Ratios: – Understanding and applying ratio relationships.

RATIOS Understanding and applying ratio relationships. Grade 7 Mathematics: Ratios – Understanding and Applying Ratio Relationships Subtopic Navigator Introduction Concept of Ratios Simplifying Ratios Equivalent Ratios Ratio in Word Problems Applications and Mixed Problems Cumulative Exercises Conclusion Lesson Objectives Understand the meaning of ratios as a comparison between two or more quantities. Learn to simplify ratios to their lowest terms. Recognize and generate equivalent ratios. Apply ratio knowledge to solve real-life word problems. Lesson Introduction A ratio is a way of comparing two or more quantities of the same kind. It is written using the colon sign (:) or as a fraction. For example, if a class has 12 boys and 8 girls, the ratio of boys to girls is [latex]12:8 = frac{12}{8} = 3:2[/latex]. Ratios are important in sharing, scaling, map reading, and solving real-life problems. Concept of Ratios A ratio compares the relative size of two or more quantities. It tells us how many times one number contains another. Ratios can be written in three forms: [latex]a:b[/latex], [latex]frac{a}{b}[/latex], or "a to b". Example 1: Write the ratio of 15 apples to 25 oranges. Solution: Ratio = [latex]15:25[/latex] = [latex]frac{15}{25} = 3:5[/latex]. Example 2: In a library, there are 40 math books and 60 English books. Write the ratio of math to English books. Solution: Ratio = [latex]40:60 = frac{40}{60} = 2:3[/latex]. Exercises (Concept of Ratios) Write the ratio of 18 pencils to 24 erasers. In a box, there are 50 blue balls and 30 red balls. Write the ratio of blue to red balls. Simplifying Ratios Just like fractions, ratios can be simplified by dividing both terms by their greatest common divisor (GCD). A simplified ratio is easier to interpret and compare. Example 3: Simplify the ratio 45:60. Solution: GCD of 45 and 60 is 15. [latex]frac{45}{60} = frac{3}{4}[/latex]. So ratio = [latex]3:4[/latex]. Example 4: Simplify 28:42. Solution: GCD of 28 and 42 is 14. [latex]frac{28}{42} = frac{2}{3}[/latex]. Ratio = [latex]2:3[/latex]. Exercises (Simplifying Ratios) Simplify the ratio 64:80. Simplify the ratio 81:108. Equivalent Ratios Two ratios are equivalent if they represent the same comparison. They can be generated by multiplying or dividing both terms of a ratio by the same number. Example 5: Find two ratios equivalent to 2:5. Solution: Multiply both terms by 2 → [latex]4:10[/latex]. Multiply both terms by 3 → [latex]6:15[/latex]. Thus, equivalent ratios are [latex]4:10[/latex] and [latex]6:15[/latex]. Example 6: Are the ratios 12:18 and 20:30 equivalent? Solution: Simplify 12:18 → [latex]2:3[/latex]. Simplify 20:30 → [latex]2:3[/latex]. Since both simplify to the same ratio, they are equivalent. Exercises (Equivalent Ratios) Find two ratios equivalent to 5:7. Are the ratios 9:12 and 15:20 equivalent? Ratio in Word Problems Ratios are often used in sharing quantities, scaling recipes, or interpreting maps. Solving word problems with ratios requires setting up correct relationships. Example 7: Share ₦180 in the ratio 2:3. Solution: Total parts = [latex]2+3=5[/latex]. Value of 1 part = [latex]frac{180}{5} = 36[/latex]. First share = [latex]2 times 36 = ₦72[/latex]. Second share = [latex]3 times 36 = ₦108[/latex]. Example 8: A map scale is 1:50,000. What real distance is represented by 4 cm on the map? Solution: [latex]1 text{ cm} : 50,000 text{ cm}[/latex]. [latex]4 text{ cm} : (4 times 50,000) = 200,000 text{ cm}[/latex]. Convert to km: [latex]200,000 div 100,000 = 2 text{ km}[/latex]. Exercises (Word Problems) Share ₦360 in the ratio 5:7. A recipe uses flour and sugar in the ratio 3:2. If 12 cups of flour are used, how many cups of sugar are needed? Applications and Mixed Problems Example 9: In a class, the ratio of boys to girls is 7:5. If there are 84 boys, how many girls are there? Solution: [latex]frac{7}{5} = frac{84}{x}[/latex]. Cross multiply: [latex]7x = 84 times 5 = 420[/latex]. [latex]x = 60[/latex]. So there are 60 girls. Example 10: A piece of land is divided in the ratio 4:6:10 among three brothers. If the land is 400 hectares, find each share. Solution: Total parts = 20. Value of 1 part = [latex]frac{400}{20} = 20[/latex]. Shares = [latex]4 times 20 = 80[/latex], [latex]6 times 20 = 120[/latex], [latex]10 times 20 = 200[/latex]. Exercises (Applications) A map uses a scale of 1:25,000. What real distance does 6 cm represent? The ages of two brothers are in the ratio 5:7. If the older is 28 years, how old is the younger? Cumulative Exercises Write the ratio of 24 red balls to 36 green balls in simplest form. Find two ratios equivalent to 8:12. Are 14:21 and 20:30 equivalent ratios? Simplify 100:120. A sum of ₦450 is divided in the ratio 2:3. Find each share. A map scale is 1:100,000. What distance does 3 cm represent? The ratio of boys to girls is 9:11. If there are 99 boys, how many girls are there? In a school, the ratio of teachers to students is 1:25. If there are 40 teachers, how many students are there? Write the ratio of 45 minutes to 1 hour in simplest form. A recipe requires sugar and milk in the ratio 2:5. If 250g of milk is used, how much sugar is required? Conclusion/Recap Ratios help us compare quantities and solve real-life problems involving sharing, scaling, and proportional reasoning. By simplifying ratios, finding equivalent forms, and applying them to word problems, students can develop strong problem-solving skills. Ratios are also foundational in topics like percentages, probability, and geometry. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c