Ratios: Understanding and calculating ratios.

Understanding Ratios. Grade 7 Mathematics: Ratios - Understanding and Calculating Ratios Subtopic Navigator Introduction to Ratios Writing and Simplifying Ratios Equivalent Ratios and Scaling Ratio Tables Comparing Two Quantities with Ratios Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Define a ratio and identify ratios in real-world situations. Write ratios in three different forms (a:b, a to b, a/b). Simplify ratios to their simplest form. Generate and recognize equivalent ratios using scaling. Use ratio tables to solve for missing values. Apply ratios to solve practical problems like mixing paint or comparing speeds. Introduction to Ratios Have you ever followed a recipe, mixed juice concentrate, or compared the number of boys to girls in your class? If so, you've already used ratios! A ratio is a way to compare two or more quantities. It tells us how much of one thing there is compared to another. Ratios are everywhere—in maps (scale), in cooking, in building plans, and even in sports statistics. Understanding ratios helps us make sense of relationships between numbers and solve problems in a proportional way. Writing and Simplifying Ratios A ratio compares quantities of the same kind. The quantities must be in the same units to be compared meaningfully. We can write a ratio in three ways: using the word "to" (3 to 4), using a colon (3:4), or as a fraction (3/4). All three are read as "three to four". The order matters! The ratio of apples to oranges is different from the ratio of oranges to apples. Key Principle: To simplify a ratio, divide both parts by their Greatest Common Factor (GCF). [latex]text{Simplified Ratio} = frac{a div text{GCF}}{b div text{GCF}}[/latex] Where: a = first quantity in the ratio b = second quantity in the ratio GCF = Greatest Common Factor of a and b Example 1: Basic Level In a basket, there are 12 apples and 8 oranges. What is the ratio of apples to oranges? Write it in all three forms and simplify. Solution: Apples to Oranges = 12 to 8. Forms: 12:8, 12 to 8, [latex]frac{12}{8}[/latex]. To simplify, find the GCF of 12 and 8, which is 4. [latex]12 div 4 = 3[/latex] [latex]8 div 4 = 2[/latex] Simplified Ratio = 3:2, 3 to 2, [latex]frac{3}{2}[/latex]. Example 2: Unit Conversion The length of a room is 4 meters, and its width is 300 centimeters. Find the ratio of length to width in simplest form. Solution: First, convert to the same unit. 1 meter = 100 cm, so 4 m = 400 cm. Ratio (length to width) = 400 cm to 300 cm. Simplify by dividing by 100: 4 to 3. Find GCF of 400 and 300 = 100. [latex]400 div 100 = 4[/latex] [latex]300 div 100 = 3[/latex] Simplified Ratio = 4:3. Practice Problems In a garden, there are 15 rose plants and 10 lily plants. Write the ratio of roses to lilies in simplest form. A recipe calls for 2 cups of flour and 4 cups of milk. What is the ratio of flour to milk? Simplify the ratio 24:18. A class has 18 girls and 12 boys. What is the ratio of boys to girls? Simplify. Write the ratio of 50 minutes to 2 hours in simplest form (convert hours to minutes first). Equivalent Ratios and Scaling Equivalent ratios are different ratios that represent the same relationship. You can create equivalent ratios by multiplying or dividing both parts of a ratio by the same non-zero number. This is called scaling. For example, 1:2 is equivalent to 2:4, 3:6, and 50:100. They all mean "one part for every two parts". Key Principle: If [latex]a:b[/latex] is a ratio, then [latex]a times k : b times k[/latex] (where k is any number) is an equivalent ratio. Similarly, [latex]a div k : b div k[/latex] (k ≠ 0) is also equivalent. Example 1: Finding Equivalent Ratios Find three ratios equivalent to 5:7. Solution: Multiply both parts by 2: [latex]5 times 2 : 7 times 2 = 10:14[/latex] Multiply both parts by 3: [latex]15:21[/latex] Multiply both parts by 10: [latex]50:70[/latex] So, 10:14, 15:21, and 50:70 are all equivalent to 5:7. Example 2: Checking for Equivalence Are the ratios 16:20 and 36:45 equivalent? Solution: Simplify both ratios to their simplest form and compare. 16:20 simplifies (divide by 4) to 4:5. 36:45 simplifies (divide by 9) to 4:5. Both simplify to 4:5, so they are equivalent ratios. Original Ratio (Flour:Sugar) Scale Factor (k) Equivalent Ratio 2:1 2 4:2 2:1 5 10:5 2:1 0.5 1:0.5 Table: Scaling a 2:1 ratio creates equivalent ratios for different batch sizes. Practice Problems Give two equivalent ratios for 3:8. Are 21:14 and 12:8 equivalent? Show your work. Multiply the ratio 2:5 by 6 to create an equivalent ratio. Divide the ratio 100:60 by 20 to create an equivalent ratio. Which of the following is NOT equivalent to 9:12? a) 3:4 b) 18:24 c) 27:30 d) 36:48 Ratio Tables A ratio table is a tool that lists equivalent ratios in an organized way. It helps us find missing values when we scale a ratio up or down. To complete a ratio table, multiply or divide each term in a column by the same number to move to the next column. Example 1: Completing a Ratio Table If 3 bags of popcorn cost $6, use a ratio table to find the cost of 7 bags. Solution: Set up the ratio Bags : Cost as 3 : 6. To find the unit rate, divide both by 3: 1 bag costs $2. Now, scale up: Multiply by 7 to find cost for 7 bags. [latex]1 times 7 : 2 times 7 = 7 : 14[/latex] So, 7 bags cost $14. Ratio Table: Bags317 Cost ($)6214 Example 2: Finding a Missing Value The ratio of red to blue marbles in a jar is 4:5. If there are 20 red marbles, how many blue marbles are there? Use a ratio table. Solution: Red : Blue = 4 : 5. We know Red = 20. How do we get from 4 to 20? Multiply by 5. So, multiply the blue side by the same factor (5). [latex]5 times 5 = 25[/latex] There are 25 blue marbles. Red Marbles420 Blue Marbles525 Practice Problems If 5 pencils cost $1.50, use a ratio table to find the cost of 12 pencils. The ratio of wins to losses for a team is 7:3. If they have 21 wins, how many losses do they have? Complete the ratio table: 24?16 51020? The top row is related to the bottom row by the same ratio. A car travels 150 km on 10 liters of fuel. How far can it travel on 25 liters? Use a table. For every 4 meters of fabric, 3 dresses can be made. How many dresses can be made from 28 meters of fabric? Comparing with Ratios Ratios are powerful tools for direct comparison. We often use them to see which mixture is stronger, which recipe is sweeter, or which class has a higher proportion of students. To compare two ratios, express them in the same simplified form or scale them so one of the terms is identical. Example 1: Comparing Concentrations Mixture A is made with 2 cups of juice concentrate and 5 cups of water. Mixture B is made with 3 cups of concentrate and 8 cups of water. Which mixture is stronger (has a higher concentrate-to-water ratio)? Solution: Find the ratio of concentrate to water for each. Mixture A: 2:5 or [latex]frac{2}{5} = 0.4[/latex] Mixture B: 3:8 or [latex]frac{3}{8} = 0.375[/latex] Compare 0.4 and 0.375. 0.4 > 0.375. Therefore, Mixture A has a higher concentrate ratio and is stronger. Example 2: Part-to-Whole Comparison In a survey, 15 out of 25 students in Class 7A like soccer. In Class 7B, 12 out of 20 students like soccer. Which class has a greater proportion of soccer fans? Solution: Find the ratio of fans to total students for each class (a part-to-whole ratio). Class 7A: 15:25 simplifies to 3:5 or [latex]frac{3}{5} = 0.6[/latex] Class 7B: 12:20 simplifies to 3:5 or [latex]frac{3}{5} = 0.6[/latex] The proportions are equal (0.6). Both classes have the same proportion of soccer fans. Practice Problems Team X scores 40 goals in 10 games. Team Y scores 56 goals in 14 games. Which team scores more goals per game? Which is a better deal: 3 books for $12 or 5 books for $18? Compare using ratios. In a box of 30 candies, 12 are chocolates. In another box of 45 candies, 20 are chocolates. Which box has a greater fraction of chocolates? Compare the ratios 7:11 and 5:8. Which is larger? Two paint shades use yellow and blue. Shade 1 uses 3 parts yellow to 4 parts blue. Shade 2 uses 5 parts yellow to 7 parts blue. Which shade is yellower? Real-World Applications Ratios are not just math class problems; they are used daily. Chefs use ratios for recipes. Builders use scale ratios on blueprints. Map readers use ratios to understand distance. Artists use ratios for proportions in drawings. Understanding ratios allows you to adapt recipes, read maps correctly, and mix accurate solutions. Example 1: Map Scales The scale on a map is 1:50,000. This means 1 cm on the map represents 50,000 cm in real life. How many kilometers does 4 cm on the map represent? Solution: Map : Real = 1 cm : 50,000 cm. 4 cm on map = [latex]4 times 50,000 = 200,000[/latex] cm in real life. Convert cm to km (1 km = 100,000 cm). [latex]200,000 div 100,000 = 2[/latex] km. So, 4 cm on the map represents 2 kilometers. Example 2: Mixing Paint A specific green paint is made by mixing blue and yellow paint in a 2:3 ratio. If you have 10 liters of blue paint, how much yellow paint do you need to make the same shade? How much green paint will you make in total? Solution: Ratio Blue : Yellow = 2 : 3. We have 10 liters of blue. 10 is [latex]2 times 5[/latex]. So, multiply the yellow part by the same factor: [latex]3 times 5 = 15[/latex] liters of yellow needed. Total green paint = Blue + Yellow = 10 L + 15 L = 25 Liters. Practice Problems A model car is built at a scale of 1:18. If the model is 8 inches long, how long is the real car in feet? A fruit punch recipe uses cranberry juice and ginger ale in a 5:9 ratio. How much cranberry juice is needed with 36 cups of ginger ale? To make a light orange paint, you mix red and yellow in a 1:4 ratio. If you use 6 cups of red, how many cups of yellow do you need? The ratio of students to computers in a lab is 4:1. If there are 28 students, how many computers are there? A picture is 4 inches wide and 6 inches long. What is the ratio of width to length? If you enlarge it so the length is 15 inches, what will the new width be (keeping the same ratio)? Cumulative Exercises These problems combine concepts from all sections. Read carefully and decide which ratio tool is best to use. A rectangle's length and width are in the ratio 7:4. If the length is 35 cm, find its width and then its perimeter. In a bag of marbles, the ratio of red to blue marbles is 5:6. The ratio of blue to green marbles is 3:2. Find the ratio of red to green marbles. To make a certain color of purple, mix red and blue paint in a 3:5 ratio. If you want 40 liters of purple paint, how many liters of red and blue do you need separately? The ratio of Sarah's age to her mother's age is 2:7. If Sarah is 12 years old, how old is her mother? Which is a faster speed: 120 km in 2 hours or 210 km in 3.5 hours? Compare using ratio of distance to time. A recipe for 12 cookies uses 2 cups of flour. How many cups of flour are needed for 30 cookies? The scale on a drawing is 1 cm : 2 m. If a room is 8 cm long on the drawing, how long is the actual room? Simplify the ratio 1.5 : 0.5 : 2. A school has a student-teacher ratio of 25:1. If the school has 50 teachers, approximately how many students does it have? Two numbers are in the ratio 4:9. If the larger number is 108, find the smaller number. Show/Hide Answers Problem: Rectangle ratio 7:4, length=35 cm. Find width and perimeter. Answer: 7:4 = 35:w. So w = (35*4)/7 = 20 cm. Perimeter = 2*(35+20) = 110 cm. Problem: Red:Blue = 5:6, Blue:Green = 3:2. Find Red:Green. Answer: Make Blue the same in both ratios. Red:Blue = 5:6 and Blue:Green = 3:2 = 6:4 (multiplied by 2). So Red:Blue:Green = 5:6:4. Therefore, Red:Green = 5:4. Problem: Red:Blue = 3:5, total purple = 40 L. Answer: Total parts = 3+5=8. Red = (3/8)*40 = 15 L. Blue = (5/8)*40 = 25 L. Problem: Sarah:Mother = 2:7, Sarah = 12. Answer: 2:7 = 12:M. M = (12*7)/2 = 42 years old. Problem: Compare speeds 120km/2hr vs 210km/3.5hr. Answer: First speed = 120/2 = 60 km/hr. Second speed = 210/3.5 = 60 km/hr. They are the same speed. Problem: 12 cookies need 2 cups flour. Flour for 30 cookies? Answer: Cookies:Flour = 12:2 = 6:1. For 30 cookies, flour = 30/6 = 5 cups. Problem: Scale 1 cm : 2 m. Drawing length 8 cm. Answer: Actual length = 8 cm * 2 m/cm = 16 m. Problem: Simplify 1.5 : 0.5 : 2. Answer: Multiply all by 2 to eliminate decimals: 3 : 1 : 4. This is the simplest form. Problem: Student:Teacher = 25:1. 50 teachers. Answer: Students = 25 * 50 = 1250 students. Problem: Two numbers ratio 4:9, larger=108. Answer: 4:9 = smaller:108. Smaller = (108*4)/9 = 48. Conclusion/Recap In this lesson, we explored the fundamental concept of ratios. We learned that a ratio is a comparative relationship between two or more quantities, and it can be written in three forms: using "to", a colon, or as a fraction. The order in which we write the ratio is crucial because it specifies what is being compared to what. Mastering ratios is the first step toward understanding more complex concepts like rates, proportions, and percentages, which you will encounter in later math studies and in everyday life. Keep looking for ratios around you—they’re hidden in recipes, sports statistics, and even in the design of your favorite video games! Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c