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Fraction Fusion

A comprehensive lesson mastering fraction arithmetic and complex fraction simplification with structured worked models, student practice, and fully worked solutions.

MMMonica Michalak

Fraction Fusion Practice Worksheet

Math Mastery Fractions Unit

Fraction Fusion: Operations & Strategy

Name:

Date: Period:

1

Addition & Subtraction (Common Denominators)

LCM Strategy

Worked Example Model

Model A (Addition with unlike denominators):

Find \(\frac{2}{3} + \frac{1}{6}\). LCD for 3 and 6 is 6.

\(\frac{2 \times 2}{3 \times 2} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}\)

Model B (Mixed number subtraction):

Find \(2\frac{1}{4} - \frac{5}{6}\). Convert to improper fractions: LCD is 12.

\(\frac{9}{4} - \frac{5}{6} = \frac{27}{12} - \frac{10}{12} = \frac{17}{12} = 1\frac{5}{12}\)

Problem 1 Add

\(\frac{3}{8} + \frac{5}{12}\)

Answer:

Problem 2 Subtract

\(\frac{5}{6} - \frac{3}{8}\)

Answer:

Problem 3 Mixed Number

\(1\frac{2}{3} + \frac{3}{4}\)

Answer:

2

Multiplication of Fractions (Cross-Simplifying)

Cross-Cancel First

Worked Example Model

Model C (Cross-simplifying factors):

Find \(\frac{3}{8} \times \frac{4}{9}\). Divide 3 and 9 by 3; divide 4 and 8 by 4.

\(\frac{3}{8} \times \frac{4}{9} = \frac{1}{2} \times \frac{1}{3} = \frac{1 \times 1}{2 \times 3} = \frac{1}{6}\)

Model D (Multiplying with a mixed number):

Find \(1\frac{1}{2} \times \frac{4}{5}\). Convert \(1\frac{1}{2} = \frac{3}{2}\) then cross-cancel 2 & 4.

\(\frac{3}{2} \times \frac{4}{5} = \frac{3}{1} \times \frac{2}{5} = \frac{6}{5} = 1\frac{1}{5}\)

Problem 4 Multiply

\(\frac{5}{12} \times \frac{8}{15}\)

Answer:

Problem 5 Mixed Number

\(1\frac{3}{5} \times \frac{5}{12}\)

Answer:

Problem 6 Cross-Cancel

\(\frac{7}{9} \times 2\frac{1}{7}\)

Answer:

Always simplify all answers to lowest terms or proper mixed numbers. Page 1 of 3

Fraction Fusion | Part II: Division & Complex Fractions

Student Copy

3

Division of Fractions (Keep, Change, Flip)

Rule: Multiply by Reciprocal

Worked Example Model

Model E (Fraction Division):

Find \(\frac{5}{6} \div \frac{2}{3}\). Keep \(\frac{5}{6}\), change \(\div\) to \(\times\), flip to \(\frac{3}{2}\).

\(\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{5 \times 1}{2 \times 2} = \frac{5}{4} = 1\frac{1}{4}\)

Model F (Dividing with Mixed Numbers):

Find \(2\frac{2}{5} \div 1\frac{1}{5}\). Convert both to improper fractions first.

\(\frac{12}{5} \div \frac{6}{5} = \frac{12}{5} \times \frac{5}{6} = \frac{2 \times 1}{1 \times 1} = 2\)

Problem 7 Divide & Simplify

\(\frac{7}{10} \div \frac{14}{15}\)

Answer:

Problem 8 Mixed Number Division

\(3\frac{1}{3} \div \frac{5}{6}\)

Answer:

4

Simplifying Complex Fractions

Main Bar = Division Sign

Two Core Strategies & Worked Models

Model G (Single fraction over single fraction):

Simplify \(\frac{\frac{2}{3}}{\frac{5}{9}}\). Rewrite as horizontal division.

\(\frac{\frac{2}{3}}{\frac{5}{9}} = \frac{2}{3} \div \frac{5}{9} = \frac{2}{3} \times \frac{9}{5} = \frac{2 \times 3}{1 \times 5} = \frac{6}{5} = 1\frac{1}{5}\)

Model H (Operations inside numerator/denominator):

Simplify \(\frac{\frac{1}{2} + \frac{1}{3}}{\frac{5}{6}}\). Combine the numerator first.

Numerator: \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\) → \(\frac{\frac{5}{6}}{\frac{5}{6}} = \frac{5}{6} \times \frac{6}{5} = 1\)

Problem 9 Basic Complex Fraction

\[\frac{\frac{3}{4}}{\frac{9}{16}}\]

Answer:

Problem 10 Difference in Numerator

\[\frac{\frac{4}{5} - \frac{1}{2}}{\frac{3}{10}}\]

Answer:

Problem 11 Sum in Numerator

\[\frac{\frac{1}{3} + \frac{1}{2}}{\frac{5}{8}}\]

Answer:

Problem 12 (Challenge) Compound Operations

\[\frac{1 - \frac{1}{4}}{\frac{3}{8} + \frac{1}{2}}\]

Answer:

Tip: Multiply numerator and denominator by the overall LCD to clear all inner fractions in one step! Page 2 of 3

Independent Mastery Part III • Unassisted Practice

Independent Challenge: All Operations & Complex Fractions

No Hints Provided Problems 13–24

Solve each problem independently. Simplify completely to lowest terms or proper mixed numbers.

1 pt each

Section A: Addition & Subtraction

Independent

Problem 13

\(\frac{5}{9} + \frac{7}{12}\)

Ans:

Problem 14

\(\frac{11}{15} - \frac{3}{10}\)

Ans:

Problem 15

\(3\frac{1}{6} - 1\frac{4}{9}\)

Ans:

Problem 16

\(\frac{1}{2} - \frac{1}{3} + \frac{3}{4}\)

Ans:

Section B: Multiplication & Division

Independent

Problem 17

\(\frac{9}{14} \times \frac{21}{27}\)

Ans:

Problem 18

\(2\frac{2}{3} \times 1\frac{7}{8}\)

Ans:

Problem 19

\(\frac{8}{15} \div \frac{12}{25}\)

Ans:

Problem 20

\(4\frac{1}{2} \div 3\frac{3}{8}\)

Ans:

Section C: Complex Fractions

Independent

Problem 21 Quotient

\[\frac{\frac{5}{18}}{\frac{10}{27}}\]

Answer:

Problem 22 Difference in Numerator

\[\frac{\frac{5}{6} - \frac{1}{4}}{\frac{7}{12}}\]

Answer:

Problem 23 Compound Operations

\[\frac{\frac{2}{5} + \frac{3}{10}}{1 - \frac{1}{6}}\]

Answer:

Problem 24 (Mastery Challenge) Multi-Step

\[\frac{\frac{3}{4} + \frac{2}{3}}{\frac{5}{6} - \frac{1}{8}}\]

Answer:

Independent Assessment • Fraction Fusion Page 3 of 3

Fraction Fusion Solution Key

Answer Key & Solutions Teacher / Self-Check Guide

Fraction Fusion: Worked Solutions (Part I)

Scoring Total

24 Problems Total

1

Addition & Subtraction Solutions (Problems 1–3)

Finding LCD & Equivalent Fractions

Problem 1 LCD = 24

\(\frac{3}{8} + \frac{5}{12}\)

• \(\frac{3 \times 3}{8 \times 3} = \frac{9}{24}\)

• \(\frac{5 \times 2}{12 \times 2} = \frac{10}{24}\)

• Sum: \(\frac{9 + 10}{24} = \frac{19}{24}\)

Answer: \(\frac{19}{24}\)

Problem 2 LCD = 24

\(\frac{5}{6} - \frac{3}{8}\)

• \(\frac{5 \times 4}{6 \times 4} = \frac{20}{24}\)

• \(\frac{3 \times 3}{8 \times 3} = \frac{9}{24}\)

• Diff: \(\frac{20 - 9}{24} = \frac{11}{24}\)

Answer: \(\frac{11}{24}\)

Problem 3 LCD = 12

\(1\frac{2}{3} + \frac{3}{4}\)

• \(1\frac{2}{3} = \frac{5}{3} = \frac{20}{12}\)

• \(\frac{3}{4} = \frac{9}{12}\)

• Sum: \(\frac{20 + 9}{12} = \frac{29}{12} = 2\frac{5}{12}\)

Answer: \(\frac{29}{12}\) or \(2\frac{5}{12}\)

Common Pitfall: Remind students never to add or subtract denominators (e.g., \(\frac{3}{8} + \frac{5}{12} \neq \frac{8}{20}\)). Denominators must be matched using common multiples first.

2

Multiplication Solutions (Problems 4–6)

Cross-Simplification Walkthrough

Problem 4 Simplify

\(\frac{5}{12} \times \frac{8}{15}\)

• Cancel 5 & 15 by 5: \(\to 1\) & \(3\)

• Cancel 8 & 12 by 4: \(\to 2\) & \(3\)

• Multiply: \(\frac{1 \times 2}{3 \times 3} = \frac{2}{9}\)

Answer: \(\frac{2}{9}\)

Problem 5 Mixed #

\(1\frac{3}{5} \times \frac{5}{12}\)

• Convert: \(1\frac{3}{5} = \frac{8}{5}\)

• Cancel 5 & 5 by 5: \(\to 1\) & \(1\)

• Cancel 8 & 12 by 4: \(\to 2\) & \(3\)

• Multiply: \(\frac{2 \times 1}{1 \times 3} = \frac{2}{3}\)

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

Problem 6 Cross-Cancel

\(\frac{7}{9} \times 2\frac{1}{7}\)

• Convert: \(2\frac{1}{7} = \frac{15}{7}\)

• Cancel 7 & 7 by 7: \(\to 1\) & \(1\)

• Cancel 15 & 9 by 3: \(\to 5\) & \(3\)

• Result: \(\frac{1 \times 5}{3 \times 1} = \frac{5}{3} = 1\frac{2}{3}\)

Answer: \(\frac{5}{3}\) or \(1\frac{2}{3}\)

Teaching Note: Cross-simplifying before multiplying keeps numbers manageable and prevents errors when reducing large fractions at the end (e.g. \(40/180\) simplified vs. \(2/9\) directly).

Fraction Fusion Solutions • Grade 6-8 Pre-Algebra / Algebra 1 Page 1 of 3

Fraction Fusion Solutions | Part II: Division & Complex Fractions

Scaffolded Key

3

Division Solutions (Problems 7–8)

Multiply by Reciprocal

Problem 7 \(\frac{7}{10} \div \frac{14}{15}\)

• Reciprocal: \(\frac{7}{10} \times \frac{15}{14}\)

• Cancel 7 & 14: \(\to 1\) & \(2\); Cancel 10 & 15: \(\to 2\) & \(3\)

• Multiply: \(\frac{1 \times 3}{2 \times 2} = \frac{3}{4}\)

Answer: \(\frac{3}{4}\)

Problem 8 \(3\frac{1}{3} \div \frac{5}{6}\)

• Convert: \(3\frac{1}{3} = \frac{10}{3}\) → \(\frac{10}{3} \times \frac{6}{5}\)

• Cancel 10 & 5: \(\to 2\) & \(1\); Cancel 6 & 3: \(\to 2\) & \(1\)

• Multiply: \(\frac{2 \times 2}{1 \times 1} = 4\)

Answer: \(4\)

4

Complex Fraction Solutions (Problems 9–12)

Multi-Method Breakdown

Problem 9: \(\frac{\frac{3}{4}}{\frac{9}{16}}\)

Method 1 (Division Sign):

\(\frac{3}{4} \div \frac{9}{16} = \frac{3}{4} \times \frac{16}{9} = \frac{1 \times 4}{1 \times 3} = \frac{4}{3} = 1\frac{1}{3}\)

Method 2 (Clear by LCD 16):

\(\frac{\frac{3}{4} \times 16}{\frac{9}{16} \times 16} = \frac{12}{9} = \frac{4}{3} = 1\frac{1}{3}\)

Answer: \(\frac{4}{3}\) or \(1\frac{1}{3}\)

Problem 10: \(\frac{\frac{4}{5} - \frac{1}{2}}{\frac{3}{10}}\)

Step 1: Simplify Numerator:

\(\frac{4}{5} - \frac{1}{2} = \frac{8}{10} - \frac{5}{10} = \frac{3}{10}\)

Step 2: Divide by Denominator:

\(\frac{\frac{3}{10}}{\frac{3}{10}} = \frac{3}{10} \times \frac{10}{3} = 1\)

Answer: \(1\)

Problem 11: \(\frac{\frac{1}{3} + \frac{1}{2}}{\frac{5}{8}}\)

Step 1: Simplify Numerator:

\(\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}\)

Step 2: Multiply by Reciprocal:

\(\frac{5}{6} \div \frac{5}{8} = \frac{5}{6} \times \frac{8}{5} = \frac{1 \times 4}{3 \times 1} = \frac{4}{3} = 1\frac{1}{3}\)

Answer: \(\frac{4}{3}\) or \(1\frac{1}{3}\)

Problem 12: \(\frac{1 - \frac{1}{4}}{\frac{3}{8} + \frac{1}{2}}\)

Step 1: Numerator: \(1 - \frac{1}{4} = \frac{3}{4}\)

Step 2: Denominator: \(\frac{3}{8} + \frac{4}{8} = \frac{7}{8}\)

Step 3: Invert & Multiply:

\(\frac{3}{4} \times \frac{8}{7} = \frac{3 \times 2}{1 \times 7} = \frac{6}{7}\)

Answer: \(\frac{6}{7}\)

Pedagogical Tip for Complex Fractions:

Students struggling with multi-step complex fractions often try to flip terms prematurely. Emphasize that the numerator and denominator must each be condensed into a single fraction before applying "Keep, Change, Flip", OR students can multiply the entire numerator and denominator by the overall LCD (\(\text{LCD of } 4, 8, 2 = 8\) in Problem 12: \(\frac{8 - 2}{3 + 4} = \frac{6}{7}\)).

Fraction Fusion Solutions • Grade 6-8 Pre-Algebra / Algebra 1 Page 2 of 3

Part III Solutions Independent Mastery Key

Worked Solutions: Problems 13–24

Full Solutions 12 Independent Items

Section A: Addition & Subtraction

Prob 13: \(\frac{5}{9} + \frac{7}{12}\)

LCD = 36
\(\frac{20}{36} + \frac{21}{36} = \frac{41}{36}\)

\(\frac{41}{36}\) or \(1\frac{5}{36}\)

Prob 14: \(\frac{11}{15} - \frac{3}{10}\)

LCD = 30
\(\frac{22}{30} - \frac{9}{30} = \frac{13}{30}\)

\(\frac{13}{30}\)

Prob 15: \(3\frac{1}{6} - 1\frac{4}{9}\)

\(\frac{19}{6} - \frac{13}{9} = \frac{57 - 26}{18}\)
\(= \frac{31}{18}\)

\(\frac{31}{18}\) or \(1\frac{13}{18}\)

Prob 16: \(\frac{1}{2} - \frac{1}{3} + \frac{3}{4}\)

LCD = 12
\(\frac{6 - 4 + 9}{12} = \frac{11}{12}\)

\(\frac{11}{12}\)

Section B: Multiplication & Division

Prob 17: \(\frac{9}{14} \times \frac{21}{27}\)

\(\frac{9}{27} \to \frac{1}{3}\); \(\frac{21}{14} \to \frac{3}{2}\)
\(\frac{1 \times 3}{2 \times 3} = \frac{1}{2}\)

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

Prob 18: \(2\frac{2}{3} \times 1\frac{7}{8}\)

\(\frac{8}{3} \times \frac{15}{8}\)
Cancel 8s; \(\frac{15}{3} = 5\)

\(5\)

Prob 19: \(\frac{8}{15} \div \frac{12}{25}\)

\(\frac{8}{15} \times \frac{25}{12} = \frac{2 \times 5}{3 \times 3}\)
\(= \frac{10}{9}\)

\(\frac{10}{9}\) or \(1\frac{1}{9}\)

Prob 20: \(4\frac{1}{2} \div 3\frac{3}{8}\)

\(\frac{9}{2} \times \frac{8}{27} = \frac{1 \times 4}{1 \times 3}\)
\(= \frac{4}{3}\)

\(\frac{4}{3}\) or \(1\frac{1}{3}\)

Section C: Complex Fractions

Problem 21: \(\frac{\frac{5}{18}}{\frac{10}{27}}\)

• Rewrite: \(\frac{5}{18} \div \frac{10}{27} = \frac{5}{18} \times \frac{27}{10}\)

• Cancel 5 & 10 \(\to 1\) & \(2\); Cancel 27 & 18 by 9 \(\to 3\) & \(2\)

• Multiply: \(\frac{1 \times 3}{2 \times 2} = \frac{3}{4}\)

Answer: \(\frac{3}{4}\)

Problem 22: \(\frac{\frac{5}{6} - \frac{1}{4}}{\frac{7}{12}}\)

• Numerator: \(\frac{10}{12} - \frac{3}{12} = \frac{7}{12}\)

• Divide: \(\frac{7}{12} \div \frac{7}{12} = \frac{7}{12} \times \frac{12}{7} = 1\)

• Alternate: Multiply top & bottom by 12: \(\frac{10 - 3}{7} = \frac{7}{7} = 1\)

Answer: \(1\)

Problem 23: \(\frac{\frac{2}{5} + \frac{3}{10}}{1 - \frac{1}{6}}\)

• Numerator: \(\frac{4}{10} + \frac{3}{10} = \frac{7}{10}\)

• Denominator: \(\frac{6}{6} - \frac{1}{6} = \frac{5}{6}\)

• Divide: \(\frac{7}{10} \times \frac{6}{5} = \frac{7 \times 3}{5 \times 5} = \frac{21}{25}\)

Answer: \(\frac{21}{25}\)

Problem 24: \(\frac{\frac{3}{4} + \frac{2}{3}}{\frac{5}{6} - \frac{1}{8}}\)

• Numerator: \(\frac{9 + 8}{12} = \frac{17}{12}\)

• Denominator: \(\frac{20 - 3}{24} = \frac{17}{24}\)

• Divide: \(\frac{17}{12} \times \frac{24}{17} = \frac{24}{12} = 2\)

Answer: \(2\)

Fraction Fusion Solutions • Grade 6-8 Pre-Algebra / Algebra 1 Page 3 of 3

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