Problem Solving and Rational Equations MATH 018 Combined Algebra S. Rook

2 Overview Section 7.6 in the textbook: –Solving proportions –Using proportions to solve word problems –Solving word problems about numbers –Solving word problems about work

Solving Proportions

4 Ratios & Proportions Ratio – a quotient of two quantities e.g. 1 / 5 or x / (x – 1) –An expression Proportion – a mathematical equation of two equal ratios Ex: 2 / 3 = 6 / 9 or x / (x – 1) = 7 / 8 –Only one rational expression on each side of the = We can solve a proportion as a rational equation We can also cross-multiply to solve a proportion e.g. x / 4 = 1 / 2

Solving Proportions (Example) Ex 1: Solve: a) b) 5

Using Proportions to Solve Word Problems

7 Key is to extract information from the word problem to set up proportions Proportions compare two units –e.g. cups of sugar to batches, number of cards to people, etc Align the units –e.g. Put cups of sugar is in the numerator and batches in the denominator for both sides of the proportion –DO NOT mix up the units on each side

Using Proportions to Solve Word Problems (Example) Ex 2: Solve: a) If 2 boxes of staples can fill 3 staplers, how many boxes of staples are needed to fill 9 staplers? b) On a certain map 1.5 inches represents 30 miles. How many inches on this same map would be used to represent 75 miles? 8

Solving Word Problems about Numbers

10 Solving Word Problems about Numbers Look for key words to help set up the problem Solve the resulting rational equation

Solving Word Problems about Numbers (Example) Ex 3: Solve: a) The quotient of three more than a number AND two times the number plus five is equal to three-fifths. Set up an equation and find the number. b) The quotient of twelve and a number equals two-thirds. Set up an equation and find the number. 11

Solving Word Problems about Work

13 Solving Word Problems about Work Allows us to consider situations where two people or objects perform some task –Work problems answer one of the following two questions: How long does it take a person or object to finish the task alone How long does it take both people or objects to finish the task together Key is to identify how much of a task can be completed in 1 hour –If a job can be finished in 2 hours, what part of the job has been completed in 1 hour?

Solving Word Problems about Work (Example) Ex 4: Solve: a) If pump A can fill a swimming pool in 3 hours and pump B can fill the same swimming pool in 2 hours, how long would it take pumps A and B to fill the pool if they worked together? b) A bricklayer’s apprentice can finish a job in 8 hours whereas the bricklayer and his apprentice can finish the same job together in 2 hours. How long would it take the bricklayer to finish the same job alone? 14

15 Summary After studying these slides, you should know how to do the following: –Understand the concept of a proportion –Solve a proportion –Solve word problems involving proportions –Solve these types of word problems that involve rational equations: Numbers Work Additional Practice –See the list of suggested problems for 7.6 Next lesson –Simplifying Complex Fractions (Section 7.7)