Module 5 Ratios, Rates, & Proportions Section 5.1: Ratios and Rates A ratio is the comparison of two quantities that have the same units. 18 6 We can express this ratio three different ways: 18 to 6 18:6 18 6 P. 1 of 6 CQ-5-02. Write 0:84 as a ratio (in fraction) in simplest form. 1 1 27 7 5 14 P. 4 of 6 CQ-5-01. One day, a veterinary clinic treated 6 male dogs, 10 female dogs, male cats, and 4 female cats. What is the ratio of dogs to cats that were seen that day 1. 71. to 16 to. 16. to 4. 104. to 16 7 2 1 P. 2 of 6 Ratios Dorothy earns $500 weekly. Out of that $500 gross pay, $125 is withheld for federal taxes and $60 is withheld for state taxes. What is the ratio of the amount withheld for state taxes to gross pay state taxes 60 gross pay 500 25 $ of it is withheld for state taxes for every $25 Dorothy makes. P. 5 of 6 Simplest Form of a Ratio A ratio is in simplest form when the two numbers do not have a common factor and both numbers are whole numbers. Rates A rate is a comparison of two quantities that have different units. $1.50 18 6 18 1 6 2 Simplest form P. of 6 18 The rate of dollars to is dollars 1.50 18 P. 6 of 6 1 of 6
CQ-5-0. Write 252 per 8 gallons as a rate in simplest form. 1. 28 11. gallon 6 2 gallons 125 4 gallons 64 gallons P. 7 of 6 CQ-5-05. Write $6,188 for 82 shares of stock as a unit price. Round to the nearest cent (hundredth), if necessary. $1.25/ share $69.24/ share $80.92/ share $75.46/ share P. 10 of 6 Unit Rates A unit rate is the rate for a single unit. $1.50 dollars 1.50 18 18 The rate of dollars to is 0.25 0.08. 1 The unit rate is approximately $0.08 per ounce. Section 5.2 The Concept of Proportions Denominator is 1. P. 8 of 6 P. 11 of 6 CQ-5-04. Write 48 gallons in 14 days as a unit rate. Round to the nearest hundredth, if necessary. 1..4 1. 2. 42...14 292 P. 9 of 6 Proportions A proportion states that two ratios or rates are equal. 16 4 12 sixteen-twelfths equals four-thirds or sixteen is to twelve as four is to three Write the proportion 5.6 is to 4.4 as 112 is to 88. 5.6 112 4.4 88 P. 12 of 6 2 of 6
Equality Test for Proportions To determine whether a statement is a proportion, the equality test for proportions is used. This method is also called finding cross products. Equality Test for Fractions For any two fractions where b 0 and d 0, a c if and only if, then a d = b c. b d 1 4 2 8 = 2 4 8 1 8 8 The products are equal, 1 4 therefore. 2 8 P. 1 of 6 CQ-5-07. Determine which equation is a true statement 11 12 15 22 8 12 1.75 15 7 15.5 8 18 2.5 1.6 6 4 P. 16 of 6 Equality Test for Proportions Is the rate 75 105 = 5 7 75 5 hours equal to the rate 5 105 525 75 7 525 105 7 hours The two rates are equal. This is a proportion. Section 5. Solving Proportions P. 14 of 6 P. 17 of 6 CQ-5-06. Determine which equation is a true statement 12 10 5 1 4 42 2 9 56 10 11 10 48 40 48 P. 15 of 6 Variable & Equation A variable is a letter used to represent a number we do not yet know. 8 n 72 An equation has an equal sign. This indicates that the values on each side are equivalent. This will not change the value of n in the equation. 8 n 72 8 n 72 8 8 8 n 9 8 We divide both sides of the equation of the form a n = b by the number that is multiplied by n. 1 n 9 Therefore, n = 9. P. 18 of 6 of 6
Solving for a Variable Solve for n. n 11.4 = 57 n 11.4 = 57 n 11.4 57 = 11. 4 11.4 11.4 n = 5 11.4 n = 5 Check: 5 11.4 = 57 CQ-5-08. Solve for x. Round to the nearest tenth, if necessary. x 12 x 9.5 x 11 x 10.9 8 x 45 P. 19 of 6 P. 22 of 6 Finding Missing Numbers in a Proportion Sometimes one of the pieces of a proportion is unknown. We can use an equation of the form a n = b and solve for n to find the unknown quantity. 15 n 4 6 To Solve for a Missing Number in a Proportion 1. Find the cross products. 2. Divide each side of the equation by the number multiplied by n.. Simplify the result. 4. Check your answer. P. 20 of 6 CQ-5-09. Solve for x. Round to the nearest tenth, if necessary. 14 x 21.2 15 x 7.8 x 20.2 x 9.9 x 19.8 P. 2 of 6 Solving for a Variable Solving for a Variable Find the value of n. 15 n 4 6 4 n 15 6 Find the cross products. 4 n 90 4 n 90 4 4 n 22.5 Divide each side by 4. Check your answer: 15 22.5 4 6 4 22.5 15 6 90 P. 21 of 6 Find the value of n. 12 quarters 87 quarters dollars n dollars 87 12 n Find the cross products. 261 12 n 261 12 n 12 12 21.75 n Divide each side by 12. n = $21.75 Check your answer: 12 87 21.75 87 12 21.75 261 P. 24 of 6 4 of 6
CQ-5-10. Solve for x. Round to the nearest tenth, if necessary. 1. x 1. 2. x 2.. x. 4. x 4. 10.6 14.4 9.8 15.2 $9.75 24 $5.85 x P. 25 of 6 Problem Solving Steps 1. Understand the problem. a) Read the problem carefully. b) Draw a picture if this is helpful. c) Fill in the so that you have the facts and a method of proceeding in this situation. 2. Solve and state the answer. a) Perform the calculations. b) State the answer, including the unit of measure.. Check. a) Estimate the answer. b) Compare the exact answer with the estimate to see if your answer is reasonable. P. 28 of 6 CQ-5-11. Solve for x. Round to the nearest tenth, if necessary. 14.2 cups of flour 20 cups of flour 6 of bread x of bread 1. x 1. 2. x 2.. x. 4. x 4. 47. 11.8 8.5 10.7 P. 26 of 6 The Mathematical Blueprint is simply a sheet of paper with four columns. Each column tells you something to do. for Problem Solving Gather the Facts What Am I Asked to Do How Do I Proceed Key Points to Remember P. 29 of 6 Section 5.4 Solving Applied Problems Involving Proportions A baseball pitcher gave up 52 earned runs in 260 innings of pitching. At this rate, how many runs would he give up in a 9-inning game (This decimal is called the pitcher s earned run average, ERA.) for Problem Solving Gather the Facts 52 runs were given up in 260 innings. What Am I Asked to Do Find the number of runs in 9 innings. How Do I Proceed Set up a proportion comparing runs to innings Key Points to Remember One fraction represents the total innings and one represents the 9 innings. P. 27 of 6 Example continues. P. 0 of 6 5 of 6
A baseball pitcher gave up 52 earned runs in 260 innings of pitching. At this rate, how many runs would he give up in a 9-inning game (This decimal is called the pitcher s earned run average, ERA.) earned runs 52 n innings 260 9 260 n 52 9 260 n 468 260 260 n 1.8 The pitcher will give up 1.8 runs in a 9-inning game. P. 1 of 6 CQ-5-12. Mark traveled 455 in 5 hours. At this rate, how far could he travel in 7.5 hours 682.5 728 117.5 650 P. 4 of 6 It is recommended that 2 gallons of paint are used for every 750 square feet of wall. A painter is going to paint 7,875 square feet of wall with a paint that costs $8.50 per gallon. How much will the painter spend for paint for Problem Solving Gather the Facts 2 gal per 750 sq. ft.; 7875 sq. ft. total to be painted; cost is 8.50 per gallon What Am I Asked to Do How Do I Proceed Find the total cost Set up a for the paint. proportion comparing gals to ft.; multiply the answer by $8.50. Key Points to Remember One fraction represents the recommended paint and one represents the needed paint. Example continues. P. 2 of 6 CQ-5-1. If 2 centimeters on a map represents 86, what distance does 5 centimeters represent 1. 92 1.. 258. 4. 215 4. 4.4 P. 5 of 6 It is recommended that 2 gallons of paint is used for every 750 square feet of wall. A painter is going to paint 7,875 square feet of wall with a paint that costs $8.50 per gallon. How much will the painter spend for paint gallons of paint 2 n square feet 750 7875 750 n 2 7875 21 gallons of paint will be needed. 750 n 15750 750 750 n 21 1700 178.5 The total cost for the paint is $178.50. 8.5 21 85 P. of 6 CQ-5-14. At a bakery, for every 55 baked, have unacceptable texture. If the bakery makes 95 per month, how many have unacceptable texture 48 2. 512.. 156. 4. 854. P. 6 of 6 6 of 6