Fractional Calculations. Grade 7 Mathematics: Fractional Calculations - Computing with Fractions and Mixed Numbers Subtopic Navigator Understanding Fractional Operations Adding Fractions and Mixed Numbers Subtracting Fractions and Mixed Numbers Multiplying Fractions and Mixed Numbers Dividing Fractions and Mixed Numbers Mixed Operations with Fractions Complex Fraction Simplification Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Add and subtract fractions and mixed numbers with unlike denominators Multiply fractions and mixed numbers using proper techniques Divide fractions and mixed numbers using reciprocal multiplication Simplify complex fractions and mixed operation expressions Apply fractional operations to solve multi-step word problems Develop strategies for efficient fraction computation Fractional Operations Mastery of fractional calculations is essential for advanced mathematics and real-world problem-solving. Fractions represent parts of wholes, and operations with fractions follow specific rules that maintain mathematical consistency. Understanding these operations requires careful attention to denominators, simplification, and conversion between mixed numbers and improper fractions. Adding Fractions and Mixed Numbers When adding fractions with different denominators, we must first find a common denominator, usually the Least Common Multiple (LCM) of the denominators. Mixed numbers should be converted to improper fractions or handled by adding whole numbers and fractions separately. Example 1: Complex Fraction Addition Calculate: $frac{3}{4} + frac{5}{6} + frac{7}{8}$ Solution: Find LCM of denominators 4, 6, 8: Prime factors: $4 = 2^2$, $6 = 2 × 3$, $8 = 2^3$ LCM = $2^3 × 3 = 24$ Convert to common denominator: $frac{3}{4} = frac{18}{24}$ $frac{5}{6} = frac{20}{24}$ $frac{7}{8} = frac{21}{24}$ Add: $frac{18}{24} + frac{20}{24} + frac{21}{24} = frac{59}{24}$ Convert to mixed number: $frac{59}{24} = 2frac{11}{24}$ Example 2: Mixed Number Addition with Regrouping Calculate: $3frac{2}{5} + 4frac{7}{10} + 2frac{3}{4}$ Solution: Method 1: Convert to improper fractions $3frac{2}{5} = frac{17}{5}$ $4frac{7}{10} = frac{47}{10}$ $2frac{3}{4} = frac{11}{4}$ Find LCM of 5, 10, 4: LCM = 20 Convert: $frac{17}{5} = frac{68}{20}$, $frac{47}{10} = frac{94}{20}$, $frac{11}{4} = frac{55}{20}$ Add: $frac{68}{20} + frac{94}{20} + frac{55}{20} = frac{217}{20} = 10frac{17}{20}$ Method 2: Add whole numbers and fractions separately Whole numbers: $3 + 4 + 2 = 9$ Fractions: $frac{2}{5} + frac{7}{10} + frac{3}{4}$ LCM of 5, 10, 4 = 20 $frac{8}{20} + frac{14}{20} + frac{15}{20} = frac{37}{20} = 1frac{17}{20}$ Total: $9 + 1frac{17}{20} = 10frac{17}{20}$ Addition Problems Calculate: $frac{5}{6} + frac{7}{9} + frac{3}{4}$ Find: $2frac{3}{8} + 3frac{5}{6} + 1frac{7}{12}$ Add: $frac{11}{15} + frac{7}{10} + frac{4}{5}$ What is the sum of $4frac{2}{3}$, $3frac{5}{6}$, and $2frac{7}{9}$? If $frac{a}{b} + frac{c}{d} = frac{13}{12}$ and $frac{a}{b} = frac{3}{4}$, find $frac{c}{d}$ Subtracting Fractions and Mixed Numbers Subtraction of fractions follows similar rules to addition, requiring common denominators. With mixed numbers, we may need to regroup (borrow) from the whole number when the fractional part being subtracted is larger than the fractional part being subtracted from. Example 1: Complex Fraction Subtraction Calculate: $frac{7}{8} - frac{2}{3} + frac{1}{6}$ Solution: Since we have addition and subtraction, work left to right with common denominator: LCM of 8, 3, 6 = 24 Convert: $frac{7}{8} = frac{21}{24}$, $frac{2}{3} = frac{16}{24}$, $frac{1}{6} = frac{4}{24}$ Perform operations in order: $frac{21}{24} - frac{16}{24} = frac{5}{24}$ $frac{5}{24} + frac{4}{24} = frac{9}{24} = frac{3}{8}$ Example 2: Mixed Number Subtraction with Regrouping Calculate: $5frac{1}{4} - 2frac{5}{6}$ Solution: Method 1: Convert to improper fractions $5frac{1}{4} = frac{21}{4}$ $2frac{5}{6} = frac{17}{6}$ Find common denominator: LCM of 4 and 6 = 12 $frac{21}{4} = frac{63}{12}$, $frac{17}{6} = frac{34}{12}$ Subtract: $frac{63}{12} - frac{34}{12} = frac{29}{12} = 2frac{5}{12}$ Method 2: Subtract with regrouping Write: $5frac{1}{4} - 2frac{5}{6}$ Cannot subtract $frac{5}{6}$ from $frac{1}{4}$, so regroup: $5frac{1}{4} = 4 + 1frac{1}{4} = 4 + frac{5}{4}$ Now: $4frac{5}{4} - 2frac{5}{6}$ Subtract whole numbers: $4 - 2 = 2$ Subtract fractions: $frac{5}{4} - frac{5}{6} = frac{15}{12} - frac{10}{12} = frac{5}{12}$ Result: $2frac{5}{12}$ Subtraction Problems Calculate: $frac{11}{12} - frac{5}{8} + frac{1}{3}$ Find: $6frac{2}{3} - 3frac{7}{8}$ Subtract: $4frac{3}{5} - 2frac{4}{7}$ What is $7frac{1}{6} - 4frac{5}{9}$? If $frac{x}{y} - frac{3}{8} = frac{5}{12}$, find $frac{x}{y}$ Multiplying Fractions and Mixed Numbers To multiply fractions, multiply numerators together and denominators together, then simplify. Mixed numbers must be converted to improper fractions before multiplication. Canceling common factors before multiplying can simplify the process. Example 1: Complex Fraction Multiplication Calculate: $frac{3}{4} × frac{8}{9} × frac{15}{16}$ Solution: Method 1: Multiply then simplify Multiply numerators: $3 × 8 × 15 = 360$ Multiply denominators: $4 × 9 × 16 = 576$ $frac{360}{576}$ simplifies by dividing numerator and denominator by 72: $frac{360 ÷ 72}{576 ÷ 72} = frac{5}{8}$ Method 2: Cancel common factors first $frac{3}{4} × frac{8}{9} × frac{15}{16} = frac{3}{4} × frac{8}{9} × frac{15}{16}$ Cancel 3 and 9: $frac{1}{4} × frac{8}{3} × frac{15}{16}$ Cancel 8 and 16: $frac{1}{4} × frac{1}{3} × frac{15}{2}$ Cancel 15 and 3: $frac{1}{4} × frac{1}{1} × frac{5}{2}$ Multiply: $frac{1 × 1 × 5}{4 × 1 × 2} = frac{5}{8}$ Example 2: Mixed Number Multiplication Calculate: $2frac{2}{3} × 3frac{3}{5} × 1frac{1}{4}$ Solution: Convert to improper fractions: $2frac{2}{3} = frac{8}{3}$ $3frac{3}{5} = frac{18}{5}$ $1frac{1}{4} = frac{5}{4}$ Multiply: $frac{8}{3} × frac{18}{5} × frac{5}{4}$ Cancel common factors: Cancel 8 and 4: $frac{2}{3} × frac{18}{5} × frac{5}{1}$ Cancel 18 and 3: $frac{2}{1} × frac{6}{5} × frac{5}{1}$ Cancel 5's: $frac{2}{1} × frac{6}{1} × frac{1}{1} = 12$ Multiplication Problems Calculate: $frac{5}{6} × frac{9}{10} × frac{4}{15}$ Find: $3frac{1}{2} × 2frac{2}{3} × 1frac{1}{5}$ Multiply: $frac{7}{8} × frac{12}{14} × frac{16}{21}$ What is $4frac{2}{5} × 3frac{3}{7} × frac{5}{11}$? If $frac{a}{b} × frac{3}{4} = frac{9}{20}$, find $frac{a}{b}$ Dividing Fractions and Mixed Numbers To divide by a fraction, multiply by its reciprocal (flipped version). Mixed numbers must be converted to improper fractions before finding reciprocals. Complex division problems may involve multiple fractions. Example 1: Complex Fraction Division Calculate: $frac{3}{4} ÷ frac{5}{6} ÷ frac{7}{8}$ Solution: Division is not associative, so we must work left to right: $frac{3}{4} ÷ frac{5}{6} = frac{3}{4} × frac{6}{5} = frac{18}{20} = frac{9}{10}$ $frac{9}{10} ÷ frac{7}{8} = frac{9}{10} × frac{8}{7} = frac{72}{70} = frac{36}{35} = 1frac{1}{35}$ Example 2: Mixed Number Division Chain Calculate: $2frac{1}{2} ÷ 1frac{1}{4} ÷ 1frac{2}{3}$ Solution: Convert to improper fractions: $2frac{1}{2} = frac{5}{2}$ $1frac{1}{4} = frac{5}{4}$ $1frac{2}{3} = frac{5}{3}$ Work left to right: $frac{5}{2} ÷ frac{5}{4} = frac{5}{2} × frac{4}{5} = frac{20}{10} = 2$ $2 ÷ frac{5}{3} = 2 × frac{3}{5} = frac{6}{5} = 1frac{1}{5}$ Division Problems Calculate: $frac{7}{8} ÷ frac{3}{4} ÷ frac{5}{6}$ Find: $3frac{1}{3} ÷ 2frac{1}{2} ÷ 1frac{1}{4}$ Divide: $frac{9}{10} ÷ frac{3}{5} ÷ frac{2}{3}$ What is $4frac{4}{5} ÷ 2frac{2}{3} ÷ 1frac{1}{2}$? If $frac{a}{b} ÷ frac{3}{5} = frac{10}{9}$, find $frac{a}{b}$ Mixed Operations with Fractions When expressions contain multiple operations, we follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Example 1: Complex Mixed Operations Calculate: $frac{3}{4} + frac{2}{3} × frac{5}{6} - frac{1}{2}$ Solution: Follow PEMDAS: Multiplication before addition/subtraction $frac{2}{3} × frac{5}{6} = frac{10}{18} = frac{5}{9}$ Now: $frac{3}{4} + frac{5}{9} - frac{1}{2}$ Find common denominator: LCM of 4, 9, 2 = 36 Convert: $frac{3}{4} = frac{27}{36}$, $frac{5}{9} = frac{20}{36}$, $frac{1}{2} = frac{18}{36}$ $frac{27}{36} + frac{20}{36} - frac{18}{36} = frac{29}{36}$ Example 2: Multi-step Fraction Expression Calculate: $(frac{3}{4} + frac{2}{5}) × (frac{5}{6} - frac{1}{3}) ÷ frac{7}{10}$ Solution: First parentheses: $frac{3}{4} + frac{2}{5} = frac{15}{20} + frac{8}{20} = frac{23}{20}$ Second parentheses: $frac{5}{6} - frac{1}{3} = frac{5}{6} - frac{2}{6} = frac{3}{6} = frac{1}{2}$ Multiply: $frac{23}{20} × frac{1}{2} = frac{23}{40}$ Divide: $frac{23}{40} ÷ frac{7}{10} = frac{23}{40} × frac{10}{7} = frac{230}{280} = frac{23}{28}$ Mixed Operations Problems Calculate: $frac{5}{6} - frac{1}{3} × frac{3}{4} + frac{2}{5}$ Find: $(frac{2}{3} + frac{3}{4}) ÷ (frac{5}{6} - frac{1}{2})$ Simplify: $frac{7}{8} × frac{4}{5} + frac{3}{10} ÷ frac{2}{3}$ What is $2frac{1}{2} - 1frac{1}{3} × frac{3}{4} + frac{5}{6}$? Calculate: $(frac{3}{5} + frac{1}{4}) × (frac{2}{3} - frac{1}{6}) ÷ frac{7}{10}$ Complex Fraction Simplification Complex fractions have fractions in the numerator, denominator, or both. To simplify, find a common denominator for the numerator and denominator separately, or multiply the entire complex fraction by the LCM of all denominators involved. Example 1: Complex Fraction with Fractions in Both Parts Simplify: $frac{frac{2}{3} + frac{1}{4}}{frac{5}{6} - frac{1}{3}}$ Solution: Method 1: Simplify numerator and denominator separately Numerator: $frac{2}{3} + frac{1}{4} = frac{8}{12} + frac{3}{12} = frac{11}{12}$ Denominator: $frac{5}{6} - frac{1}{3} = frac{5}{6} - frac{2}{6} = frac{3}{6} = frac{1}{2}$ Now: $frac{frac{11}{12}}{frac{1}{2}} = frac{11}{12} ÷ frac{1}{2} = frac{11}{12} × frac{2}{1} = frac{22}{12} = frac{11}{6} = 1frac{5}{6}$ Method 2: Multiply by LCM of all denominators Denominators: 3, 4, 6, 3 → LCM = 12 Multiply numerator and denominator by 12: $frac{12(frac{2}{3} + frac{1}{4})}{12(frac{5}{6} - frac{1}{3})} = frac{8 + 3}{10 - 4} = frac{11}{6} = 1frac{5}{6}$ Example 2: Nested Complex Fraction Simplify: $frac{2frac{1}{2} - 1frac{1}{3}}{frac{3}{4} + frac{1}{6}}$ Solution: Convert mixed numbers to improper fractions: $2frac{1}{2} = frac{5}{2}$, $1frac{1}{3} = frac{4}{3}$ Numerator: $frac{5}{2} - frac{4}{3} = frac{15}{6} - frac{8}{6} = frac{7}{6}$ Denominator: $frac{3}{4} + frac{1}{6} = frac{9}{12} + frac{2}{12} = frac{11}{12}$ Now: $frac{frac{7}{6}}{frac{11}{12}} = frac{7}{6} ÷ frac{11}{12} = frac{7}{6} × frac{12}{11} = frac{84}{66} = frac{14}{11} = 1frac{3}{11}$ Complex Fraction Problems Simplify: $frac{frac{3}{4} + frac{2}{5}}{frac{7}{10} - frac{1}{4}}$ Find: $frac{3frac{1}{2} - 2frac{1}{3}}{frac{5}{6} + frac{1}{4}}$ Simplify: $frac{frac{5}{6} × frac{3}{4}}{frac{2}{3} ÷ frac{4}{5}}$ What is $frac{2frac{3}{4} + 1frac{1}{2}}{3frac{1}{3} - 2frac{1}{6}}$? Simplify: $frac{frac{2}{3} + frac{3}{4} - frac{1}{2}}{frac{5}{6} × frac{2}{5}}$ Real-World Applications Fractional calculations are essential in everyday situations such as cooking, construction, financial planning, and measurement conversions. These applications demonstrate the practical importance of mastering fraction operations. Example 1: Recipe Adjustment A recipe calls for $2frac{1}{2}$ cups of flour, $frac{3}{4}$ cup of sugar, and $1frac{1}{3}$ cups of milk. If you want to make $frac{2}{3}$ of the recipe, how much of each ingredient do you need? Solution: Multiply each ingredient by $frac{2}{3}$: Flour: $2frac{1}{2} × frac{2}{3} = frac{5}{2} × frac{2}{3} = frac{10}{6} = frac{5}{3} = 1frac{2}{3}$ cups Sugar: $frac{3}{4} × frac{2}{3} = frac{6}{12} = frac{1}{2}$ cup Milk: $1frac{1}{3} × frac{2}{3} = frac{4}{3} × frac{2}{3} = frac{8}{9}$ cup Example 2: Construction Measurement A carpenter needs to cut a board that is $12frac{3}{4}$ feet long into 5 equal pieces. What is the length of each piece? If each cut wastes $frac{1}{8}$ inch of material, what is the total length wasted? Solution: Length of each piece: $12frac{3}{4} ÷ 5 = frac{51}{4} ÷ 5 = frac{51}{4} × frac{1}{5} = frac{51}{20} = 2frac{11}{20}$ feet Total cuts: 4 cuts (to make 5 pieces) Total waste: $4 × frac{1}{8} = frac{4}{8} = frac{1}{2}$ inch Convert to feet: $frac{1}{2}$ inch = $frac{1}{24}$ foot (since 1 foot = 12 inches) Real-World Application Problems A car travels $15frac{1}{2}$ miles on $frac{3}{4}$ gallon of gas. How many miles per gallon is this? If $frac{3}{5}$ of a class are boys and there are 24 students, how many are girls? A recipe requires $2frac{1}{4}$ cups of flour to make 18 cookies. How much flour is needed for 30 cookies? A tank is $frac{2}{3}$ full. After adding $5frac{1}{2}$ gallons, it becomes $frac{5}{6}$ full. What is the tank's capacity? John spends $frac{1}{4}$ of his day sleeping, $frac{1}{3}$ at school, and $frac{1}{8}$ doing homework. What fraction of the day remains? Cumulative Exercises Calculate: $3frac{1}{2} + 2frac{2}{3} - 1frac{5}{6}$ Find: $frac{5}{6} × frac{9}{10} ÷ frac{3}{4}$ Simplify: $frac{frac{2}{3} + frac{3}{4}}{frac{5}{6} - frac{1}{3}}$ Calculate: $4frac{2}{5} - 2frac{3}{4} + 1frac{7}{10}$ Find: $(frac{3}{4} + frac{2}{5}) × (frac{5}{6} - frac{1}{3})$ Simplify: $frac{3frac{1}{3} × 2frac{1}{4}}{1frac{1}{2} + 2frac{1}{3}}$ Calculate: $frac{7}{8} - frac{2}{3} × frac{3}{4} + frac{5}{12}$ Find: $2frac{1}{2} ÷ 1frac{1}{4} × 3frac{1}{3}$ Simplify: $frac{frac{5}{6} - frac{1}{3}}{frac{3}{4} + frac{1}{2}} × frac{2}{3}$ If $frac{a}{b} + frac{c}{d} = frac{7}{6}$ and $frac{a}{b} - frac{c}{d} = frac{1}{6}$, find $frac{a}{b}$ and $frac{c}{d}$ Show/Hide Answers Problem: Calculate: $3frac{1}{2} + 2frac{2}{3} - 1frac{5}{6}$ Answer: Convert to improper fractions: $frac{7}{2} + frac{8}{3} - frac{11}{6}$ Common denominator: 6 $frac{21}{6} + frac{16}{6} - frac{11}{6} = frac{26}{6} = frac{13}{3} = 4frac{1}{3}$ Problem: Find: $frac{5}{6} × frac{9}{10} ÷ frac{3}{4}$ Answer: $frac{5}{6} × frac{9}{10} = frac{45}{60} = frac{3}{4}$ $frac{3}{4} ÷ frac{3}{4} = 1$ Problem: Simplify: $frac{frac{2}{3} + frac{3}{4}}{frac{5}{6} - frac{1}{3}}$ Answer: Numerator: $frac{8}{12} + frac{9}{12} = frac{17}{12}$ Denominator: $frac{5}{6} - frac{2}{6} = frac{3}{6} = frac{1}{2}$ $frac{frac{17}{12}}{frac{1}{2}} = frac{17}{12} × frac{2}{1} = frac{34}{12} = frac{17}{6} = 2frac{5}{6}$ Problem: Calculate: $4frac{2}{5} - 2frac{3}{4} + 1frac{7}{10}$ Answer: Convert: $frac{22}{5} - frac{11}{4} + frac{17}{10}$ Common denominator: 20 $frac{88}{20} - frac{55}{20} + frac{34}{20} = frac{67}{20} = 3frac{7}{20}$ Problem: Find: $(frac{3}{4} + frac{2}{5}) × (frac{5}{6} - frac{1}{3})$ Answer: First parentheses: $frac{15}{20} + frac{8}{20} = frac{23}{20}$ Second parentheses: $frac{5}{6} - frac{2}{6} = frac{3}{6} = frac{1}{2}$ Multiply: $frac{23}{20} × frac{1}{2} = frac{23}{40}$ Problem: Simplify: $frac{3frac{1}{3} × 2frac{1}{4}}{1frac{1}{2} + 2frac{1}{3}}$ Answer: Convert: $frac{frac{10}{3} × frac{9}{4}}{frac{3}{2} + frac{7}{3}}$ Numerator: $frac{10}{3} × frac{9}{4} = frac{90}{12} = frac{15}{2}$ Denominator: $frac{3}{2} + frac{7}{3} = frac{9}{6} + frac{14}{6} = frac{23}{6}$ $frac{frac{15}{2}}{frac{23}{6}} = frac{15}{2} × frac{6}{23} = frac{90}{46} = frac{45}{23} = 1frac{22}{23}$ Problem: Calculate: $frac{7}{8} - frac{2}{3} × frac{3}{4} + frac{5}{12}$ Answer: Multiplication first: $frac{2}{3} × frac{3}{4} = frac{6}{12} = frac{1}{2}$ Now: $frac{7}{8} - frac{1}{2} + frac{5}{12}$ Common denominator: 24 $frac{21}{24} - frac{12}{24} + frac{10}{24} = frac{19}{24}$ Problem: Find: $2frac{1}{2} ÷ 1frac{1}{4} × 3frac{1}{3}$ Answer: Convert: $frac{5}{2} ÷ frac{5}{4} × frac{10}{3}$ Left to right: $frac{5}{2} ÷ frac{5}{4} = frac{5}{2} × frac{4}{5} = frac{20}{10} = 2$ $2 × frac{10}{3} = frac{20}{3} = 6frac{2}{3}$ Problem: Simplify: $frac{frac{5}{6} - frac{1}{3}}{frac{3}{4} + frac{1}{2}} × frac{2}{3}$ Answer: Numerator: $frac{5}{6} - frac{2}{6} = frac{3}{6} = frac{1}{2}$ Denominator: $frac{3}{4} + frac{2}{4} = frac{5}{4}$ $frac{frac{1}{2}}{frac{5}{4}} = frac{1}{2} × frac{4}{5} = frac{4}{10} = frac{2}{5}$ $frac{2}{5} × frac{2}{3} = frac{4}{15}$ Problem: If $frac{a}{b} + frac{c}{d} = frac{7}{6}$ and $frac{a}{b} - frac{c}{d} = frac{1}{6}$, find $frac{a}{b}$ and $frac{c}{d}$ Answer: Add equations: $2(frac{a}{b}) = frac{7}{6} + frac{1}{6} = frac{8}{6} = frac{4}{3}$ So $frac{a}{b} = frac{2}{3}$ Subtract equations: $2(frac{c}{d}) = frac{7}{6} - frac{1}{6} = frac{6}{6} = 1$ So $frac{c}{d} = frac{1}{2}$ Conclusion/Recap Fractional calculations form the foundation for advanced mathematical concepts and real-world problem-solving. Mastery of addition, subtraction, multiplication, and division with fractions and mixed numbers enables efficient computation and accurate solutions to complex problems. The ability to work with unlike denominators, simplify complex fractions, and apply order of operations correctly is essential for success in algebra, geometry, and beyond. These skills have practical applications in science, engineering, finance, and everyday life. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
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