Lesson Notes By Weeks and Term v3 - Junior Secondary 2
Fractions
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Fractions
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Subject: General Mathematics
Class: Junior Secondary 2
Term: 1st Term
Week: 9
Theme: Numbers And Numeration
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Convert simple fractions to ratios, decimals and percentages and vice versa Solve quantitative reasoning problems related to conversion of fractions to ratios, decimals and percentages;
This section provides a detailed explanation of the interconversion between fractions, ratios, decimals, and percentages. A. Fractions A fraction represents a part of a whole or a collection of objects. It is expressed in the form $\frac{a}{b}$, where 'a' is the numerator (the number of parts being considered) and 'b' is the denominator (the total number of equal parts the whole is divided into). The denominator 'b' cannot be zero.
B. Conversions
1. Fraction to Ratio and Vice Versa Fraction to Ratio: A fraction $\frac{a}{b}$ can be expressed as a ratio $a:b$. It represents a part-to-whole relationship (a parts out of b total parts) or, more commonly in a ratio, a comparison of two quantities (a to b). When expressing a fraction as a ratio, ensure the ratio is in its simplest form.
Method: Write the numerator as the first term and the denominator as the second term, separated by a colon (:). Simplify the ratio by dividing both terms by their highest common factor (HCF).
Example 1: Convert $\frac{3}{5}$ to a ratio.
Solution: $\frac{3}{5}$ is equivalent to the ratio $3:5$.
Example 2: In a class, $\frac{2}{3}$ of the students are boys. Express this as a ratio of boys to total students.
Solution: The fraction $\frac{2}{3}$ means 2 parts (boys) out of 3 total parts (students). So the ratio of boys to total students is $2:3$.
Ratio to Fraction: A ratio $a:b$ can be expressed as a fraction $\frac{a}{b}$ (if comparing two distinct quantities, like apples to oranges) or $\frac{a}{a+b}$ (if 'a' is a part of a total 'a+b'). For consistency with the definition of a fraction as a part of a whole, consider the part-to-whole relationship when converting ratios to fractions in this context. If the ratio is given as part-to-part (e.g., boys to girls), sum the parts to get the whole.
Method: If the ratio represents part-to-whole (e.g., boys to total students), write the first term as the numerator and the second term as the denominator. If the ratio represents part-to-part (e.g., boys to girls), sum the terms to find the total, then write the desired part over the total.
Example 3: Convert the ratio $2:7$ (part-to-whole) to a fraction.
Solution: The ratio $2:7$ means 2 parts out of a total of 7 parts. The fraction is $\frac{2}{7}$.
Example 4: The ratio of oranges to mangoes in a basket is $3:4$. What fraction of the fruits are oranges?
Solution: Total parts = $3 \text{ (oranges)} + 4 \text{ (mangoes)} = 7$. The fraction of oranges is $\frac{3 \text{ (oranges)}}{7 \text{ (total fruits)}}$.
2. Fraction to Decimal and Vice Versa Fraction to Decimal: A decimal is another way of representing a fraction, particularly those with denominators that are powers of 10 (10, 100, 1000, etc.).
Method: Divide the numerator by the denominator.
Example 5: Convert $\frac{3}{4}$ to a decimal.
Solution: $3 \div 4 = 0.75$.
Example 6: Convert $\frac{5}{8}$ to a decimal.
Solution: $5 \div 8 = 0.625$.
Decimal to Fraction: Method: Write the decimal as a fraction with a denominator that is a power of 10, corresponding to the number of decimal places. Then simplify the fraction to its lowest terms. 1 decimal place: denominator is 10 2 decimal places: denominator is 100 3 decimal places: denominator is 1000, and so on.
Example 7: Convert $0.6$ to a fraction.
Solution: $0.6 = \frac{6}{10}$. Simplify by dividing numerator and denominator by their HCF (2): $\frac{6 \div 2}{10 \div 2} = \frac{3}{5}$.
Example 8: Convert $0.125$ to a fraction.
Solution: $0.125 = \frac{125}{1000}$.
Simplify: $\frac{125 \div 25}{1000 \div 25} = \frac{5}{40}$. $\frac{5 \div 5}{40 \div 5} = \frac{1}{8}$.
3. Fraction to Percentage and Vice Versa Fraction to Percentage: A percentage is a fraction where the denominator is
1
0
0. The symbol '%' means "per hundred" or "out of 100".
Method: Multiply the fraction by 100%.
Example 9: Convert $\frac{3}{5}$ to a percentage.
Solution: $\frac{3}{5} \times 100\% = \frac{3 \times 100}{5}\% = \frac{300}{5}\% = 60\%$.
Example 10: Convert $\frac{1}{4}$ to Example 8: Convert $0.125$ to a fraction.
Solution: $0.125 = \frac{125}{1000}$.
Simplify: $\frac{125 \div 25}{1000 \div 25} = \frac{5}{40}$. $\frac{5 \div 5}{40 \div 5} = \frac{1}{8}$.
3. Fraction to Percentage and Vice Versa Fraction to Percentage: A percentage is a fraction where the denominator is
1
0
0. The symbol '%' means "per hundred" or "out of 100".
Method: Multiply the fraction by 100%.
Example 9: Convert $\frac{3}{5}$ to a percentage.
Solution: $\frac{3}{5} \times 100\% = \frac{3 \times 100}{5}\% = \frac{300}{5}\% = 60\%$.
Example 10: Convert $\frac{1}{4}$ to a percentage.
Solution: $\frac{1}{4} \times 100\% = 25\%$.
Percentage to Fraction: Method: Divide the percentage by 100 and simplify the resulting fraction to its lowest terms.
Example 11: Convert $75\%$ to a fraction.
Solution: $75\% = \frac{75}{100}$.
Simplify by dividing by HCF (25): $\frac{75 \div 25}{100 \div 25} = \frac{3}{4}$.
Example 12: Convert $12.5\%$ to a fraction.
Solution: $12.5\% = \frac{12.5}{100}$. To remove the decimal, multiply numerator and denominator by 10: $\frac{12.5 \times 10}{100 \times 10} = \frac{125}{1000}$.
Simplify: $\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$.
4. Decimal to Percentage and Vice Versa Decimal to Percentage: Method: Multiply the decimal by 100%. This shifts the decimal point two places to the right.
Example 13: Convert $0.8$ to a percentage.
Solution: $0.8 \times 100\% = 80\%$.
Example 14: Convert $0.045$ to a percentage.
Solution: $0.045 \times 100\% = 4.5\%$.
Percentage to Decimal: Method: Divide the percentage by
1
0
0. This shifts the decimal point two places to the left.
Example 15: Convert $30\%$ to a decimal.
Solution: $30\% = \frac{30}{100} = 0.30 \text{ or } 0.3$.
Example 16: Convert $7.25\%$ to a decimal. * Solution: $7.25\% = \frac{7.25}{100} = 0.0725$. C. Quantitative Reasoning Problems These problems require the application of the above conversion skills in real-world scenarios, often involving word problems or patterns. Students must first identify the quantities involved, choose the appropriate conversion, and then solve the problem systematically.
Phase 1: Introduction and Recall (10 minutes)
Teacher Activity: Begins by briefly revising the concept of a fraction from JSS1, using simple real-life examples like sharing a loaf of bread or dividing a piece of land in a community. Asks probing questions to gauge students' prior knowledge: "What is a fraction?", "Give an example of when you use fractions at home or in the market." Introduces the lesson topic: "Today, we will learn how fractions relate to other forms of numbers like ratios, decimals, and percentages, and how to convert between them." Student Activity: Participates in the revision, answering questions based on prior knowledge. Recalls and provides examples of fractions.
Phase 2: Concept Development and Explanations (30 minutes)
Teacher Activity: Systematically explains each conversion type (Fraction to Ratio, Decimal, Percentage; and vice versa for all), writing clear steps and working through the examples provided in Section 2 on the whiteboard. For each conversion, uses Nigerian-relevant examples.
For instance: "If Mr. Audu shared his land such that his son got $\frac{2}{5}$ of it, how do we express this as a ratio of the son's share to the total land?" "A tailor bought fabric for N150.00 and sold it for N180.
0
0. What is the profit as a fraction, decimal, and percentage of the cost price?" "In a community meeting, 0.7 of the attendees were women. What fraction and percentage of attendees were women?" Encourages questions and clarifies any misconceptions immediately. Demonstrates how to handle quantitative reasoning by breaking down a simple problem into steps involving conversion.
Student Activity: Pays close attention to the explanations and examples. Copies notes and worked examples into their notebooks. Asks questions for clarification. Attempts to follow the steps of the examples.
Phase 3: Guided Practice (20 minutes)
Teacher Activity: Distributes a few practice questions (from Section 4) to students. Guides students through solving the problems, perhaps by calling on students to contribute steps. Walks around the classroom, providing individual assistance and feedback. Corrects solutions on the board, ensuring students understand the correct procedure.
Student Activity: Works individually or in small groups on the guided practice questions. Applies the learned conversion methods. Discusses solutions with peers and the teacher. Corrects their work based on the teacher's guidance and board solutions.
Phase 4: Independent Practice and Conclusion (10 minutes)
Teacher Activity: Assigns a few independent practice questions (from Section 5) for students to attempt before the end of the lesson or as homework. Summarizes the key learning points of the lesson, reiterating the importance of interconverting between fractions, ratios, decimals, and percentages. Emphasizes the real-life relevance of these skills in various Nigerian contexts.
Student Activity: Begins working on the independent practice questions. Listens attentively to the summary. Notes down homework if applicable.
Question 1: In a local government election, Mallam Idris won $\frac{7}{10}$ of the total votes. a) Express Mallam Idris's share of votes as a decimal. b) Express Mallam Idris's share of votes as a percentage. c) What is the ratio of votes for Mallam Idris to the total votes?
Solution 1: a)
Fraction to Decimal: Divide the numerator by the denominator. $\frac{7}{10} = 7 \div 10 = 0.7$ b)
Fraction to Percentage: Multiply the fraction by 100%. $\frac{7}{10} \times 100\% = \frac{700}{10}\% = 70\%$ c)
Fraction to Ratio: Write the numerator as the first term and the denominator as the second. $\frac{7}{10}$ is equivalent to $7:10$.
Commentary: This question tests direct conversion from a fraction to its decimal, percentage, and ratio forms, a core objective.
Question 2: A bag of rice weighing 50kg has a price tag that says "15% discount for bulk purchase". a) Express the discount percentage as a fraction in its simplest form. b) Express the discount percentage as a decimal.
Solution 2: a)
Percentage to Fraction: Divide the percentage by 100 and simplify. $15\% = \frac{15}{100}$ To simplify, divide both numerator and denominator by their HCF, which is 5: $\frac{15 \div 5}{100 \div 5} = \frac{3}{20}$ b)
Percentage to Decimal: Divide the percentage by 100. $15\% = \frac{15}{100} = 0.15$
Commentary: This question focuses on converting a percentage to a fraction and a decimal, using a common Nigerian market scenario.
Question 3: Mrs. Ngozi spent N3,500 out of her N5,000 weekly budget on food items. a) What fraction of her budget did she spend on food? (Simplify your answer) b) Express this fraction as a percentage.
Solution 3: a)
Identify the fraction: Fraction spent on food = $\frac{\text{Amount spent}}{\text{Total budget}} = \frac{N3,500}{N5,000}$ Simplify the fraction: Divide both by common factors. $\frac{3500}{5000} = \frac{35}{50}$ (Dividing by 100) $\frac{35 \div 5}{50 \div 5} = \frac{7}{10}$ (Dividing by 5) So, she spent $\frac{7}{10}$ of her budget on food. b)
Fraction to Percentage: Multiply the fraction by 100%. $\frac{7}{10} \times 100\% = 7 \times 10\% = 70\%$
Commentary: This problem integrates fraction formation from real data, simplification, and then conversion to percentage, addressing quantitative reasoning.
Question 4: The ratio of boys to girls in a JSS2 class is $3:5$. a) What fraction of the class are boys? b) What percentage of the class are girls?
Solution 4: a)
Ratio to Fraction (Part-to-whole): Total parts = $3 \text{ (boys)} + 5 \text{ (girls)} = 8$ parts. Fraction of boys = $\frac{\text{Number of boy parts}}{\text{Total parts}} = \frac{3}{8}$. b) Ratio to Fraction (Part-to-whole) then to Percentage: Fraction of girls = $\frac{\text{Number of girl parts}}{\text{Total parts}} = \frac{5}{8}$. Convert $\frac{5}{8}$ to a percentage: $\frac{5}{8} \times 100\% = \frac{500}{8}\% = 62.5\%$
Commentary: This addresses quantitative reasoning using ratios, requiring finding the total parts before converting to fractions and then percentages.
Budgeting and Household Finance: Families in Nigeria use fractions, decimals, and percentages to manage their monthly budgets. For example, allocating $\frac{1}{3}$ of income to rent, $0.4$ to food, and the remaining as savings or other expenses. These conversions help determine the actual amount of money allocated to each category and allow for easy comparison. Understanding discounts (e.g., "10% off during Black Friday sales" at Shoprite or local markets) or interest rates on loans requires converting percentages to decimals or fractions for calculations.
Market Transactions and Entrepreneurship: Small business owners (e.g., a seamstress, a petty trader in the market) need these skills to calculate profit margins (e.g., "my profit is 25% of my cost price"), discounts offered to customers, or proportions of ingredients in a recipe. A farmer selling produce might state that $\frac{3}{4}$ of his yam harvest has been sold, which can easily be converted to $75\%$ to communicate sales progress.
Community Development and Demographics: Understanding community statistics often involves these concepts. For example, if a report states that $0.6$ of the children in a community are enrolled in primary school, converting this to $60\%$ or $\frac{3}{5}$ makes it more relatable and easier to understand for community leaders or parents. Ratios are used to compare populations (e.g., ratio of men to women in a town) or resource allocation among different age groups.