Conver sions. Grade 7 Mathematics: Conversions Between Fractions, Decimals, and Percentages Subtopic Navigator Understanding Number Conversions Fraction to Decimal Conversion Decimal to Fraction Conversion Fraction to Percentage Conversion Percentage to Fraction Conversion Decimal to Percentage Conversion Percentage to Decimal Conversion Repeating Decimal Conversions Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Convert fractions to decimals using division and place value understanding Convert decimals to fractions using place value analysis Convert between percentages and fractions using equivalent fractions Convert between percentages and decimals using multiplication and division Handle repeating decimals in conversion problems Apply conversion skills to solve complex real-world problems Understanding Number Conversions Fractions, decimals, and percentages are different ways of representing the same mathematical values. Mastering conversions between these forms is essential for mathematical fluency and problem-solving. Each representation has its advantages in different contexts: fractions for exact values, decimals for calculations, and percentages for comparisons. Fraction to Decimal Conversion To convert a fraction to a decimal, divide the numerator by the denominator. The result may be a terminating decimal (ends after a certain number of digits) or a repeating decimal (has a pattern that repeats infinitely). Example 1: Complex Fraction to Decimal Convert $frac{13}{16}$ to a decimal. Show your division process. Solution: Divide 13 by 16: 16 goes into 13.0: 0.8 (16 × 0.8 = 12.8) Subtract: 13.0 - 12.8 = 0.2 Bring down 0: 0.20 16 goes into 0.20: 0.0125 (16 × 0.0125 = 0.2) $frac{13}{16} = 0.8125$ Alternative method using equivalent fractions: $frac{13}{16} = frac{13 times 625}{16 times 625} = frac{8125}{10000} = 0.8125$ Example 2: Repeating Decimal Result Convert $frac{5}{11}$ to a decimal and identify the repeating pattern. Solution: Divide 5 by 11: 11 goes into 5.0: 0.4 (11 × 0.4 = 4.4) Subtract: 5.0 - 4.4 = 0.6 Bring down 0: 0.60 11 goes into 0.60: 0.05 (11 × 0.05 = 0.55) Subtract: 0.60 - 0.55 = 0.05 Bring down 0: 0.050 11 goes into 0.050: 0.0045 (11 × 0.0045 = 0.0495) This continues: $frac{5}{11} = 0.454545... = 0.overline{45}$ The repeating pattern is "45" Fraction to Decimal Problems Convert $frac{7}{8}$ to a decimal using long division Convert $frac{5}{12}$ to a decimal and identify if it's terminating or repeating What decimal is equivalent to $frac{9}{16}$? Convert $frac{11}{15}$ to a decimal and show the repeating pattern Which fraction gives a longer repeating pattern: $frac{1}{7}$ or $frac{2}{13}$? Decimal to Fraction Conversion To convert a decimal to a fraction, use place value to determine the denominator, then simplify the resulting fraction. For terminating decimals, the denominator is a power of 10 (10, 100, 1000, etc.). Example 1: Complex Decimal to Fraction Convert 0.875 to a fraction in simplest form using two different methods. Solution: Method 1: Place Value 0.875 = $frac{875}{1000}$ Simplify by dividing numerator and denominator by 125: $frac{875 ÷ 125}{1000 ÷ 125} = frac{7}{8}$ Method 2: Equivalent Fractions 0.875 = $frac{875}{1000}$ Prime factorize: 875 = $5^3 × 7$, 1000 = $10^3 = 2^3 × 5^3$ Cancel common factors: $frac{5^3 × 7}{2^3 × 5^3} = frac{7}{2^3} = frac{7}{8}$ Example 2: Decimal with Many Places Convert 0.3125 to a fraction in simplest form. Solution: 0.3125 = $frac{3125}{10000}$ Find GCD of 3125 and 10000: 625 $frac{3125 ÷ 625}{10000 ÷ 625} = frac{5}{16}$ Check: $5 ÷ 16 = 0.3125$ ✓ Decimal to Fraction Problems Convert 0.625 to a fraction in simplest form using two methods Convert 0.1875 to a fraction and verify by converting back to decimal What fraction is equivalent to 0.36? Simplify completely Convert 1.75 to a mixed number in simplest form Which decimal has the simplest fraction form: 0.25, 0.375, or 0.4375? Fraction to Percentage Conversion To convert a fraction to a percentage, first convert it to a decimal, then multiply by 100 and add the % symbol. Alternatively, find an equivalent fraction with denominator 100. Example 1: Complex Fraction to Percentage Convert $frac{5}{6}$ to a percentage, rounded to two decimal places. Solution: Method 1: Decimal Conversion $frac{5}{6} = 5 ÷ 6 ≈ 0.833333...$ Multiply by 100: $0.833333... × 100 = 83.3333...$% Rounded to two decimal places: 83.33% Method 2: Equivalent Fraction We need denominator 100: $frac{5}{6} = frac{x}{100}$ Cross-multiply: $5 × 100 = 6x$ $500 = 6x$ $x = 500 ÷ 6 = 83.333...$ So $frac{5}{6} = 83.overline{3}$% ≈ 83.33% Example 2: Multiple Fraction Comparison Convert $frac{3}{8}$, $frac{5}{12}$, and $frac{7}{16}$ to percentages and arrange them from smallest to largest. Solution: $frac{3}{8} = 0.375 = 37.5%$ $frac{5}{12} ≈ 0.4167 = 41.67%$ $frac{7}{16} = 0.4375 = 43.75%$ In order: 37.5% < 41.67% < 43.75% So: $frac{3}{8}$ < $frac{5}{12}$ < $frac{7}{16}$ Fraction to Percentage Problems Convert $frac{7}{9}$ to a percentage rounded to one decimal place Which is greater as a percentage: $frac{4}{7}$ or $frac{5}{8}$? Convert $frac{11}{15}$ to a percentage with repeating decimal notation Arrange these fractions as percentages: $frac{2}{5}$, $frac{3}{7}$, $frac{4}{9}$ What percentage is equivalent to $frac{13}{20}$? Percentage to Fraction Conversion To convert a percentage to a fraction, write the percentage as a fraction with denominator 100, then simplify. For percentages with decimals, first convert to a fraction with denominator 1000, 10000, etc., then simplify. Example 1: Complex Percentage to Fraction Convert 87.5% to a fraction in simplest form. Solution: 87.5% = $frac{87.5}{100}$ Multiply numerator and denominator by 10 to eliminate decimal: $frac{87.5 × 10}{100 × 10} = frac{875}{1000}$ Simplify by dividing by 125: $frac{875 ÷ 125}{1000 ÷ 125} = frac{7}{8}$ Check: $frac{7}{8} = 0.875 = 87.5%$ ✓ Example 2: Percentage with Repeating Decimal Convert 33.$overline{3}$% to a fraction in simplest form. Solution: 33.$overline{3}$% = 33$frac{1}{3}$% = $frac{100}{3}$% Convert percentage to fraction: $frac{frac{100}{3}}{100} = frac{100}{3} × frac{1}{100} = frac{1}{3}$ So 33.$overline{3}$% = $frac{1}{3}$ Check: $frac{1}{3} = 0.333... = 33.333...%$ ✓ Percentage to Fraction Problems Convert 62.5% to a fraction in simplest form Convert 16.$overline{6}$% to a fraction Which has a simpler fraction form: 45% or 37.5%? Convert 125% to a mixed number in simplest form What fraction is equivalent to 28.75%? Decimal to Percentage Conversion To convert a decimal to a percentage, multiply by 100 and add the % symbol. This is equivalent to moving the decimal point two places to the right. Example 1: Complex Decimal to Percentage Convert 0.428571... to a percentage with the repeating pattern indicated. Solution: 0.428571... × 100 = 42.8571...% Identify the repeating pattern: 428571 repeats So: 0.428571... = 42.$overline{857142}$% Note: This is $frac{3}{7}$ as a percentage Example 2: Multiple Decimal Conversion Convert 0.1875, 0.3125, and 0.4375 to percentages and arrange from largest to smallest. Solution: 0.1875 × 100 = 18.75% 0.3125 × 100 = 31.25% 0.4375 × 100 = 43.75% From largest to smallest: 43.75% > 31.25% > 18.75% So: 0.4375 > 0.3125 > 0.1875 Decimal to Percentage Problems Convert 0.8333... to a percentage with repeating notation What percentage is equivalent to 0.0625? Convert 2.375 to a percentage Which is greater as a percentage: 0.571 or 0.5714? Convert 0.09 to a percentage Percentage to Decimal Conversion To convert a percentage to a decimal, divide by 100 or move the decimal point two places to the left. Remove the % symbol after conversion. Example 1: Complex Percentage to Decimal Convert 33.$overline{3}$% to a decimal. Solution: Divide by 100: 33.$overline{3}$% ÷ 100 = 0.33$overline{3}$ Alternative: Move decimal point two places left: 33.$overline{3}$ → 0.33$overline{3}$ So: 33.$overline{3}$% = 0.$overline{3}$ (since 0.33$overline{3}$ = 0.$overline{3}$) Check: 0.$overline{3}$ × 100 = 33.$overline{3}$% ✓ Example 2: Percentage with Many Decimal Places Convert 87.5% and 87.55% to decimals and find their difference. Solution: 87.5% = 87.5 ÷ 100 = 0.875 87.55% = 87.55 ÷ 100 = 0.8755 Difference: 0.8755 - 0.875 = 0.0005 Or as fractions: $frac{8755}{10000} - frac{8750}{10000} = frac{5}{10000} = 0.0005$ Percentage to Decimal Problems Convert 66.$overline{6}$% to a decimal What decimal is equivalent to 12.5%? Convert 225% to a decimal Which is smaller as a decimal: 0.5% or 0.05%? Convert 33.75% to a decimal Repeating Decimal Conversions Repeating decimals require special techniques for conversion to fractions. We use algebraic methods to eliminate the repeating pattern and solve for the fraction. Example 1: Repeating Decimal to Fraction Convert 0.$overline{36}$ to a fraction in simplest form. Solution: Let x = 0.363636... Multiply by 100: 100x = 36.363636... Subtract: 100x - x = 36.363636... - 0.363636... 99x = 36 x = $frac{36}{99} = frac{4}{11}$ Check: $4 ÷ 11 = 0.363636...$ ✓ Example 2: Complex Repeating Decimal Convert 0.4$overline{6}$ to a fraction in simplest form. Solution: Let x = 0.4666... Multiply by 10: 10x = 4.666... Multiply by 100: 100x = 46.666... Subtract: 100x - 10x = 46.666... - 4.666... 90x = 42 x = $frac{42}{90} = frac{7}{15}$ Check: $7 ÷ 15 = 0.4666...$ ✓ Repeating Decimal Problems Convert 0.$overline{18}$ to a fraction Convert 0.1$overline{6}$ to a fraction in simplest form What fraction is equivalent to 0.$overline{285714}$? Convert 0.8$overline{3}$ to a fraction Which repeating decimal is larger: 0.$overline{3}$ or 0.$overline{33}$? Real-World Applications Conversion skills are essential in real-world contexts such as shopping discounts, interest rates, recipe adjustments, and statistical data interpretation. Example 1: Shopping Discount A store offers a discount of $frac{1}{5}$ off the original price. What percentage discount is this? If an item originally costs $80, what is the sale price? Solution: $frac{1}{5} = 0.2 = 20%$ discount Discount amount: $80 × 0.2 = $16 Sale price: $80 - $16 = $64 Or: $80 × 0.8 = $64 (since 100% - 20% = 80%) Example 2: Test Score Analysis Maria scored 42 out of 50 on a test. Express her score as: a) A fraction in simplest form b) A decimal c) A percentage Solution: a) Fraction: $frac{42}{50} = frac{21}{25}$ b) Decimal: $42 ÷ 50 = 0.84$ c) Percentage: $0.84 × 100 = 84%$ All three represent the same value: $frac{21}{25} = 0.84 = 84%$ Real-World Application Problems A recipe calls for 0.375 cups of butter. What fraction of a cup is this in simplest form? In a survey, 15 out of 24 people preferred option A. What percentage preferred option A? A bank offers 3.75% annual interest. Express this as a decimal and as a fraction in simplest form. If you completed 0.8 of a project and your partner completed $frac{5}{6}$, who completed more? A shirt is on sale for 30% off. If the original price was $45, what fraction of the original price are you paying? Cumulative Exercises Convert $frac{9}{16}$ to a decimal and percentage Convert 0.5625 to a fraction in simplest form and as a percentage Convert 33.$overline{3}$% to a fraction and decimal Convert 0.8$overline{3}$ to a fraction and percentage Which is larger: $frac{5}{8}$ as a decimal or 62.5% as a decimal? Convert 1.375 to a mixed number and percentage What fraction is equivalent to 0.0$overline{6}$? Convert 87.5% to a fraction and decimal Arrange from smallest to largest: 0.625, $frac{5}{8}$, 62.5% Create three different representations of the value 0.75 Show/Hide Answers Problem: Convert $frac{9}{16}$ to a decimal and percentage Answer: Decimal: $9 ÷ 16 = 0.5625$ Percentage: $0.5625 × 100 = 56.25%$ Problem: Convert 0.5625 to a fraction in simplest form and as a percentage Answer: Fraction: $0.5625 = frac{5625}{10000} = frac{9}{16}$ Percentage: $0.5625 × 100 = 56.25%$ Problem: Convert 33.$overline{3}$% to a fraction and decimal Answer: Fraction: $33.overline{3}% = frac{1}{3}$ Decimal: $33.overline{3}% ÷ 100 = 0.overline{3}$ Problem: Convert 0.8$overline{3}$ to a fraction and percentage Answer: Let $x = 0.8333...$, then $10x = 8.333...$ and $100x = 83.333...$ Subtract: $100x - 10x = 83.333... - 8.333... = 75$ $90x = 75$, so $x = frac{75}{90} = frac{5}{6}$ Percentage: $frac{5}{6} × 100 = 83.overline{3}%$ Problem: Which is larger: $frac{5}{8}$ as a decimal or 62.5% as a decimal? Answer: $frac{5}{8} = 0.625$, 62.5% = 0.625 They are equal: $0.625 = 0.625$ Problem: Convert 1.375 to a mixed number and percentage Answer: Mixed number: $1.375 = 1frac{375}{1000} = 1frac{3}{8}$ Percentage: $1.375 × 100 = 137.5%$ Problem: What fraction is equivalent to 0.0$overline{6}$? Answer: Let $x = 0.0666...$, then $10x = 0.666...$ and $100x = 6.666...$ Subtract: $100x - 10x = 6.666... - 0.666... = 6$ $90x = 6$, so $x = frac{6}{90} = frac{1}{15}$ Problem: Convert 87.5% to a fraction and decimal Answer: Fraction: $87.5% = frac{87.5}{100} = frac{875}{1000} = frac{7}{8}$ Decimal: $87.5% ÷ 100 = 0.875$ Problem: Arrange from smallest to largest: 0.625, $frac{5}{8}$, 62.5% Answer: All are equal: $0.625 = frac{5}{8} = 62.5%$ Problem: Create three different representations of the value 0.75 Answer: 1. Fraction: $frac{3}{4}$ 2. Decimal: 0.75 3. Percentage: 75% Also: $frac{6}{8}$, $frac{9}{12}$, 75.0%, 0.750, etc. Conclusion/Recap Mastering conversions between fractions, decimals, and percentages is a fundamental mathematical skill that enables flexible problem-solving and clear communication of numerical information. These interconnected representations allow us to choose the most appropriate form for different contexts, from exact measurements with fractions to comparative analysis with percentages. Proficiency in these conversions forms the foundation for more advanced mathematical concepts and real-world applications in finance, science, and data analysis. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
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