Whole Number Concepts: – Factors, multiples, primes, squares, cubes.

FRACTIONS Equivalent fractions and simplification techniques. Grade 7 Mathematics: Fractions – Equivalent Fractions and Simplification Techniques Subtopic Navigator Introduction Equivalent Fractions Simplifying Fractions Comparing and Ordering Fractions Complex Fraction Simplification Applications and Mixed Problems Cumulative Exercises Conclusion Lesson Objectives Define and generate equivalent fractions. Apply techniques to simplify fractions to their lowest terms. Compare and order fractions using different strategies. Work with complex fractions and apply simplification in problem contexts. Lesson Introduction Fractions are an essential concept in mathematics representing parts of a whole. Two key ideas when working with fractions are equivalence (different fractions that represent the same value) and simplification (reducing fractions to their simplest form). Mastering these techniques helps in comparing, adding, subtracting, and applying fractions in real-life and examination settings. Equivalent Fractions Equivalent fractions are fractions that represent the same value, even though their numerators and denominators may be different. They can be formed by multiplying or dividing both numerator and denominator by the same non-zero number. Example 1: Write two fractions equivalent to [latex]tfrac{5}{7}[/latex]. Solution: Multiply numerator and denominator by 2: [latex]tfrac{5 times 2}{7 times 2} = tfrac{10}{14}[/latex]. Multiply by 3: [latex]tfrac{5 times 3}{7 times 3} = tfrac{15}{21}[/latex]. So, [latex]tfrac{10}{14}[/latex] and [latex]tfrac{15}{21}[/latex] are equivalent to [latex]tfrac{5}{7}[/latex]. Example 2: Show that [latex]tfrac{18}{24}[/latex] and [latex]tfrac{3}{4}[/latex] are equivalent. Solution: Simplify [latex]tfrac{18}{24}[/latex] by dividing numerator and denominator by 6: [latex]tfrac{18 div 6}{24 div 6} = tfrac{3}{4}[/latex]. Hence, they are equivalent fractions. Example 3: Find a fraction equivalent to [latex]tfrac{12}{15}[/latex] with denominator 60. Solution: Factor: [latex]15 times 4 = 60[/latex]. Multiply numerator by 4: [latex]12 times 4 = 48[/latex]. So the required fraction is [latex]tfrac{48}{60}[/latex]. Exercises (Equivalent Fractions) Find two fractions equivalent to [latex]tfrac{7}{9}[/latex]. Express [latex]tfrac{9}{12}[/latex] as an equivalent fraction with denominator 36. Simplifying Fractions Simplification means reducing a fraction to its lowest terms by dividing numerator and denominator by their greatest common divisor (GCD). Example 4: Simplify [latex]tfrac{84}{126}[/latex]. Solution: Find GCD(84, 126) = 42. Divide numerator and denominator: [latex]tfrac{84 div 42}{126 div 42} = tfrac{2}{3}[/latex]. Example 5: Simplify [latex]tfrac{154}{198}[/latex]. Solution: GCD(154, 198) = 22. Divide: [latex]tfrac{154 div 22}{198 div 22} = tfrac{7}{9}[/latex]. Example 6: Simplify [latex]tfrac{350}{490}[/latex]. Solution: GCD(350, 490) = 70. Divide: [latex]tfrac{350 div 70}{490 div 70} = tfrac{5}{7}[/latex]. Exercises (Simplifying Fractions) Simplify [latex]tfrac{168}{252}[/latex]. Simplify [latex]tfrac{225}{360}[/latex]. Comparing and Ordering Fractions To compare fractions, we may bring them to a common denominator or use cross multiplication. Example 7: Which is greater: [latex]tfrac{7}{12}[/latex] or [latex]tfrac{5}{8}[/latex]? Solution: Cross multiply: [latex]7 times 8 = 56[/latex], [latex]5 times 12 = 60[/latex]. Since 60 > 56, [latex]tfrac{5}{8} > tfrac{7}{12}[/latex]. Example 8: Arrange [latex]tfrac{3}{5}, tfrac{7}{10}, tfrac{2}{3}[/latex] in ascending order. Solution: Common denominator: 30. [latex]tfrac{3}{5} = tfrac{18}{30}, tfrac{7}{10} = tfrac{21}{30}, tfrac{2}{3} = tfrac{20}{30}[/latex]. Order: [latex]tfrac{18}{30} 153, [latex]tfrac{11}{17} > tfrac{9}{14}[/latex]. Exercises (Comparing Fractions) Arrange [latex]tfrac{4}{7}, tfrac{5}{9}, tfrac{3}{8}[/latex] in ascending order. Which is greater: [latex]tfrac{13}{20}[/latex] or [latex]tfrac{19}{30}[/latex]? Complex Fraction Simplification Complex fractions are fractions where the numerator, denominator, or both are themselves fractions. Simplify by finding a common denominator or multiplying by reciprocal. Example 10: Simplify [latex]tfrac{tfrac{3}{4}}{tfrac{5}{6}}[/latex]. Solution: [latex]tfrac{3}{4} div tfrac{5}{6} = tfrac{3}{4} times tfrac{6}{5} = tfrac{18}{20} = tfrac{9}{10}[/latex]. Example 11: Simplify [latex]tfrac{tfrac{7}{8}}{tfrac{21}{32}}[/latex]. Solution: [latex]tfrac{7}{8} div tfrac{21}{32} = tfrac{7}{8} times tfrac{32}{21} = tfrac{224}{168} = tfrac{4}{3}[/latex]. Example 12: Simplify [latex]tfrac{tfrac{12}{25}}{tfrac{18}{35}}[/latex]. Solution: [latex]tfrac{12}{25} div tfrac{18}{35} = tfrac{12}{25} times tfrac{35}{18} = tfrac{420}{450} = tfrac{14}{15}[/latex]. Exercises (Complex Fractions) Simplify [latex]tfrac{tfrac{15}{28}}{tfrac{25}{49}}[/latex]. Simplify [latex]tfrac{tfrac{9}{10}}{tfrac{27}{40}}[/latex]. Applications and Mixed Problems Example 13: A recipe requires [latex]tfrac{2}{3}[/latex] cup of sugar, but you only have a [latex]tfrac{1}{4}[/latex] cup measure. How many of these measures are needed? Solution: [latex]tfrac{2}{3} div tfrac{1}{4} = tfrac{2}{3} times tfrac{4}{1} = tfrac{8}{3} = 2 tfrac{2}{3}[/latex]. You need 2 full measures and [latex]tfrac{2}{3}[/latex] of another. Example 14: If a worker completes [latex]tfrac{5}{8}[/latex] of a job in a day, how many days will it take to complete the full job? Solution: Days needed = [latex]1 div tfrac{5}{8} = tfrac{8}{5} = 1 tfrac{3}{5}[/latex] days. Example 15: A tank is [latex]tfrac{3}{5}[/latex] full and holds 120 liters when full. How much water is in the tank? Solution: [latex]tfrac{3}{5} times 120 = 72[/latex] liters. Exercises (Applications) If a rope is [latex]tfrac{7}{8}[/latex] m long, how many pieces of [latex]tfrac{1}{16}[/latex] m can be cut from it? A farmer harvested [latex]tfrac{5}{12}[/latex] of his field in the morning and [latex]tfrac{1}{3}[/latex] in the afternoon. What fraction of the field is harvested in total? Cumulative Exercises Find three fractions equivalent to [latex]tfrac{11}{13}[/latex]. Write [latex]tfrac{15}{18}[/latex] in simplest form. Arrange [latex]tfrac{2}{5}, tfrac{7}{12}, tfrac{5}{9}[/latex] in ascending order. Compare [latex]tfrac{14}{21}[/latex] and [latex]tfrac{18}{27}[/latex]. Simplify [latex]tfrac{tfrac{21}{32}}{tfrac{7}{16}}[/latex]. Find an equivalent fraction to [latex]tfrac{9}{11}[/latex] with denominator 77. Simplify [latex]tfrac{96}{120}[/latex]. Which is larger: [latex]tfrac{19}{24}[/latex] or [latex]tfrac{23}{30}[/latex]? If [latex]tfrac{3}{4}[/latex] of a classroom are boys and there are 24 boys, how many pupils are in the class? A tap fills [latex]tfrac{2}{7}[/latex] of a tank in one hour. How many hours to fill the tank? Conclusion/Recap In this lesson, we explored equivalent fractions, simplification, comparison, and complex fractions. These skills are crucial in handling fractions efficiently in higher operations like addition, subtraction, algebra, and real-world applications. Mastering these ensures accuracy in exams and builds strong mathematical foundations. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c