \left(-\frac{3}{10}\right) =}\)

\(\displaystyle {-\frac{4}{9} \times \left(-\frac{3}{8}\right) =}\)

\(\displaystyle {\frac{7}{12} \times \frac{4}{21} =}\)

\(\displaystyle {-\frac{5}{14} \times \frac{7}{15} =}\)

\(\displaystyle {\frac{8}{15} \times \left(-\frac{5}{16}\right) =}\)

\(\displaystyle {-\frac{9}{16} \times \left(-\frac{4}{27}\right) =}\)

\(\displaystyle {-\frac{6}{25} \times \frac{10}{21} =}\)

\(\displaystyle {-1\frac{1}{5} \times \frac{5}{8} =}\)

\(\displaystyle {2\frac{2}{3} \times \left(-\frac{9}{16}\right) =}\)

\(\displaystyle {-1\frac{7}{8} \times \left(-1\frac{1}{5}\right) =}\)

\(\displaystyle {-3\frac{1}{3} \times 1\frac{1}{20} =}\)

Answers

Mixed Fraction Cancellation Answer Key

Multiplying Fractions with Cross Cancelling

Answer Key

Step-by-step cross-cancelling solutions. Answers provided as simplified fractions and mixed numbers.

Ans: \(-\frac{1}{4}\)

\(\displaystyle {-\frac{2}{3} \times \frac{3}{8}}\)

Cancel 3 & 3 (1, 1); 2 & 8 (1, 4)
\(= -\frac{1 \times 1}{1 \times 4} = \mathbf{-\frac{1}{4}}\)

Ans: \(-\frac{1}{4}\)

\(\displaystyle {\frac{5}{6} \times \left(-\frac{3}{10}\right)}\)

Cancel 5 & 10 (1, 2); 3 & 6 (1, 2)
\(= -\frac{1 \times 1}{2 \times 2} = \mathbf{-\frac{1}{4}}\)

Ans: \(\frac{1}{6}\)

\(\displaystyle {-\frac{4}{9} \times \left(-\frac{3}{8}\right)}\)

Cancel 4 & 8 (1, 2); 3 & 9 (1, 3)
\(= \frac{1 \times 1}{3 \times 2} = \mathbf{\frac{1}{6}}\)

Ans: \(\frac{1}{9}\)

\(\displaystyle {\frac{7}{12} \times \frac{4}{21}}\)

Cancel 7 & 21 (1, 3); 4 & 12 (1, 3)
\(= \frac{1 \times 1}{3 \times 3} = \mathbf{\frac{1}{9}}\)

Ans: \(-\frac{1}{6}\)

\(\displaystyle {-\frac{5}{14} \times \frac{7}{15}}\)

Cancel 5 & 15 (1, 3); 7 & 14 (1, 2)
\(= -\frac{1 \times 1}{2 \times 3} = \mathbf{-\frac{1}{6}}\)

Ans: \(-\frac{1}{6}\)

\(\displaystyle {\frac{8}{15} \times \left(-\frac{5}{16}\right)}\)

Cancel 8 & 16 (1, 2); 5 & 15 (1, 3)
\(= -\frac{1 \times 1}{3 \times 2} = \mathbf{-\frac{1}{6}}\)

Ans: \(\frac{1}{12}\)

\(\displaystyle {-\frac{9}{16} \times \left(-\frac{4}{27}\right)}\)

Cancel 9 & 27 (1, 3); 4 & 16 (1, 4)
\(= \frac{1 \times 1}{4 \times 3} = \mathbf{\frac{1}{12}}\)

Ans: \(-\frac{4}{35}\)

\(\displaystyle {-\frac{6}{25} \times \frac{10}{21}}\)

Cancel 6 & 21 by 3; 10 & 25 by 5
\(= -\frac{2 \times 2}{5 \times 7} = \mathbf{-\frac{4}{35}}\)

Ans: \(-\frac{3}{4}\)

\(\displaystyle {-1\frac{1}{5} \times \frac{5}{8} = -\frac{6}{5} \times \frac{5}{8}}\)

Cancel 5 & 5 (1, 1); 6 & 8 (3, 4)
\(= -\frac{3 \times 1}{1 \times 4} = \mathbf{-\frac{3}{4}}\)

Ans: \(-\frac{3}{2}\) or \(-1\frac{1}{2}\)

\(\displaystyle {2\frac{2}{3} \times \left(-\frac{9}{16}\right) = \frac{8}{3} \times \left(-\frac{9}{16}\right)}\)

Cancel 8 & 16 (1, 2); 3 & 9 (1, 3)
\(= -\frac{1 \times 3}{1 \times 2} = \mathbf{-\frac{3}{2}}\text{ or }\mathbf{-1\frac{1}{2}}\)

Cross Reduction Factors Worksheet

Cross Reduction Factor Pairs Worksheet

Accompanying Factor Pair Organizer for 12-Question Mixed Cross-Cancelling

Name: Date:

Directions: Under each diagonal pair, list the factor pairs for both numbers (e.g., 1 × 8, 2 × 4) to find the GCF.

Pair 1 (Num 1 & Den 2) Pair 2 (Num 2 & Den 1)

\(\displaystyle {-\frac{2}{3} \times \frac{3}{8}}\)

Pair 1: 2 & 8 GCF:

Factor pairs (2):

Factor pairs (8):

Pair 2: 3 & 3 GCF:

Factor pairs (3):

\(\displaystyle {\frac{5}{6} \times \left(-\frac{3}{10}\right)}\)

Pair 1: 5 & 10 GCF:

Factor pairs (5):

Factor pairs (10):

Pair 2: 3 & 6 GCF:

Factor pairs (3):

Factor pairs (6):

\(\displaystyle {-\frac{4}{9} \times \left(-\frac{3}{8}\right)}\)

Pair 1: 4 & 8 GCF:

Factor pairs (4):

Factor pairs (8):

Pair 2: 3 & 9 GCF:

Factor pairs (3):

Factor pairs (9):

\(\displaystyle {\frac{7}{12} \times \frac{4}{21}}\)

Pair 1: 7 & 21 GCF:

Factor pairs (7):

Factor pairs (21):

Pair 2: 4 & 12 GCF:

Factor pairs (4):

Factor pairs (12):

\(\displaystyle {-\frac{5}{14} \times \frac{7}{15}}\)

Pair 1: 5 & 15 GCF:

Factor pairs (5):

Factor pairs (15):

Pair 2: 7 & 14 GCF:

Factor pairs (7):

Factor pairs (14):

\(\displaystyle {\frac{8}{15} \times \left(-\frac{5}{16}\right)}\)

Pair 1: 8 & 16 GCF:

Factor pairs (8):

Factor pairs (16):

Pair 2: 5 & 15 GCF:

Factor pairs (5):

Factor pairs (15):

\(\displaystyle {-\frac{9}{16} \times \left(-\frac{4}{27}\right)}\)

Pair 1: 9 & 27 GCF:

Factor pairs (9):

Factor pairs (27):

Pair 2: 4 & 16 GCF:

Factor pairs (4):

Factor pairs (16):

\(\displaystyle {-\frac{6}{25} \times \frac{10}{21}}\)

Pair 1: 6 & 21 GCF:

Factor pairs (6):

Factor pairs (21):

Cross Cancelling Fractions Answer Key

Multiplying Fractions with Cross Cancelling

Answer Key

Step-by-step cross-cancelling solutions for all 18 problems. 18-Question Edition

Ans: \(-\frac{2}{15}\)

\(\displaystyle {-\frac{1}{6} \times \frac{4}{5}}\)

Cancel 6 & 4 by 2 (3, 2) \(\rightarrow \mathbf{-\frac{2}{15}}\)

Ans: \(-\frac{1}{3}\)

\(\displaystyle {\frac{1}{2} \times \left(-\frac{2}{3}\right)}\)

Cancel 2 & 2 (1, 1) \(\rightarrow \mathbf{-\frac{1}{3}}\)

Ans: \(\frac{1}{5}\)

\(\displaystyle {-\frac{1}{3} \times \left(-\frac{3}{5}\right)}\)

Cancel 3 & 3 (1, 1) \(\rightarrow \mathbf{\frac{1}{5}}\)

Ans: \(\frac{7}{10}\)

\(\displaystyle {\frac{4}{5} \times \frac{7}{8}}\)

Cancel 4 & 8 by 4 (1, 2) \(\rightarrow \mathbf{\frac{7}{10}}\)

Ans: \(-\frac{1}{14}\)

\(\displaystyle {-\frac{4}{7} \times \frac{1}{8}}\)

Cancel 4 & 8 by 4 (1, 2) \(\rightarrow \mathbf{-\frac{1}{14}}\)

Ans: \(-\frac{1}{4}\)

\(\displaystyle {\frac{1}{3} \times \left(-\frac{3}{4}\right)}\)

Cancel 3 & 3 (1, 1) \(\rightarrow \mathbf{-\frac{1}{4}}\)

Ans: \(\frac{1}{3}\)

\(\displaystyle {-\frac{3}{7} \times \left(-\frac{7}{9}\right)}\)

Cancel 7 & 7 (1, 1); 3 & 9 (1, 3) \(\rightarrow \mathbf{\frac{1}{3}}\)

Ans: \(-\frac{1}{3}\)

\(\displaystyle {-\frac{2}{5} \times \frac{5}{6}}\)

Cancel 5 & 5 (1, 1); 2 & 6 (1, 3) \(\rightarrow \mathbf{-\frac{1}{3}}\)

Ans: \(\frac{1}{2}\)

\(\displaystyle {\frac{2}{3} \times \frac{3}{4}}\)

Cancel 3 & 3 (1, 1); 2 & 4 (1, 2) \(\rightarrow \mathbf{\frac{1}{2}}\)

Ans: \(\frac{1}{48}\)

\(\displaystyle {-\frac{5}{54} \times \left(-\frac{9}{40}\right)}\)

Cancel 5 & 40 (1, 8); 9 & 54 (1, 6) \(\rightarrow \mathbf{\frac{1}{48}}\)

Ans: \(-\frac{1}{63}\)

\(\displaystyle {\frac{6}{27} \times \left(-\frac{3}{42}\right)}\)

Cancel 6 & 42 (1, 7); 3 & 27 (1, 9) \(\rightarrow \mathbf{-\frac{1}{63}}\)

Ans: \(-\frac{1}{30}\)

\(\displaystyle {-\frac{9}{30} \times \frac{3}{27}}\)

Cancel 9 & 27 (1, 3); 3 & 3 (1, 1) \(\rightarrow \mathbf{-\frac{1}{30}}\)

Mixed Fractions Cancelling Worksheet

Multiplying Fractions with Cross Cancelling

Mixed Numbers & Factor Pairs Practice with Model Example

Name: Date:

Directions: Convert mixed numbers to improper fractions, list factor pairs to find the GCF, cross-cancel, and multiply. See #2 for Model Example

\(\displaystyle {-1\frac{1}{5} \times \frac{5}{8} =}\)

Your Work

1. Convert to improper fraction & cross-cancel:

2. Diagonal Factor Pairs:

Pair 1 (Factor pairs):

Pair 2 (Factor pairs):

3. Final Answer:

\(\displaystyle {2\frac{2}{3} \times \left(-\frac{9}{16}\right) =}\)

Example

1. Convert to improper fraction & set up:

\(2\frac{2}{3} = \frac{8}{3} \quad \rightarrow \quad \displaystyle {\frac{8}{3} \times \left(-\frac{9}{16}\right)}\)

2. Find diagonal factor pairs & GCF:

Pair: 8 & 16 (GCF = 8) 8: 1×8, 2×4 16: 1×16, 2×8, 4×4 \(\div 8 \rightarrow \mathbf{1 \text{ & } 2}\)

Pair: 3 & 9 (GCF = 3) 3: 1×3 9: 1×9, 3×3 \(\div 3 \rightarrow \mathbf{1 \text{ & } 3}\)

3. Multiply: \(-\frac{1 \times 3}{1 \times 2}\) Ans: \(-\frac{3}{2}\) or \(-1\frac{1}{2}\)

\(\displaystyle {-1\frac{7}{8} \times \left(-1\frac{1}{5}\right) =}\)

Your Work

1. Convert to improper fractions & cross-cancel:

2. Diagonal Factor Pairs:

Pair 1 (Factor pairs):

Pair 2 (Factor pairs):

3. Final Answer:

\(\displaystyle {\frac{7}{12} \times 2\frac{2}{7} =}\)

Your Work

1. Convert to improper fraction & cross-cancel:

2. Diagonal Factor Pairs:

Pair 1 (Factor pairs):

Pair 2 (Factor pairs):

3. Final Answer:

\(\displaystyle {-3\frac{1}{3} \times 1\frac{1}{20} =}\)

Your Work

1. Convert to improper fractions & cross-cancel:

2. Diagonal Factor Pairs:

Pair 1 (Factor pairs):

Pair 2 (Factor pairs):

3. Final Answer:

\(\displaystyle {-2\frac{1}{4} \times \left(-1\frac{7}{9}\right) =}\)

Your Work

1. Convert to improper fractions & cross-cancel:

2. Diagonal Factor Pairs:

Pair 1 (Factor pairs):

Pair 2 (Factor pairs):

3. Final Answer:

Fraction Multiplication Quiz

Fraction Multiplication & Cross-Cancelling Quiz

Determine factor pairs to test for diagonal cross-reduction before multiplying.

Name: Date:

Score: / 100

Directions (Part 1): List factor pairs for each diagonal pair to find the GCF. Cross-cancel if possible (\(\text{GCF} > 1\)), then multiply. Write answers in simplest form. 8 pts each

Q1 \(\displaystyle {\frac{3}{4} \times \frac{8}{9} =}\)

Answer:

Diagonal 1: (3 & 9) GCF:

Factors of 3:

Factors of 9:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (4 & 8) GCF:

Factors of 4:

Factors of 8:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

Q2 \(\displaystyle {-\frac{5}{6} \times \frac{3}{10} =}\)

Answer:

Diagonal 1: (5 & 10) GCF:

Factors of 5:

Factors of 10:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (3 & 6) GCF:

Factors of 3:

Factors of 6:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

Q3 \(\displaystyle {\frac{7}{12} \times \left(-\frac{4}{21}\right) =}\)

Answer:

Diagonal 1: (7 & 21) GCF:

Factors of 7:

Factors of 21:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (4 & 12) GCF:

Factors of 4:

Factors of 12:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

Q4 \(\displaystyle {-\frac{2}{7} \times \left(-\frac{5}{8}\right) =}\)

Answer:

Diagonal 1: (2 & 8) GCF:

Factors of 2:

Factors of 8:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (5 & 7) GCF:

Factors of 5:

Factors of 7:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

Q5 \(\displaystyle {\frac{9}{16} \times \frac{8}{15} =}\)

Answer:

Diagonal 1: (9 & 15) GCF:

Factors of 9:

Factors of 15:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (8 & 16) GCF:

Factors of 8:

Factors of 16:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

* Remember: Both diagonal pairs must be evaluated before multiplying across. Page 1 of 3

Fraction Multiplication Quiz Answer Key

Fraction Multiplication Quiz

Answer Key

Complete factor pairs, GCF identification, cross-cancellations, and solutions.

Teacher Reference Guide

Scoring Criteria (Q1–Q10): 3 pts for factor pairs & GCFs; 3 pts for cross-reduction work; 2 pts for correct simplified product (8 pts each). Total: 100 pts

Q1 \(\displaystyle {\frac{3}{4} \times \frac{8}{9} =}\)

Answer: \(\mathbf{\frac{2}{3}}\)

Diagonal 1: (3 & 9) GCF: 3

3: 1 × 3

9: 1 × 9, 3 × 3

Cross-reduce? [✓] Yes (\(\div 3 \rightarrow 1, 3\))

Diagonal 2: (4 & 8) GCF: 4

4: 1 × 4, 2 × 2

8: 1 × 8, 2 × 4

Cross-reduce? [✓] Yes (\(\div 4 \rightarrow 1, 2\))

Cross-Cancel & Solution:

\(\frac{1}{1} \times \frac{2}{3} = \frac{2}{3}\)

Q2 \(\displaystyle {-\frac{5}{6} \times \frac{3}{10} =}\)

Answer: \(\mathbf{-\frac{1}{4}}\)

Diagonal 1: (5 & 10) GCF: 5

5: 1 × 5

10: 1 × 10, 2 × 5

Cross-reduce? [✓] Yes (\(\div 5 \rightarrow 1, 2\))

Diagonal 2: (3 & 6) GCF: 3

3: 1 × 3

6: 1 × 6, 2 × 3

Cross-reduce? [✓] Yes (\(\div 3 \rightarrow 1, 2\))

Cross-Cancel & Solution:

\(-\frac{1}{2} \times \frac{1}{2} = -\frac{1}{4}\)

Q3 \(\displaystyle {\frac{7}{12} \times \left(-\frac{4}{21}\right) =}\)

Answer: \(\mathbf{-\frac{1}{9}}\)

Diagonal 1: (7 & 21) GCF: 7

7: 1 × 7

21: 1 × 21, 3 × 7

Cross-reduce? [✓] Yes (\(\div 7 \rightarrow 1, 3\))

Diagonal 2: (4 & 12) GCF: 4

4: 1 × 4, 2 × 2

12: 1 × 12, 2 × 6, 3 × 4

Cross-reduce? [✓] Yes (\(\div 4 \rightarrow 1, 3\))

Cross-Cancel & Solution:

\(\frac{1}{3} \times \left(-\frac{1}{3}\right) = -\frac{1}{9}\)

Q4 \(\displaystyle {-\frac{2}{7} \times \left(-\frac{5}{8}\right) =}\)

Answer: \(\mathbf{\frac{5}{28}}\)

Diagonal 1: (2 & 8) GCF: 2

2: 1 × 2

8: 1 × 8, 2 × 4

Cross-reduce? [✓] Yes (\(\div 2 \rightarrow 1, 4\))

Diagonal 2: (5 & 7) GCF: 1

5: 1 × 5

7: 1 × 7

Cross-reduce? [✓] No (\(\text{GCF}=1\), keep 5 & 7)

Cross-Cancel & Solution:

\(-\frac{1}{7} \times \left(-\frac{5}{4}\right) = +\frac{5}{28}\)

Q5 \(\displaystyle {\frac{9}{16} \times \frac{8}{15} =}\)

Fraction Division Slides

Math 6–7 Fraction Operations

Unit 4 • Lesson 2

Division by Fractions

Keep, Change, Flip

Mastering fraction division by turning division into familiar multiplication using reciprocals and cross-cancelling.

1

Understand K-C-F

2

The Reciprocal

3

Solve & Simplify

The Strategy

Keep • Change • Flip (KCF)

Step-by-Step

K

KEEP

Keep the first fraction exactly as it is.

First fraction: \( \frac{3}{4} \) Stays \( \frac{3}{4} \)

C

CHANGE

Change the division sign into multiplication.

Sign change: \( \div \;\rightarrow\; \times \) Multiply instead

F

FLIP

Flip the second fraction upside down.

Second fraction: \( \frac{2}{5} \;\rightarrow\; \frac{5}{2} \) Take the reciprocal

Key takeaway: Never actually divide! Turn division into fraction multiplication.

Official Mathematical Language

Multiply by the Reciprocal

Mathematical Definition

The Official Math Rule

"Keep Change Flip" is our helpful shortcut. The official mathematical term is:

"To divide by a fraction, multiply by its reciprocal."

\( \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} \)

Why? Any number multiplied by its reciprocal equals 1: \( \frac{2}{5} \times \frac{5}{2} = 1 \)

Finding the Reciprocal

Swap the numerator and denominator:

\( \frac{3}{5} \) \( \frac{5}{3} \)

\( \frac{1}{8} \) \( \frac{8}{1} = 8 \)

\( 6 = \frac{6}{1} \) \( \frac{1}{6} \)

Every whole number like 6 is \( \frac{6}{1} \), so its reciprocal is \( \frac{1}{6} \).

When tests or textbooks ask for the formal rule, say: "Multiply by the reciprocal."

Quick Check

Practice: Finding the Reciprocal

Your Turn

What is the reciprocal of each number? Write your answer on your board or paper!

1. Proper Fraction

\( \frac{4}{7} \)

Reciprocal:

?

2. Unit Fraction

\( \frac{1}{9} \)

Reciprocal:

Dividing Fractions Worksheet

Dividing Fractions with Keep Change Flip

Name:

Multiply by the reciprocal (Keep Change Flip). Cross-cancel where possible. Simplify answers completely.

\(\displaystyle {\frac{2}{3} \div \frac{4}{9} =}\)

\(\displaystyle {\frac{5}{8} \div \left(-\frac{15}{16}\right) =}\)

\(\displaystyle {-\frac{7}{10} \div \left(-\frac{14}{25}\right) =}\)

\(\displaystyle {-\frac{4}{15} \div \frac{8}{21} =}\)

\(\displaystyle {\frac{9}{14} \div \left(-\frac{3}{7}\right) =}\)

\(\displaystyle {-\frac{6}{25} \div \left(-\frac{9}{20}\right) =}\)

\(\displaystyle {\frac{8}{27} \div \frac{4}{9} =}\)

\(\displaystyle {-\frac{5}{12} \div 10 =}\)

\(\displaystyle {12 \div \left(-\frac{8}{5}\right) =}\)

\(\displaystyle {-1\frac{1}{4} \div \frac{5}{6} =}\)

\(\displaystyle {2\frac{2}{5} \div \left(-1\frac{1}{15}\right) =}\)

\(\displaystyle {-3\frac{1}{3} \div \left(-2\frac{2}{9}\right) =}\)

Answers

Dividing Fractions Answer Key

Dividing Fractions with Keep Change Flip

Answer Key

Step-by-step KCF and cross-cancelling solutions. Answers provided as simplified fractions and mixed numbers.

Ans: \(\frac{3}{2}\) or \(1\frac{1}{2}\)

\(\displaystyle {\frac{2}{3} \times \frac{9}{4}}\)

Cancel 2 & 4 (1, 2); 3 & 9 (1, 3)
\(= \frac{1 \times 3}{1 \times 2} = \mathbf{\frac{3}{2}\text{ or }1\frac{1}{2}}\)

Ans: \(-\frac{2}{3}\)

\(\displaystyle {\frac{5}{8} \times \left(-\frac{16}{15}\right)}\)

Cancel 5 & 15 (1, 3); 8 & 16 (1, 2)
\(= -\frac{1 \times 2}{1 \times 3} = \mathbf{-\frac{2}{3}}\)

Ans: \(\frac{5}{4}\) or \(1\frac{1}{4}\)

\(\displaystyle {-\frac{7}{10} \times \left(-\frac{25}{14}\right)}\)

Cancel 7 & 14 (1, 2); 10 & 25 (2, 5)
\(= \frac{1 \times 5}{2 \times 2} = \mathbf{\frac{5}{4}\text{ or }1\frac{1}{4}}\)

Ans: \(-\frac{7}{10}\)

\(\displaystyle {-\frac{4}{15} \times \frac{21}{8}}\)

Cancel 4 & 8 (1, 2); 15 & 21 (5, 7)
\(= -\frac{1 \times 7}{5 \times 2} = \mathbf{-\frac{7}{10}}\)

Ans: \(-\frac{3}{2}\) or \(-1\frac{1}{2}\)

\(\displaystyle {\frac{9}{14} \times \left(-\frac{7}{3}\right)}\)

Cancel 9 & 3 (3, 1); 14 & 7 (2, 1)
\(= -\frac{3 \times 1}{2 \times 1} = \mathbf{-\frac{3}{2}\text{ or }-1\frac{1}{2}}\)

Ans: \(\frac{8}{15}\)

\(\displaystyle {-\frac{6}{25} \times \left(-\frac{20}{9}\right)}\)

Cancel 6 & 9 (2, 3); 25 & 20 (5, 4)
\(= \frac{2 \times 4}{5 \times 3} = \mathbf{\frac{8}{15}}\)

Ans: \(\frac{2}{3}\)

\(\displaystyle {\frac{8}{27} \times \frac{9}{4}}\)

Cancel 8 & 4 (2, 1); 27 & 9 (3, 1)
\(= \frac{2 \times 1}{3 \times 1} = \mathbf{\frac{2}{3}}\)

Ans: \(-\frac{1}{24}\)

\(\displaystyle {-\frac{5}{12} \times \frac{1}{10}}\)

Cancel 5 & 10 (1, 2)
\(= -\frac{1 \times 1}{12 \times 2} = \mathbf{-\frac{1}{24}}\)

Ans: \(-\frac{15}{2}\) or \(-7\frac{1}{2}\)

\(\displaystyle {\frac{12}{1} \times \left(-\frac{5}{8}\right)}\)

Cancel 12 & 8 (3, 2)
\(= -\frac{3 \times 5}{1 \times 2} = \mathbf{-\frac{15}{2}\text{ or }-7\frac{1}{2}}\)

Ans: \(-\frac{3}{2}\) or \(-1\frac{1}{2}\)

\(\displaystyle {-\frac{5}{4} \times \frac{6}{5}}\)

Ans: \(\frac{9}{4}\) or \(2\frac{1}{4}\)

\(\displaystyle {\left(-\frac{15}{8}\right) \times \left(-\frac{6}{5}\right)}\)

Cancel 15 & 5 (3, 1); 6 & 8 (3, 4)
\(= \frac{3 \times 3}{4 \times 1} = \mathbf{\frac{9}{4}}\text{ or }\mathbf{2\frac{1}{4}}\)

Ans: \(-\frac{7}{2}\) or \(-3\frac{1}{2}\)

\(\displaystyle {\left(-\frac{10}{3}\right) \times \frac{21}{20}}\)

Cancel 10 & 20 (1, 2); 3 & 21 (1, 7)
\(= -\frac{1 \times 7}{1 \times 2} = \mathbf{-\frac{7}{2}}\text{ or }\mathbf{-3\frac{1}{2}}\)

Answers

\(-\frac{1}{4}\)
\(-\frac{1}{4}\)
\(\frac{1}{6}\)
\(\frac{1}{9}\)
\(-\frac{1}{6}\)
\(-\frac{1}{6}\)
\(\frac{1}{12}\)
\(-\frac{4}{35}\)
\(-\frac{3}{4}\)
\(-\frac{3}{2}\) / \(-1\frac{1}{2}\)
\(\frac{9}{4}\) / \(2\frac{1}{4}\)
\(-\frac{7}{2}\) / \(-3\frac{1}{2}\)

Pair 2: 10 & 25 GCF:

Factor pairs (10):

Factor pairs (25):

\(-1\frac{1}{5} \rightarrow -\frac{6}{5} \times \frac{5}{8}\)

Pair 1: 6 & 8 GCF:

Factor pairs (6):

Factor pairs (8):

Pair 2: 5 & 5 GCF:

Factor pairs (5):

\(2\frac{2}{3} \rightarrow \frac{8}{3} \times \left(-\frac{9}{16}\right)\)

Pair 1: 8 & 16 GCF:

Factor pairs (8):

Factor pairs (16):

Pair 2: 9 & 3 GCF:

Factor pairs (9):

Factor pairs (3):

\(-\frac{15}{8} \times \left(-\frac{6}{5}\right)\)

Pair 1: 15 & 5 GCF:

Factor pairs (15):

Factor pairs (5):

Pair 2: 6 & 8 GCF:

Factor pairs (6):

Factor pairs (8):

\(-\frac{10}{3} \times \frac{21}{20}\)

Pair 1: 10 & 20 GCF:

Factor pairs (10):

Factor pairs (20):

Pair 2: 21 & 3 GCF:

Factor pairs (21):

Factor pairs (3):

* For mixed numbers (9–12), convert to improper fractions first to reveal the diagonal numerator and denominator pairs. Factor Pairs Organizer

Ans: \(-\frac{25}{7}\)

\(\displaystyle {-\frac{5}{2} \times \frac{10}{7}}\)

Cancel 2 & 10 by 2 (1, 5) \(\rightarrow \mathbf{-\frac{25}{7}\text{ or }-3\frac{4}{7}}\)

Ans: \(-\frac{65}{6}\)

\(\displaystyle {\frac{10}{3} \times \left(-\frac{13}{4}\right)}\)

Cancel 10 & 4 by 2 (5, 2) \(\rightarrow \mathbf{-\frac{65}{6}\text{ or }-10\frac{5}{6}}\)

Ans: \(\frac{85}{14}\)

\(\displaystyle {-\frac{17}{8} \times \left(-\frac{20}{7}\right)}\)

Cancel 8 & 20 by 4 (2, 5) \(\rightarrow \mathbf{\frac{85}{14}\text{ or }6\frac{1}{14}}\)

Ans: \(-\frac{33}{14}\)

\(\displaystyle {-\frac{11}{8} \times \frac{12}{7}}\)

Cancel 8 & 12 by 4 (2, 3) \(\rightarrow \mathbf{-\frac{33}{14}\text{ or }-2\frac{5}{14}}\)

Ans: \(-\frac{5}{4}\)

\(\displaystyle {\frac{7}{8} \times \left(-\frac{10}{7}\right)}\)

Cancel 7 & 7 (1, 1); 8 & 10 (4, 5) \(\rightarrow \mathbf{-\frac{5}{4}\text{ or }-1\frac{1}{4}}\)

Ans: \(\frac{25}{18}\)

\(\displaystyle {-\frac{10}{9} \times \left(-\frac{5}{4}\right)}\)

Cancel 10 & 4 by 2 (5, 2) \(\rightarrow \mathbf{\frac{25}{18}\text{ or }1\frac{7}{18}}\)

Answers

\(-\frac{2}{15}\)
\(-\frac{1}{3}\)
\(\frac{1}{5}\)
\(\frac{7}{10}\)
\(-\frac{1}{14}\)
\(-\frac{1}{4}\)
\(\frac{1}{3}\)
\(-\frac{1}{3}\)
\(\frac{1}{2}\)
\(\frac{1}{48}\)
\(-\frac{1}{63}\)
\(-\frac{1}{30}\)
\(-\frac{25}{7}\) / \(-3\frac{4}{7}\)
\(-\frac{65}{6}\) / \(-10\frac{5}{6}\)
\(\frac{85}{14}\) / \(6\frac{1}{14}\)
\(-\frac{33}{14}\) / \(-2\frac{5}{14}\)
\(-\frac{5}{4}\) / \(-1\frac{1}{4}\)
\(\frac{25}{18}\) / \(1\frac{7}{18}\)

Answers

\(-\frac{3}{2}\)

* Use model problem #2 as your guide to convert mixed numbers, find diagonal factor pairs, and cross-cancel. 6-Problem Practice Sheet

Fraction Multiplication Quiz (Questions 6–10)

Name: Questions 9 & 10 contain mixed numbers

Q6 \(\displaystyle {-\frac{6}{25} \times \frac{15}{16} =}\)

Answer:

Diagonal 1: (6 & 16) GCF:

Factors of 6:

Factors of 16:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (15 & 25) GCF:

Factors of 15:

Factors of 25:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

Q7 \(\displaystyle {-\frac{14}{15} \times \left(-\frac{5}{21}\right) =}\)

Answer:

Diagonal 1: (14 & 21) GCF:

Factors of 14:

Factors of 21:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (5 & 15) GCF:

Factors of 5:

Factors of 15:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

Q8 \(\displaystyle {\frac{7}{10} \times \frac{5}{11} =}\)

Answer:

Diagonal 1: (7 & 11) GCF:

Factors of 7:

Factors of 11:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (5 & 10) GCF:

Factors of 5:

Factors of 10:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

Q9 \(\displaystyle {-1\frac{1}{5} \times \frac{5}{9} =}\) Improper:

Answer:

Diagonal 1: (6 & 9) GCF:

Factors of 6:

Factors of 9:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (5 & 5) GCF:

Factors of 5:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

Q10 \(\displaystyle {1\frac{7}{8} \times \left(-1\frac{1}{5}\right) =}\) Improper:

Answer:

Diagonal 1: (15 & 5) GCF:

Factors of 15:

Factors of 5:

Cross-reduce? [ ] Yes [ ] No

Diagonal 2: (6 & 8) GCF:

Factors of 6:

Factors of 8:

Cross-reduce? [ ] Yes [ ] No

Cross-Cancel & Show Work:

* Make sure all signs (\(+\) or \(-\)) and simplest fraction forms are accurately shown. Page 2 of 3

Part 2: Integer Multiplication & Mixed Number Reference

Determine the sign of integer products, then study the mixed number transformation guide.

Name: Q11–Q15: 4 pts each

Questions 11–15: Determine whether each integer product is POSITIVE or NEGATIVE, then solve. Check exactly one box per question

Q11 \((-8) \times (-7)\)

Sign of final answer: