Understanding Ratio and Proportion: Definition, Examples, and Applications, Slides of Mathematics

Download Understanding Ratio and Proportion: Definition, Examples, and Applications and more Slides Mathematics in PDF only on Docsity! CONTENT STANDARD: At the end of the lesson, the learners demonstrate an understanding of the key concepts of ratio and proportion. PERFORMANCE STANDARD: The learners shall be able (1) formulate and solve problems involving ratio and proportion (2) use the concept of proportion in making life decisions. LEARNING COMPETENCY: Identify the different kinds of proportions and write examples of real-life situation for each CONCEPT: Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal. Here, a and b are any two integers. The ratio and proportion are the two important concepts, and it is the foundation to understand the various concepts in mathematics as well as in science. In our daily life activities, we use the concept of ratio and proportion such as in business while dealing with money, cooking, and so on. Sometimes, students get confused with the concept of ratio and proportion. Define what a ratio is. A ratio is a comparison of two quantities by division. The ratio of a to b can be written in two ways: a. as a:b (odds notation), where the symbol “:” is read as “is to”; or b. 𝑎 𝑏 , b≠ 0 (fractional notation) Example 1: The ratio of 10 to 12 may be written as 10:12 (odds notation) or 10 12 (fractional notation). https://tinyurl.com/yysw35du BUSINESS MATH Quarter 1, Week 3 Module Example 2: Simplify the following ratios to their simplest form. a. 1 3 : 3 4 b. 1 1 3 : 3 4 c. 0.09: 0.12 Solution: To simplify a ratio involving rational numbers, we multiply each of the quantities by the LCM of their denominators. a. The LCM of 3 and 4 is 12; hence, 1 3 : 3 4 → 12( 1 3 ) : 12( 3 4 ) → 4 : 9. Alternative solution: (Reduce/simplify the way you would a complex fraction.) 1 3 : 3 4 = 1 3 3 4 = 1 3  4 3 = 4 9 or 4 : 9 b. The LCM of the denominators of 1 1 3 : 3 4 is 12. 1 1 3 : 3 4 → 4 3 : 3 4 → 12( 4 3 ) : 12( 3 4 ) → 16 : 9. Alternative solution: 1 1 3 : 3 4 = 1 1 3 3 4 = 4 3 3 4 = 4 3  4 3 = 16 9 or 16 : 9 c. When a ratio is in decimal form, write its equivalent form without the decimals then reduce the ratio as needed. The ratio 0.09 : 0.12 may be cleared of decimals by multiplying both the numerator and denominator by 100. That is, 0.09 0.12 = 0.09  100 0.12  100 = 9 12 = 3 4 or 3 : 4. Alternative solution: Convert to fraction so as to show which power of 10 is to be multiplied to the ratio. 0.09 0.12 = 9 100 12 100 = 9 12 = 3 4 or 3 : 4. if a b = c d , then ad = bc. Example 1: We use the Fundamental Property of Proportions to verify that 7 8 = 14 16 = . Equating the cross products of the term gives 7 x 16 = 8 x 4. That is, 112 = 112 Example 2: Do the ratios 8 10 and 18 22 form a proportion? Explain. Solution: We compute for the cross of 8 10 and 18 22 . If they are equal, then 8 10 and 18 22 form a proportion. We have 8 x 22 = 176; while 10 x 8 = 180. Since 176≠180, 8 10 and 18 22 do not form a proportion. Example 3: We use the Fundamental Property of Proportions to find the missing term in a proportion as shown below. Given 7 108 = n 16 . We set cross products equal 8n = 7(16) or n = 14. Example 4: Solve for n: n+4 5 = n+2 3 Solution: The cross products are equated as shown below. 5(n – 2) = 3(n + 4) 5n – 10 = 3n + 12 2n = 22 n = 11 Check: Verify that 11 is the solution. Proportions can be used to solve a variety of word problems. The following examples show some of these word problems. In each case, we will translate the word problem into a proportion, and then solve the proportion using the fundamental property of proportions. Example 1: Eight tea bags are needed to make 5 liters of iced tea. How many tea bags are needed to make 15 liters of iced tea? Solution: Let t represent the number of tea bags. The problem translates to the following proportion: 𝑡 15 = 8 5 Note that both ratios in the proportion compare the same quantities; that is, both ratios compare number of tea bags to liters of iced tea. In words, the proportion says “t tea bags to 15 liters of iced tea as 8 tea bags to 5 liters of iced tea”. Using the fundamental property of proportions, we obtain the following: 𝑡 15 = 8 5 → 5 (t) = 15(8) t = 24 tea bags 15 5 Example 2: A manufacturer knows that during an average production run, out of 1,000 items produced by a certain machine, 25 will be defective. If the machine produces 2,030 items, how many can be expected to be defective? Solution: We let x represent the number of defective items and solve the following proportion: 𝑥 2030 = 25 1000 Example 3: If 1 out of 6 people buy a particular branded item, how many people can be expected to buy this item in a community of 6,000 people? Solution: Let p = the number of people buying the branded item. The ratio 𝑝 1000 defines the number of people p out of 6000 buying the branded item. This ratio is equal to 1 to 6. These two ratios are equal; that is, they form a proportion as given below. 𝑝 1000 = 1 6 Solving for p, we get p = 1000. So, 1000 people can be expected to buy the particular branded item. NAME: _______________________________________ GRADE AND SECTION: ___________ NAME: _____________________________________ DATE: ____________ SCORE: __________ SKILLS: Solve problems involving ratios. Directions: Solve and show your solutions for each given problem. 1. A sum of money is divided among Ron and Andy in the ratio 4 : 7. If Andy’s share is P616, find the total money. 2. Two numbers are in the ratio 5 : 7. On adding 1 to the first and 3 to the second, their ratio becomes 6/9 . Find the numbers. 3. The difference between two numbers is 33 and the ratio between them is 5 : 2. Find the numbers. 4. The ages of A and B are in the ratio 3 : 5. Four years later, the sum of their ages is 48. Find their present ages. 5. Ramon has notes of P100, P50 and P10, respectively. The ratio of these notes is 2: 3: 5 and the total amount is P2,00,000. Find the numbers of notes of each kind. BUSINESS MATH Quarter 1, Week 3 Activity Sheet Act. 1