5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 To solve an equation with fractions Procedure: To solve a fractional equation Step 3. Determine LCM of denominators. Step 4. Multiply all terms by LCD (clearing denominators). Step 5. Solve equation as in previous lessons. Step 2. Determine any restrictions for the fractional equation. Step 6. Compare answer to restriction. If answer is the restricted value, write no solution. If not, go on to the next Step. Step 1. Factor all denominators. Step 7. Check

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 2 Solution:1. Find the LCD LCD=18 2. Multiply all fractions by the LCD. 3. Cancel the denominators Simplify and solve. 5. Check. Your Turn Problem #1

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 3 Rational Expressions have restrictions. Restrictions are values for the variable which would make the fraction undefined. In this section, we will have fractions where the denominator may contain the variable. If the denominator contains a variable, the equation is said to have restrictions. Procedure: To solve a fractional equation (for current section) Step 2. Determine LCM of denominators (LCD). Step 3. Multiply all terms by LCD (clearing denominators). Step 4. Solve equation as in previous lessons. Step 1. Determine any restrictions for the fractional equation. Step 5. Compare answer to restriction. If answer is the restricted value, write no solution. If not, go on to the next Step. Step 6. Check all answers which are not restrictions.

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 4 Solution: 1. State restrictions LCD=12x 2. Find LCD. 3.Multiply all terms by LCD Simplify and solve. 5. Answer not equal to restriction. 6. Check. Your Turn Problem #2

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 5 Solution: 1. State restrictions LCD=3x 2. Find LCD. 3.Multiply all terms by LCD. 4. Simplify and solve. 5. Answer not equal to restriction. 6. Check: (Not Shown) Your Turn Problem #3

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 6 Solution: 1. State restrictions 2. Find LCD. 3.Multiply all terms by LCD. 4. Simplify and solve. 5. Answer not equal to restriction. 6. Check (not shown). Your Turn Problem #4 This example was solved by multiplying both sides of the equation by the LCD. Cross multiplication is another method for solving fractional equations where there is only one fraction on each side. Using the Cross-multiplication property: Next Slide

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 7 Solution: 1. State restrictions 2. Find LCD. 3.Multiply all terms by LCD. 4. Simplify and solve. 5. Answer not equal to restriction. 6. Check. Your Turn Problem #5

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 8 Proportion Application Problems To recognize a problem is a proportion problem, a sentence will be given which can be written as a fraction. Examples: 3 out of 20 smoke; 4 panels generate 1500 watts; Procedure: To solve a proportion problem 2. Find the recognizable ratio and write it as a fraction. 1. Define the variable (what you are looking for?) 3. After the fraction is written write an equal sign. Then write a fraction using the remaining information with the defined variable. Make sure the fraction is written so both of the numerators have the same units and the denominators have the same units. Examples:, 4. Solve and answer the question.

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 9 Example 6. A chef estimates that 50 lbs of vegetables will serve 130 people. Using this estimate, how many pounds will be necessary to serve 156 people? Solution: Let x = number of pounds of vegetables for 156 people. Make units match: Equation: Solve by cross multiplying: Your Turn Problem #6 A carpenter estimates that he uses 8 studs for every 10 linear feet of framed walls. At this same rate, how many studs are needed for a garage with 35 linear feet of framed walls? Answer: 28 studs are needed

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 10 Let x = federal income tax on $125,000 Make units match: Reduce, then solve by cross multiplying: 1 4 Example 7: The federal income tax on $50,000 of income is $12,500. At this rate, what is the federal income tax on $125,000 of income? Solution: Your Turn Problem #7 If a home valued at $120,000 is assessed $2160 in real estate taxes, then how much, at the same rate, are the taxes on a home valued at $200,000? Answer: The property would be $3600.

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 11 Example 8. A sum of $1400 is to be divided between two people in the ratio of 3 to 5. How much does each person receive? Solve by cross multiplying: Solution:Let x = amount first person receives Let 1400 – x = amount second person receives 525 = amount for first person. To find amount for second person, 1400 – 525 = 875. A sum of $5000 is to be divided between two people in the ratio of 3 to 7. How much does each person receive? Your Turn Problem #8 Answer: The first person receives $1500 and the second person receives $3500.

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved State restrictions 3. Find LCD. 4.Multiply all terms by LCM. 5. Simplify and solve. 6. Since x can not equal  3, cross it out as a solution. 1. Factor Denominators ( ) 7. Check x = 1 (not shown). Your Turn Problem #9

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 13 Solution: 2. State restrictions 3. Find LCD. 4.Multiply all terms by LCD. 5. Simplify and solve. 6.Answers not equal to restrictions. 1. Factor Denominators 7. Check both –11 and 5 (not shown). Your Turn Problem #10

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 14 Example:If a carpenter can build a shed in 8 hours, then his rate of work is of the task completed per hour. Also, if this carpenter works for 5 hours, then his part of task completed is Work problems are common in algebra because of the use of the rate concept as well as their application to real-life situations. Work Problems Rate of work: The portion of a task completed per one unit of time. Important: In Work Problems, the time it takes a person (or machine) to do a task is the reciprocal of his/her rate! Basic Work Formula: (Rate of work)  (Time Worked) = (Part of Task completed) In most work problems, there will always be two individuals or devices who will contribute to the completion of the task. The formula for work problems will be to add the “part of task completed” for each individual or device and set it equal to “1” (for 1 completed job). Next Slide

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 15 Procedure: To Solve Work Problems Step 1. Write down the “let statements” (what you are looking for) and any information given in the problem. Step 2. Create a chart with columns “Rate of work”, “Time worked”, and “Part of task completed.” Step 3. To find rate of work, make a fraction with a numerator equal to 1; and a denominator equal to the time it would have taken the individual or device to complete the task alone. Step 4. To find time worked, write in the actual time the individual or device worked or will work on the task. Step 5. To find Part of task completed, use the formula: (rate of work)  (time worked) = (part of task completed) Step 6. To write equation, add the “part of task completed” and set equal to 1. Step 7. Solve and answer question. Next Slide

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 16 Example 11. A carpenter can build a shed working alone in 5 hours. His assistant working alone would take 10 hours to build a similar shed. If the two work together, how long will it take them to build the shed? Let x = time to build shed together Carpenter: 5hrs., Assistant: 10 hrs. 1. Define variable (Let statements) 2.Create chart. x x assistant carpenter Rate of work Time worked Part of Task completed 3.Write in rate of work for each. 4.Write in time worked for each. 5.Write in part of task completed by multiplying rate  time. 6.Obtain equation by adding the column and setting it equal to 1. 7.Solve and answer question. Your Turn Problem #11 Maria can do a certain job in 50 minutes, whereas it takes Kristin 75 minutes to do the same job. How long would it take then to do the job together? Answer: It would take them 30 minutes to do the job together. (Multiply all terms by LCD) The End. B.R