Fractions and Decimals: Equivalent fractions, simplification, converting between fractions and decimals; performing operations with fractions and decimals; understanding percentages.

Fractions and Decimals. Grade 7 Mathematics: Fractions, Decimals, and Percentages Subtopic Navigator Understanding Fractions and Decimals Equivalent Fractions Fraction Simplification Fraction-Decimal Conversion Operations with Fractions Operations with Decimals Understanding Percentages Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Master equivalent fractions and simplification techniques Convert fluently between fractions, decimals, and percentages Perform complex operations with fractions and decimals Solve multi-step problems involving fractions, decimals, and percentages Apply fraction and decimal concepts to real-world situations Fractions and Decimals Fractions and decimals are different representations of the same mathematical concepts. Fractions express parts of a whole using numerators and denominators, while decimals use place values. Understanding the relationship between these representations is crucial for advanced mathematical problem-solving. Equivalent Fractions Equivalent fractions represent the same value despite having different numerators and denominators. They are created by multiplying or dividing both the numerator and denominator by the same non-zero number. Example 1: Finding Multiple Equivalents Find three fractions equivalent to [latex]frac{5}{8}[/latex] with denominators between 50 and 100. Solution: Multiply numerator and denominator by 7: [latex]frac{5 times 7}{8 times 7} = frac{35}{56}[/latex] Multiply by 8: [latex]frac{5 times 8}{8 times 8} = frac{40}{64}[/latex] Multiply by 9: [latex]frac{5 times 9}{8 times 9} = frac{45}{72}[/latex] Multiply by 10: [latex]frac{5 times 10}{8 times 10} = frac{50}{80}[/latex] Multiply by 11: [latex]frac{5 times 11}{8 times 11} = frac{55}{88}[/latex] Multiply by 12: [latex]frac{5 times 12}{8 times 12} = frac{60}{96}[/latex] Example 2: Complex Equivalent Fraction Problem If [latex]frac{3}{7} = frac{x}{42} = frac{45}{y}[/latex], find the values of x and y. Solution: For [latex]frac{3}{7} = frac{x}{42}[/latex]: Since [latex]7 times 6 = 42[/latex], multiply numerator by 6: [latex]3 times 6 = 18[/latex] So [latex]x = 18[/latex] For [latex]frac{3}{7} = frac{45}{y}[/latex]: Since [latex]3 times 15 = 45[/latex], multiply denominator by 15: [latex]7 times 15 = 105[/latex] So [latex]y = 105[/latex] Equivalent Fractions Problems Find four fractions equivalent to [latex]frac{7}{12}[/latex] with denominators between 60 and 120 If [latex]frac{5}{9} = frac{x}{63} = frac{40}{y}[/latex], find x and y Which fraction is not equivalent to the others: [latex]frac{12}{18}, frac{20}{30}, frac{28}{42}, frac{32}{48}[/latex]? Find the missing number: [latex]frac{8}{15} = frac{32}{x} = frac{y}{75}[/latex] Create three equivalent fractions for [latex]frac{11}{16}[/latex] where the numerator is between 50 and 70 Fraction Simplification Simplifying fractions involves reducing them to their lowest terms by dividing both numerator and denominator by their greatest common factor. This makes fractions easier to work with and compare. Example 1: Complex Simplification Simplify [latex]frac{168}{252}[/latex] to its lowest terms using prime factorization. Solution: Prime factors of 168: [latex]2^3 times 3 times 7[/latex] Prime factors of 252: [latex]2^2 times 3^2 times 7[/latex] Common factors: [latex]2^2 times 3 times 7 = 84[/latex] [latex]frac{168 div 84}{252 div 84} = frac{2}{3}[/latex] Example 2: Multiple Fraction Comparison Arrange in ascending order: [latex]frac{5}{8}, frac{7}{12}, frac{3}{5}, frac{11}{18}[/latex] Solution: Find LCM of denominators: LCM of 8, 12, 5, 18 = 360 Convert to equivalent fractions with denominator 360: [latex]frac{5}{8} = frac{225}{360}[/latex] [latex]frac{7}{12} = frac{210}{360}[/latex] [latex]frac{3}{5} = frac{216}{360}[/latex] [latex]frac{11}{18} = frac{220}{360}[/latex] In ascending order: [latex]frac{210}{360}, frac{216}{360}, frac{220}{360}, frac{225}{360}[/latex] Which is: [latex]frac{7}{12}, frac{3}{5}, frac{11}{18}, frac{5}{8}[/latex] Fraction Simplification Problems Simplify [latex]frac{315}{420}[/latex] to its lowest terms using prime factorization Arrange in descending order: [latex]frac{4}{7}, frac{5}{9}, frac{7}{12}, frac{8}{15}[/latex] Simplify the complex fraction: [latex]frac{frac{3}{4}}{frac{5}{6}}[/latex] Which fraction is in simplest form: [latex]frac{24}{36}, frac{35}{49}, frac{42}{55}, frac{51}{68}[/latex]? Simplify: [latex]frac{12}{18} times frac{15}{25} div frac{9}{30}[/latex] Fraction-Decimal Conversion Converting between fractions and decimals is essential for mathematical operations and real-world applications. Fractions can be converted to decimals by division, while decimals can be converted to fractions using place values. Example 1: Complex Decimal to Fraction Convert 0.375 to a fraction in simplest form and express 2.625 as a mixed number. Solution: 0.375 = [latex]frac{375}{1000} = frac{375 div 125}{1000 div 125} = frac{3}{8}[/latex] 2.625 = [latex]2 + frac{625}{1000} = 2 + frac{625 div 125}{1000 div 125} = 2 + frac{5}{8} = 2frac{5}{8}[/latex] Example 2: Repeating Decimal Conversion Convert 0.8333... to a fraction in simplest form. Solution: Let x = 0.8333... Multiply by 10: 10x = 8.3333... Subtract: 10x - x = 8.3333... - 0.8333... 9x = 7.5 x = [latex]frac{7.5}{9} = frac{15}{18} = frac{5}{6}[/latex] Therefore, 0.8333... = [latex]frac{5}{6}[/latex] Conversion Problems Convert 0.3125 to a fraction in simplest form Express 3.875 as a mixed number in simplest form Convert [latex]frac{7}{11}[/latex] to a decimal (round to 3 decimal places) Change 0.272727... to a fraction in simplest form Which is greater: 0.583 or [latex]frac{7}{12}[/latex]? Show your work Operations with Fractions Performing operations with fractions requires understanding of common denominators for addition and subtraction, and direct multiplication or division for other operations. Mixed numbers should be converted to improper fractions before operations. Example 1: Complex Fraction Operations Calculate: [latex]2frac{3}{4} + 3frac{5}{6} - 1frac{7}{8}[/latex] Solution: Convert to improper fractions: [latex]2frac{3}{4} = frac{11}{4}[/latex], [latex]3frac{5}{6} = frac{23}{6}[/latex], [latex]1frac{7}{8} = frac{15}{8}[/latex] Find LCM of denominators: LCM of 4, 6, 8 = 24 Convert to common denominator: [latex]frac{11}{4} = frac{66}{24}[/latex], [latex]frac{23}{6} = frac{92}{24}[/latex], [latex]frac{15}{8} = frac{45}{24}[/latex] [latex]frac{66}{24} + frac{92}{24} - frac{45}{24} = frac{113}{24} = 4frac{17}{24}[/latex] Example 2: Multi-step Fraction Problem Simplify: [latex](frac{3}{4} + frac{2}{3}) times (frac{5}{6} - frac{1}{4}) div frac{7}{8}[/latex] Solution: First parentheses: [latex]frac{3}{4} + frac{2}{3} = frac{9}{12} + frac{8}{12} = frac{17}{12}[/latex] Second parentheses: [latex]frac{5}{6} - frac{1}{4} = frac{10}{12} - frac{3}{12} = frac{7}{12}[/latex] Multiply: [latex]frac{17}{12} times frac{7}{12} = frac{119}{144}[/latex] Divide: [latex]frac{119}{144} div frac{7}{8} = frac{119}{144} times frac{8}{7} = frac{119}{126} = frac{17}{18}[/latex] Fraction Operations Problems Calculate: [latex]3frac{2}{5} + 4frac{7}{10} - 2frac{3}{4}[/latex] Simplify: [latex](frac{5}{6} - frac{1}{3}) times (frac{2}{5} + frac{3}{10}) div frac{4}{9}[/latex] Find: [latex]frac{7}{8} times frac{4}{5} div frac{14}{15}[/latex] Calculate: [latex]5frac{1}{3} - 2frac{5}{6} + 3frac{3}{4}[/latex] Simplify: [latex]frac{frac{2}{3} + frac{3}{4}}{frac{5}{6} - frac{1}{8}}[/latex] Operations with Decimals Decimal operations follow the same principles as whole number operations, with careful attention to decimal point placement. Understanding place value is crucial for accurate calculations with decimals. Example 1: Complex Decimal Operations Calculate: [latex]12.45 times 3.6 div 0.4 + 7.89[/latex] Solution: First multiplication: [latex]12.45 times 3.6 = 44.82[/latex] Then division: [latex]44.82 div 0.4 = 112.05[/latex] Finally addition: [latex]112.05 + 7.89 = 119.94[/latex] Example 2: Decimal Word Problem A recipe requires 2.25 cups of flour, 1.5 cups of sugar, and 0.75 cups of milk. If you want to make 2.5 times the recipe, how much of each ingredient do you need? Solution: Flour: [latex]2.25 times 2.5 = 5.625[/latex] cups Sugar: [latex]1.5 times 2.5 = 3.75[/latex] cups Milk: [latex]0.75 times 2.5 = 1.875[/latex] cups Decimal Operations Problems Calculate: [latex]15.78 times 4.2 div 0.6 - 8.95[/latex] Find: [latex](12.5 + 8.75) times (6.4 - 3.2) div 2.5[/latex] If 4.5 meters of cloth costs $28.35, what is the cost per meter? Calculate: [latex]23.456 + 7.89 - 12.345 times 1.2[/latex] A car travels 245.75 km using 18.5 liters of petrol. What is the fuel efficiency in km per liter? Understanding Percentages Percentages represent parts per hundred and are closely related to fractions and decimals. Understanding percentages is essential for financial calculations, statistics, and real-world problem-solving. Example 1: Complex Percentage Problems If 35% of a number is 84, what is 125% of the same number? Solution: Let the number be x [latex]35% times x = 84[/latex] [latex]0.35x = 84[/latex] [latex]x = 84 div 0.35 = 240[/latex] 125% of 240 = [latex]1.25 times 240 = 300[/latex] Example 2: Percentage Increase/Decrease A shirt originally priced at $45 is discounted by 20%. After a week, the discounted price is increased by 15%. What is the final price? Solution: Original price: $45 After 20% discount: [latex]45 times 0.8 = 36[/latex] After 15% increase: [latex]36 times 1.15 = 41.40[/latex] Final price: $41.40 Percentage Problems If 28% of a number is 56, what is 175% of the same number? A product's price increased from $80 to $92. What is the percentage increase? Express [latex]frac{7}{8}[/latex] as a percentage After a 15% discount, a book costs $25.50. What was the original price? If 45% of students are girls and there are 330 boys, how many students are there in total? Real-World Applications Fractions, decimals, and percentages are used extensively in everyday life, from cooking and shopping to finance and statistics. Understanding how to apply these concepts to real-world situations is a crucial mathematical skill. Example 1: Financial Application John spent [latex]frac{1}{4}[/latex] of his salary on rent, 30% on food, and 0.15 on transportation. If he saved the remaining $600, what is his monthly salary? Solution: Convert all to fractions or percentages: Rent: [latex]frac{1}{4} = 25%[/latex] Food: 30% Transportation: 0.15 = 15% Total spent: 25% + 30% + 15% = 70% Savings: 100% - 70% = 30% = $600 Monthly salary: [latex]600 div 0.3 = 2000[/latex] John's monthly salary is $2,000 Example 2: Measurement Application A recipe calls for [latex]2frac{1}{4}[/latex] cups of flour, but you only have a [latex]frac{1}{3}[/latex] cup measure. How many full measures do you need, and what fraction remains? Solution: Convert to improper fraction: [latex]2frac{1}{4} = frac{9}{4}[/latex] Divide by [latex]frac{1}{3}[/latex]: [latex]frac{9}{4} div frac{1}{3} = frac{9}{4} times frac{3}{1} = frac{27}{4} = 6frac{3}{4}[/latex] You need 6 full measures, with [latex]frac{3}{4}[/latex] of [latex]frac{1}{3}[/latex] cup remaining Remaining: [latex]frac{3}{4} times frac{1}{3} = frac{1}{4}[/latex] cup Real-World Application Problems In a class, [latex]frac{2}{5}[/latex] of students prefer math, 35% prefer science, and the remaining 18 prefer English. How many students are in the class? A shirt that originally cost $45 is on sale for 25% off. If you have a coupon for an additional 15% off the sale price, what is the final price? A recipe requires [latex]1frac{3}{4}[/latex] cups of sugar. If you want to make [latex]frac{2}{3}[/latex] of the recipe, how much sugar do you need? In an election, candidate A got 45% of votes, candidate B got [latex]frac{3}{8}[/latex] of votes, and the rest were invalid. If candidate A got 900 votes, how many invalid votes were there? A tank is [latex]frac{3}{5}[/latex] full. After adding 120 liters, it becomes [latex]frac{4}{5}[/latex] full. What is the capacity of the tank? Cumulative Exercises Simplify: [latex]frac{12}{18} times frac{15}{25} + frac{3}{5} - frac{2}{3}[/latex] Convert 0.5625 to a fraction in simplest form Calculate: [latex]4.25 times 3.6 div 0.9 + 7.85[/latex] If 32% of a number is 96, what is 140% of the same number? Arrange in ascending order: [latex]frac{5}{8}, 0.62, frac{7}{12}, 0.583[/latex] Simplify: [latex](frac{3}{4} + frac{2}{5}) times (frac{5}{6} - frac{1}{3}) div frac{7}{10}[/latex] Express [latex]frac{9}{16}[/latex] as a decimal and percentage Calculate: [latex]3frac{2}{3} + 4frac{5}{6} - 2frac{7}{8}[/latex] A product's price increased from $60 to $75. What is the percentage increase? If [latex]frac{3}{7} = frac{x}{56} = frac{36}{y}[/latex], find x and y Show/Hide Answers Problem: Simplify: [latex]frac{12}{18} times frac{15}{25} + frac{3}{5} - frac{2}{3}[/latex] Answer: [latex]frac{12}{18} = frac{2}{3}[/latex], [latex]frac{15}{25} = frac{3}{5}[/latex] [latex]frac{2}{3} times frac{3}{5} = frac{2}{5}[/latex] Common denominator for [latex]frac{2}{5} + frac{3}{5} - frac{2}{3}[/latex] is 15 [latex]frac{6}{15} + frac{9}{15} - frac{10}{15} = frac{5}{15} = frac{1}{3}[/latex] Problem: Convert 0.5625 to a fraction in simplest form Answer: 0.5625 = [latex]frac{5625}{10000} = frac{5625 div 625}{10000 div 625} = frac{9}{16}[/latex] Problem: Calculate: [latex]4.25 times 3.6 div 0.9 + 7.85[/latex] Answer: [latex]4.25 times 3.6 = 15.3[/latex] [latex]15.3 div 0.9 = 17[/latex] [latex]17 + 7.85 = 24.85[/latex] Problem: If 32% of a number is 96, what is 140% of the same number? Answer: Let the number be x [latex]0.32x = 96[/latex] [latex]x = 96 div 0.32 = 300[/latex] 140% of 300 = [latex]1.4 times 300 = 420[/latex] Problem: Arrange in ascending order: [latex]frac{5}{8}, 0.62, frac{7}{12}, 0.583[/latex] Answer: Convert all to decimals: [latex]frac{5}{8} = 0.625[/latex], [latex]frac{7}{12} approx 0.5833[/latex] In order: 0.583, [latex]frac{7}{12}[/latex], 0.62, [latex]frac{5}{8}[/latex] Problem: Simplify: [latex](frac{3}{4} + frac{2}{5}) times (frac{5}{6} - frac{1}{3}) div frac{7}{10}[/latex] Answer: First parentheses: [latex]frac{3}{4} + frac{2}{5} = frac{15}{20} + frac{8}{20} = frac{23}{20}[/latex] Second parentheses: [latex]frac{5}{6} - frac{1}{3} = frac{5}{6} - frac{2}{6} = frac{3}{6} = frac{1}{2}[/latex] Multiply: [latex]frac{23}{20} times frac{1}{2} = frac{23}{40}[/latex] Divide: [latex]frac{23}{40} div frac{7}{10} = frac{23}{40} times frac{10}{7} = frac{23}{28}[/latex] Problem: Express [latex]frac{9}{16}[/latex] as a decimal and percentage Answer: Decimal: [latex]9 div 16 = 0.5625[/latex] Percentage: [latex]0.5625 times 100 = 56.25%[/latex] Problem: Calculate: [latex]3frac{2}{3} + 4frac{5}{6} - 2frac{7}{8}[/latex] Answer: Convert to improper fractions: [latex]3frac{2}{3} = frac{11}{3}[/latex], [latex]4frac{5}{6} = frac{29}{6}[/latex], [latex]2frac{7}{8} = frac{23}{8}[/latex] LCM of 3, 6, 8 = 24 [latex]frac{11}{3} = frac{88}{24}[/latex], [latex]frac{29}{6} = frac{116}{24}[/latex], [latex]frac{23}{8} = frac{69}{24}[/latex] [latex]frac{88}{24} + frac{116}{24} - frac{69}{24} = frac{135}{24} = 5frac{15}{24} = 5frac{5}{8}[/latex] Problem: A product's price increased from $60 to $75. What is the percentage increase? Answer: Increase = $75 - $60 = $15 Percentage increase = [latex]frac{15}{60} times 100 = 25%[/latex] Problem: If [latex]frac{3}{7} = frac{x}{56} = frac{36}{y}[/latex], find x and y Answer: For [latex]frac{3}{7} = frac{x}{56}[/latex]: Since [latex]7 times 8 = 56[/latex], [latex]x = 3 times 8 = 24[/latex] For [latex]frac{3}{7} = frac{36}{y}[/latex]: Since [latex]3 times 12 = 36[/latex], [latex]y = 7 times 12 = 84[/latex] Conclusion/Recap Mastering fractions, decimals, and percentages is essential for mathematical proficiency and real-world problem-solving. These interconnected concepts form the foundation for algebra, statistics, and financial mathematics. The ability to convert between different representations and perform operations accurately enables students to tackle complex mathematical challenges with confidence. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c