Joe and John are planning to paint a house together. John thinks that if he worked alone, it would take him 3 times as long as it would take Joe to paint the entire house. Working together, they can complete the job in 24 hours. How long will it take each of them, working alone to complete the job? Use the formula: Rate = Let x = the time it takes Joe to complete the job. Since, it takes John 3 times as long as Joe to complete the job: 3x = the time it takes John to complete the job. The work they are trying to do is to paint 1 house. Therefore, work = 1 for both Joe and John. So, Joe’s equation would be: Rate = & John’s equation would be: Rate = So, their combined rate would be: + The problem states, it takes 24 hours for them to paint 1, whole, house together – If they can paint 1 whole house in 24 hour hours, they paint th of the house per hour, Rate = . Therefore, we can make the following combined rate equation: + We must now solve this for x. So, we multiply their combined hourly rate by the LCD of 24x to clear the fractions and determine the time it would take them to complete 1 whole house, which gives you: 24x () = 24x ( + ) So, x = + Simplify each fraction on the right by cancelling any common factors x = + x = 24 + 8 x = 32 hours
Remember : Joe = x & John = 3x So, it will take 32 hours for Joe to complete the job by himself. Since, John = 3x, It will take John : 3(32) = 96 hours to complete the job by himself. Your Problem Kim and Kelly are planning to sew a quilt together. Kim thinks if she worked alone it would take her 4 times as long as it would take Kelly to sew the quilt by herself. Working together, they can complete the job in 2 hours. How long will it take each of them, working alone, to complete the job? Use the formula: Rate =
Complete the problem below and show all work: 12) Jim and Mark can build a bookshelf in 9 hours when working together. It takes Jim 3 times longer to build a bookshelf by himself than it does Mark. How long will it take EACH of them, working alone, to build a bookshelf?
Paper For Above instruction
Solving rational equations involves understanding the relationship between the work rates of individuals and using algebra to find unknown times. In this particular problem, Joe and John are working together to paint a house, and the goal is to determine how long each would take working alone. To solve this, we first establish variables for their individual times: let x be the time (in hours) Joe takes to paint the house alone. Since John takes three times as long as Joe, his time is 3x hours. The respective work rates are then 1/x for Joe and 1/3x for John, since rate is the reciprocal of time. When working together, their combined rate is the sum of their individual rates: (1/x + 1/3x). The problem states they finish in 24 hours, meaning their combined rate is 1/24 per hour. Setting up the equation: (1/x + 1/3x) = 1/24. To solve, find a common
denominator for the left side, which is 3x, yielding (3/3x + 1/3x) = 1/24; thus, (4/3x) = 1/24. Cross-multiplied, this gives 4 * 24 = 3x; simplifying, 96 = 3x, so x = 32. Therefore, Joe takes 32 hours to paint the house alone. John, taking 3 times as long, requires 3 * 32 = 96 hours. This demonstrates how algebraic expressions reflecting individual work rates combined to solve for unknown times. Similarly, in a quilting scenario with Kim and Kelly, Kim takes four times as long as Kelly, and together they complete the quilt in 2 hours. Using the same approach, let k be Kelly’s time; Kim’s time is 4k. The rates are 1/k and 1/4k, respectively, with combined rate 1/2 (since they finish in 2 hours). Setting: (1/k + 1/4k) = 1/2.
Simplifying, (4/4k + 1/4k) = 1/2; thus, (5/4k) = 1/2. Cross-multiplied: 5 * 2 = 4k; solving gives 10 = 4k, so k = 2.5 hours. Kim’s time: 4 * 2.5 = 10 hours. These approaches show how algebra can be applied to real-world problems involving work rates and time, especially when the variables involve proportional relationships. Extending to the bookshelf problem, Jim and Mark can build a bookshelf in 9 hours working together. Jim takes three times longer than Mark to do it alone. Let m be Mark’s time: Jim’s time is 3m hours. Their work rates are 1/m and 1/3m, and their combined rate is 1/9. Set up the equation: (1/m + 1/3m) = 1/9. Simplify as before: (3/3m + 1/3m) = 1/9; (4/3m) = 1/9. Cross-multiplied: 4 * 9 = 3m; 36 = 3m; so m = 12 hours. Jim’s time: 3 * 12 = 36 hours. These solutions demonstrate the value of translating word problems into algebraic equations based on work rates, which then allows for solving unknown individual times. Each problem follows similar principles but adapts for the specific proportional relationships and total time given.
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