University:
California State University, Northridge
Course:
MATH 320 | Foundations of Higher Mathematics
Academic year:

2024
Views:

398

Pages:

25
Author:

Clark Morrison

§ 14.6 Problem Solving with Rational Equations Ratios and Rates Ratio is the quotient of two numbers or two quantities. The ratio of the numbers a and b can also be a written as a:b, or . b The units associated with the ratio are important. The units should match. If the units do not match, it is called a rate, rather than a ratio. 58 Proportions Proportion is two ratios (or rates) that are equal to each other. a c = b d We can rewrite the proportion by multiplying by the LCD, bd. This simplifies the proportion to ad = bc. This is commonly referred to as the cross product. 59 Solving Proportions Example Solve the proportion for x. x +1 5 = x+2 3 3( x + 1) = 5(x + 2 ) 3x + 3 = 5 x + 10 − 2x = 7 x = −7 2 Continued. 60 Solving Proportions Example Continued Substitute the value for x into the original equation, to check the solution. − 7 +1 5 2 = −7 + 2 3 2 −5 2=5 3 −3 2 true So the solution is x = − 7 2 61 Solving Proportions Example If a 170-pound person weighs approximately 65 pounds on Mars, how much does a 9000-pound satellite weigh? 170 - pound person on Earth 65 - pound person on Mars = 9000 - pound satellite on Earth x - pound satellite on Mars 170 x = 9000 ⋅ 65 = 585,000 x = 585000 / 170 ≈ 3441 pounds 62 Solving Proportions Example Given the following prices charged for various sizes of picante sauce, find the best buy. • 10 ounces for $0.99 • 16 ounces for $1.69 • 30 ounces for $3.29 Continued. 63 Solving Proportions Example Continued Size Price Unit Price 10 ounces $0.99 $0.99/10 = $0.099 16 ounces $1.69 $1.69/16 = $0.105625 30 ounces $3.29 $3.29/30 ≈ $0.10967 The 10 ounce size has the lower unit price, so it is the best buy. 64 Similar Triangles In similar triangles, the measures of corresponding angles are equal, and corresponding sides are in proportion. Given information about two similar triangles, you can often set up a proportion that will allow you to solve for the missing lengths of sides. 65 Similar Triangles Example Given the following triangles, find the unknown length y. 12 m 10 m 5m y Continued 66 Similar Triangles Example 1.) Understand Read and reread the problem. We look for the corresponding sides in the 2 triangles. Then set up a proportion that relates the unknown side, as well. 2.) Translate By setting up a proportion relating lengths of corresponding sides of the two triangles, we get 12 10 = 5 y Continued 67 Similar Triangles Example continued 3.) Solve 12 10 = 5 y 12 y = 5 ⋅10 = 50 y = 50 12 = 25 6 meters Continued 68 Similar Triangles Example continued 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. 12 10 60 = = 5 25 25 6 true State: The missing length of the triangle is 25 6 meters 69 Finding an Unknown Number Example The quotient of a number and 9 times its reciprocal is 1. Find the number. 1.) Understand Read and reread the problem. If we let n = the number, then 1 = the reciprocal of the number n Continued 70 Finding an Unknown Number Example continued 2.) Translate The quotient of is a number and 9 times its reciprocal n 1 9  n ÷ 1 = 1 Continued 71 Finding an Unknown Number Example continued 3.) Solve  1 n ÷ 9⋅  = 1  n 9 n ÷  =1 n n n⋅ =1 9 n2 = 9 n = 3,−3 Continued 72 Finding an Unknown Number Example continued 4.) Interpret Check: We substitute the values we found from the equation back into the problem. Note that nothing in the problem indicates that we are restricted to positive values.  1  1  3 ÷ 9 ⋅  = 1 − 3 ÷ 9⋅  =1  3  −3 3 ÷ 3 = 1 true − 3 ÷ −3 = 1 true State: The missing number is 3 or –3. 73 Solving a Work Problem Example An experienced roofer can roof a house in 26 hours. A beginner needs 39 hours to do the same job. How long will it take if the two roofers work together? 1.) Understand Read and reread the problem. By using the times for each roofer to complete the job alone, we can figure out their corresponding work rates in portion of the job done per hour. Time in hrs Experienced roofer 26 Beginner roofer 39 Together t Portion job/hr 1/26 /39 1/t Continued 74 Solving a Work Problem Example continued 2.) Translate Since the rate of the two roofers working together would be equal to the sum of the rates of the two roofers working independently, 1 1 1 + = 26 39 t Continued 75 Solving a Work Problem Example continued 3.) Solve 1 1 1 + = 26 39 t 1  1  1 78t  +  =  78t  26 39   t  3t + 2t = 78 5t = 78 t = 78 / 5 or 15.6 hours Continued 76 Solving a Work Problem Example continued 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. 1 1 1 + = 26 39 78 3 2 5 + = 78 78 78 5 true State: The roofers would take 15.6 hours working together to finish the job. 77 Solving a Rate Problem Example The speed of Lazy River’s current is 5 mph. A boat travels 20 miles downstream in the same time as traveling 10 miles upstream. Find the speed of the boat in still water. 1.) Understand Read and reread the problem. By using the formula d=rt, we can rewrite the formula to find that t = d/r. We note that the rate of the boat downstream would be the rate in still water + the water current and the rate of the boat upstream would be the rate in still water – the water current. Distance rate time = d/r Down 20 r + 5 20/(r + 5) Up 10 r–5 10/(r – 5) Continued 78 Solving a Rate Problem Example continued 2.) Translate Since the problem states that the time to travel downstairs was the same as the time to travel upstairs, we get the equation 20 10 = r +5 r −5 Continued 79 Solving a Rate Problem Example continued 3.) Solve 20 10 = r +5 r −5  20   10  (r + 5)(r − 5) (r + 5)(r − 5) =  r +5  r −5 20(r − 5) = 10(r + 5) 20r − 100 = 10r + 50 10r = 150 r = 15 mph Continued 80 Solving a Rate Problem Example continued 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. 20 10 = 15 + 5 15 − 5 20 10 = true 20 10 State: The speed of the boat in still water is 15 mph. 81

Problem Solving With Rational Equations

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