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MAT001 – Chapter 4 – Ratios, Rates, and Proportions CQ4-01. One day, a veterinary clinic treated 6 male dogs, 10 female dogs, 3 male cats, and 4 female cats. What is the ratio of dogs to cats that were seen that day? Section 4.1: Ratios and Rates Ratios A ratio is the comparison of two quantities that have the same units. 18 ounces 36 ounces 18 to 36 18:36 We can express this ratio three different ways: 18 36 1 of 36 3 4 5 6 7 8 9 10 21 22 23 24 25 26 27 28 29 30 Simplest form 13 14 15 0% 3. 16 17 4. 18 19 20 2 of 36 10 0% 0% 1. 1 2 3 4 5 6 7 8 9 10 21 22 23 24 25 26 27 28 29 30 11 12 0% 2. 13 14 0% 3. 15 16 4. 17 18 19 20 4 of 36 Rates Ratios Example: 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? 60 500 12 1 27 3 7 5 14 1 3 1. 2. 3. 4. 3 of 36 state taxes gross pay 11 0% 2. CQ4-02. Write 30:84 as a ratio (in fraction) in simplest form. 36 ounces 1 2 0% 1. 2 A ratio is in simplest form when the two numbers do not have a common factor and both numbers are whole numbers. 18 36 0% 1 Simplest Form of a Ratio 18 ounces 10 1. 7 to 16 2. 16 to 7 3. 16 to 23 4. 10 to 13 A rate is a comparison of two quantities that have different units. $1.50 18 ounces 3 25 The rate of dollars to ounces is $3 of it is withheld for state taxes for every $25 Dorothy makes. dollars ounces 5 of 36 1.50 18 6 of 36 1 of 6 MAT001 – Chapter 4 – Ratios, Rates, and Proportions CQ4-03. Write “252 miles per 8 gallons” as a rate in simplest form. 1. 2. 3. 4. Unit Rates A unit rate is the rate for a single unit. 28 miles 1 gallon 10 $1.50 63 miles 2 gallons 18 ounces 125 miles 4 gallons 1 2 3 64 miles 34 gallons 5 6 7 21 22 23 24 25 26 27 The rate of dollars to ounces is 0% 0% 1. 8 9 10 28 29 30 11 12 0% 2. 13 14 3. 15 16 dollars ounces 0% 4. 17 18 19 Denominator is 1. 20 8 of 36 CQ4-05. Write “$6,188 for 82 shares of stock” as a unit rate. Round to the nearest cent, if necessary. 10 3.43 gallons/ day 3.14 gallons/ day 292 gallons/ day 0% 0% 1. 1 2 3 4 5 6 7 8 9 10 22 23 24 25 26 27 28 29 30 11 12 0% 2. 13 14 0% 3. 15 16 10 1. $13.25 / share 2. $69.24 / share 3. $80.92 / share 4. $75.46 / share 4 gallons/ day 21 The unit rate is approximately $0.08 per ounce. 0.083 . 1 0.25 3 7 of 36 CQ4-04. Write “48 gallons in 14 days” as a unit rate. Round to the nearest hundredth, if necessary. 1. 2. 3. 4. 1.50 18 0% 4. 17 18 0% 1. 19 20 9 of 36 1 2 3 4 5 6 7 8 9 10 21 22 23 24 25 26 27 28 29 30 11 12 0% 2. 13 14 0% 3. 15 16 4. 17 18 19 20 10 of 36 Proportions Section 4.2 A proportion states that two ratios or rates are equal. 16 12 The Concept of Proportions 4 3 “sixteen-twelfths equals four-thirds” or “sixteen is to twelve as four is to three” Example: Write the proportion 5.6 is to 4.4 as 112 is to 88. 5.6 4.4 11 of 36 112 88 12 of 36 2 of 6 MAT001 – Chapter 4 – Ratios, Rates, and Proportions Equality Test for Proportions 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. Example: Is the rate 75 miles equal to the rate 105 miles ? 5 hours 7 hours Equality Test for Fractions For any two fractions where b ≠ 0 and d ≠ 0, if and only if cthen a , d a b 2 1 1=4 ? 2 8 4 8 d=b c. 75 = 105 ? 5 7 5 105 75 7 The two rates are equal. This is a proportion. 525 525 The products are equal, 1 4 therefore . 2 8 8 8 13 of 36 14 of 36 CQ4-06. Determine which equation is a true statement? 12 ? 10 42 35 1?3 2 4 10 ? 11 9 10 1. 2. 3. 4. CQ4-07. Determine which equation is a true statement? 1. 2. 3. 4. 10 48 ? 40 56 48 0% 0% 1. 1 2 3 4 5 6 7 8 9 10 21 22 23 24 25 26 27 28 29 30 11 12 0% 2. 13 14 0% 3. 15 16 4. 17 18 19 15 ? 22 8 12 10 11 13.75 12 15 7 ? 15.5 8 18 ? 2.5 ? 1.6 6 4 0% 0% 1. 20 1 2 3 4 5 6 7 8 9 10 15 of 36 21 22 23 24 25 26 27 28 29 30 11 12 0% 2. 13 14 0% 3. 15 16 4. 17 18 19 20 16 of 36 Variable & Equation Section 4.3 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. Solving Proportions This will not change the value of n in the equation. 17 of 36 8 n 72 8 n 72 8 8 8 n 9 1 n 9 8 We divide both sides of the equation of the form a n = b by the number that is multiplied by n. Therefore, n = 9. 18 of 36 3 of 6 MAT001 – Chapter 4 – Ratios, Rates, and Proportions Solving for a Variable 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. Example: Solve for n. n 11.4 = 57 n 11.4 = 57 15 4 n 11.4 57 = 11.4 11.4 11.4 n = 5 11.4 n = 5 Check: 5 n 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. 3. Simplify the result. 4. Check your answer. 11.4 = 57 19 of 36 20 of 36 CQ4-08. Solve for x. Round to the nearest tenth, if necessary. 8 33 Solving for a Variable Example: Find the value of n. 15 4 n 6 4 n 15 4 n 90 n 90 4 22.5 4 4 n 6 1. 2. 3. 4. Find the cross products. Divide each side by 4. Check your answer: 15 ? 22.5 4 6 4 22.5 15 6 90 21 of 36 1. 2. 3. 4. x 7.8 x 15 0% x 10.9 1 2 3 4 5 6 7 8 9 10 22 23 24 25 26 27 28 29 30 x 19.8 1 2 3 4 5 6 7 8 9 10 21 22 23 24 25 26 27 28 29 30 11 12 0% 2. 13 14 0% 3. 15 16 12 13 14 0% 3. 15 16 4. 17 18 19 20 22 of 36 4. 17 18 19 20 23 of 36 87 quarters n dollars 87 12 n 261 12 n 261 12 12 n 12 Divide each side by 12. n n = $21.75 21.75 1. 11 0% 2. Solving for a Variable 12 quarters 3 dollars 10 Seconds Remaining 0% 0% 1. 21 3 0% 10 Seconds Remaining x 11 Example: Find the value of n. x 20.2 x 9.9 45 x 9.5 CQ4-09. Solve for x. Round to the nearest tenth, if necessary. 14 21.2 x x 12 Find the cross products. Check your answer: 12 ? 87 3 21.75 3 87 12 21.75 261 24 of 36 4 of 6 MAT001 – Chapter 4 – Ratios, Rates, and Proportions CQ4-11. Solve for x. Round to the nearest tenth, if necessary. CQ4-10. Solve for x. Round to the nearest tenth, if necessary. $9.75 24 ounces 1. x 2. x 3. x 4. x $5.85 x 10.6 ounces 14 .2 cups of flour 6 loaves of bread 1. 2. 3. 4. 14.4 ounces 10 Seconds Remaining 9.8 ounces 0% 1 2 3 4 5 6 7 8 9 10 21 22 23 24 25 26 27 28 29 30 0% 1. 15.2 ounces 11 12 0% 2. 13 14 0% 3. 15 16 4. 17 18 19 20 cups of flour x loaves of bread x 47.3 loaves x 11.8 loaves 10 Seconds Remaining x 8.5 loaves 0% x 10.7 loaves 20 1 2 3 4 5 6 7 8 9 10 25 of 36 21 22 23 24 25 26 27 28 29 30 0% 1. 11 12 0% 2. 13 14 0% 3. 15 16 4. 17 18 19 20 26 of 36 Problem Solving Steps 1. Understand the problem. Section 4.4 Solving Applied Problems Involving Proportions a) Read the problem carefully. b) Draw a picture if this is helpful. c) Fill in the Mathematics Blueprint 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. 3. Check. a) Estimate the answer. b) Compare the exact answer with the estimate to see if your answer is reasonable. 27 of 36 28 of 36 Mathematics Blueprint Mathematics Blueprint The Mathematical Blueprint is simply a sheet of paper with four columns. Each column tells you something to do. Mathematics Blueprint for Problem Solving Gather the Facts What Am I Asked to Do? How Do I Proceed? Key Points to Remember Example: 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.) Mathematics Blueprint for Problem Solving Gather the Facts What Am I Asked to Do? How Do I Proceed? Key Points to Remember 52 runs were given up in 260 innings. Find the number of runs in 9 innings. Set up a proportion comparing runs to innings One fraction represents the total innings and one represents the 9 innings. Example continues. 29 of 36 30 of 36 5 of 6 MAT001 – Chapter 4 – Ratios, Rates, and Proportions Mathematics Blueprint Mathematics Blueprint Example: 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 9inning game? (This decimal is called the pitcher’s earned run average, ERA.) earned runs innings 260 52 260 n 52 260 n 260 n 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? n 9 Mathematics Blueprint for Problem Solving 9 468 260 1.8 The pitcher will give up 1.8 runs in a 9-inning game. 21 gallons of paint will be needed. 15750 750 21 Find the total cost Set up a for the paint. proportion comparing gals to ft.; multiply the answer by $8.50. 1. 2. 3. 4. 85 1700 178.5 $178.50. 33 of 36 1 2 3 4 5 6 7 8 9 10 21 22 23 24 25 26 27 28 29 30 11 12 0% 2. 13 14 15 16 18 1137.5 miles 0% 650 miles 0% 1. 2 3 4 5 6 7 8 9 10 22 23 24 25 26 27 28 29 30 11 12 0% 2. 13 14 0% 3. 15 16 4. 17 18 19 20 34 of 36 1. 48 cakes 2. 51 cakes 3. 156 cakes 4. 85 cakes 4. 17 10 1 0% 3. 728 miles 21 10 0% 682.5 miles CQ4-14. At a bakery, for every 55 cakes baked, 3 have unacceptable texture. If the bakery makes 935 cakes per month, how many have unacceptable texture? CQ4-13. If 2 centimeters on a map represents 86 miles, what distance does 5 centimeters represent? 1. One fraction represents the recommended paint and one represents the needed paint. CQ4-12. Mark traveled 455 miles in 5 hours. At this rate, how far could he travel in 7.5 hours? The total cost for the 0% Key Points to Remember 32 of 36 21 paint is 1. 92 miles 2. 34.4 miles 3. 258 miles 4. 215 miles 2 gal per 750 sq. ft.; 7875 sq. ft. total to be painted; cost is 8.50 per gallon 8.5 2 n 750 7875 2 7875 How Do I Proceed? Example continues. 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? 750 n 750 n What Am I Asked to Do? 31 of 36 Mathematics Blueprint gallons of paint square feet 750 n Gather the Facts 19 10 0% 0% 1. 20 1 2 3 4 5 6 7 8 9 10 35 of 36 21 22 23 24 25 26 27 28 29 30 11 12 0% 2. 13 14 0% 3. 15 16 4. 17 18 19 20 36 of 36 6 of 6