CHAPTER 6: RATIOS, RATES, & PROPORTIONS

Transcription

1 CHAPTER 6: RATIOS, RATES, & PROPORTIONS 6.1 Ratios 6.2 Rates 6.3 Unit Rates 6.4 Proportions 6.5 Applications Chapter 6 Contents 425

2 CCBC Math 081 Ratios Section pages 6.1 Ratios One of the pieces of information students are often interested in when choosing a college is the faculty student ratio. Here at CCBC, our current student to faculty ratio is 17 to 1. What exactly does that mean? What is a ratio? Ratio: A ratio is a comparison of two quantities. The quantities compared in a ratio may simply be numbers, as in 17 to 1, or the quantities may have units attached to them, as in 5 feet to 12 feet. Suppose we are comparing the number of trucks in a parking lot with the cars in the lot. We count and find that there are 7 trucks and 45 cars in the lot. We say, The ratio of trucks to cars is 7 to 45. This can be written in three different ways as shown in the box below. WRITING A RATIO Words Colon Fraction 7 to 45 7 : ORDER - The order in which we write the parts of the ratio is important. Suppose you are in a class with 11 male students and 15 female students. Notice the difference in how the two ratios are written below. Ratio of male students to female students: Male to Female 11 to 15 OR Male 11 Female 15 Ratio of female students to male students: Female to Male 15 to 11 OR Female 15 Male 11 REDUCE - Because a ratio can be written as a fraction, it makes sense to reduce a ratio to its lowest terms the same way that you would reduce a fraction. Suppose a 540-page textbook has 80 pages of problems. The ratio of total pages in the book to pages of problems is 540 to This ratio can be reduced in the following manner: Important Notes: 1. Ratios should always be reduced to lowest terms. 2. Ratios should never be written as a mixed number or a whole number. 426

3 CCBC Math 081 Ratios Section pages Example 1: Write the ratio of 5 pounds of apples to 4 pounds of oranges in three ways. Practice 1: Words: 5 to 4 Colon: 5:4 Fraction: 5 4 Notice we did not change the fraction to a mixed number. We left it as an improper fraction. Write the ratio of 6 pounds of peaches to 7 pounds of peaches in three different ways. Watch It: Answer: 6 to 7; 6 : 7; 6 7 Example 2: Write the ratio of $15 to $18 in fraction form Notice that we reduced the original fraction to lowest terms. Practice 2: Write the ratio of 12 to 20 as a fraction in simplest form. Watch It: Answer: 3 5 Example 3: There are 50 milliliters of grape juice and 45 milliliters of iced tea. Write the ratio of milliliters of grape juice to milliliters of iced tea in fraction form. Grape Juice Iced Tea Notice that we reduced the original fraction to lowest terms. Also notice that we did not change the fraction to a mixed number. We left it as an improper fraction. Practice 3: There are 35 grams of sugar and 80 grams of salt. Write the ratio of grams of sugar to grams of salt as a fraction in simplest form. 7 Watch It: Answer: 16 Example 4: There are 11 green balls and 8 red balls in the bin. Write the ratio of red balls to green balls in fraction form. Red 8 Green

4 CCBC Math 081 Ratios Section pages Practice 4: There are 15 green balls and 4 red balls in the bin. Write the ratio of red balls to green balls in fraction form. Watch It: Answer: 4 15 Example 5: Practice 5: If there are 12 black pens and 10 blue pens, write the ratio of black pens to total pens in simplest form. Black Total = = 22 Total Notice that we reduced the original fraction to lowest terms. The total contents of a bag of marbles is 14 green marbles and 20 blue marbles. Write the ratio of blue marbles to total marbles in simplest form. Watch It: Answer: Example 6: Write the ratio of the rectangle s length to its width in fraction form. L = 25 inches W = 15 inches Length Width Notice that we reduced the original fraction to lowest terms. Also notice that we did not change the fraction to a mixed number. We left it as an improper fraction. Practice 6: Write the ratio of the rectangle s length to its width in fraction form. L = 42 inches W = 28 inches Watch It: Answer: 3 2 Watch All: 428

5 CCBC Math 081 Ratios Section pages 6.1 Ratio Exercises 1. Write the following ratio as a fraction and in words. 5:12 2. Write the following ratio using a colon and in words Write the following ratio as a fraction and using a colon. 18 to 23 For Questions 4 12, simplify the ratio: 4. 3: : : to to to There are 10 black marbles and 6 red marbles in a bag. What is the ratio of black marbles to red marbles? 14. A committee consists of 10 teachers and 4 students. What is the ratio of teachers to students? 15. The XYZ Company has 80 full-time employees and 60 part-time employees. What is the ratio of part-time employees to full-time employees? 16. Mike has 8 rock CD s and 12 country CD s in his music collection. What is the ratio of rock CD s to country CD s? 17. Taylor scored 20 points in the game and Keith scored 32 points. What is the ratio of Keith s points to Taylor s points? 18. A wheelchair ramp rises 2 feet for every 12 feet it runs. What is the ratio of rise to run? 19. A tree is 18 feet high and 12 feet wide. What is the ratio of the height of the tree to the width of the tree? 20. There are 15 men and 20 women that work in the department. What is the ratio of men to the total number of employees in the department?

6 CCBC Math 081 Ratios Section pages 6.1 Ratios Exercises Answers 1. 5 ; 5 to :7; 13 to ; 18: :5 is simplified 5. 4: : to to to

7 CCBC Math 081 Rates Section pages 6.2 Rates In the last section we learned how to do problems like this: What is the ratio of 5 pounds of apples to 4 pounds of oranges? In this section we will do problems like this: What is the ratio of 5 teaspoons of salt to 4 cups of flour? Do you see the difference? Compare the units. In the first problem, the units are the same: 5 pounds of apples and 4 pounds of oranges. In the second problem, the units are different: 5 teaspoons of salt and 4 cups of flour. In this section of the chapter we will work with ratios in which the quantities have different units. This new type of ratio is a special type called a rate. Rate: A rate is a ratio that compares two quantities that have different units. A rate that you are very familiar with is the rate 55 mph. This means all of the following. 55 miles per hour = 55 miles to 1 hour = 55 miles 1 hour With rates, it is common to use the word per in place of the word to. Here s the good news: Aside from the new name rate and the use of the word per, we work with these rate problems the same way we did in the last section. There is nothing new to learn! For instance, let s say a car can be driven 216 miles on 6 gallons of gasoline. We can calculate the ratio of miles to gallons, or the rate of miles per gallon, just as we did before. This is shown below. 216 miles per 6 gallons = 216 miles 216 miles 6 36 miles 6 gallons 6 gallons 6 1 gallon = 36 miles per gallon 431

8 CCBC Math 081 Rates Section pages Example 1: Write the following rate in fraction form: $50 per 8 pounds $50 $50 2 $25 8 pounds 8 pounds 2 4 pounds Notice that we reduced to lowest terms. The answer means $25 per 4 pounds. Practice 1: Write the following rate in fraction form: $24 per 9 pounds Watch It: Answer: $8 3 pounds Example 2: Write the following rate in fraction form: 126 words in 3 minutes 126 words 126 words 3 42 words 3 minutes 3 minutes 3 1 minute Notice that we reduced to lowest terms. The answer means 42 words per one minute. Practice 2: Write the following rate in the simplest form: 10 grams of fat per 4 cookies Watch It: Answer: 5 grams of fat 2 cookies Example 3: Write the following rate in fraction form: 15 grams of fat per 6 cookies 15 grams 15 grams 3 5 grams 6 cookies 6 cookies 3 2 cookies Notice that we reduced to lowest terms. The answer means there are 5 grams of fat per 2 cookies. Practice 3: Write the following rate in the simplest form: 160 words in 2 minutes Watch It: Answer: 80 words 1 minute Watch All: 432

9 CCBC Math 081 Rates Section pages 6.2 Rate Exercises Write each rate in simplest fraction form miles per 6 gallons milliliters per 20 pounds grams per 14 milliliters pounds per 12 square inches 5. $4 per 10 boxes phone calls in 60 minutes pages in 4 hours 8. 8 meters per 10 seconds calories per 3 servings meters in 16 seconds 11. An office assistant can type 1240 words in 20 minutes. Determine the rate of words to minutes. 12. A runner can complete 28 laps around the track in 14 minutes. What is the rate of laps to minutes for this runner? 13. A patient requires 15 milligrams of medicine for every 35 pounds of body weight. What is the rate of milligrams to pounds for this patient? 14. A box of cereal contains 9 servings. The calories from the entire box of cereal are 75. What is the rate of calories to servings for this cereal? 15. A car traveled 600 miles in 9 hours. What is the rate of miles to hours? 16. On a trip a car traveled 356 miles and used 14 gallons of gas. What is the rate of miles to gallons? 17. A printer used 5 printer cartridges to print 3480 pages. What is the rate of cartridges to printed pages? 18. A grocer scans 32 items in 48 seconds. What is the rate of items scanned to seconds? 19. A student read 852 pages in 12 hours. What is the rate of pages read to hours? 20. A 15 page paper took 27 hours to write. What is the rate of hours to pages? 433

10 CCBC Math 081 Rates Section pages 6.2 Rates Exercises Answers 1. 6 miles 1 gallon words 1 minute 2. 1 milliliter 2 pounds laps 1 minute 3. 2 grams 1 milliliter milligrams 7 pounds 4. 7 pounds 1 square inch calories 3 servings 5. $2 5 boxes miles 3 hours 6. 1 call 5 minutes miles 7 gallons pages 1 hour cartridge 696 pages 8. 4 meters 5 seconds items 3 seconds calories 1 serving pages 1 hour meters 1 second hours 5 pages 434

11 CCBC Math 081 Unit Rates Section pages 6.3 Unit Rates Unit Rate: A unit rate is a rate where the second quantity or the denominator is 1. The following are examples of unit rates: 5 men to 1 car = 5 men per car = 5 men / car 30 miles to 1 gallon = 30 miles per gallon = 30 mpg $3.49 to 1 pound = $3.49 per pound HOW TO DETERMINE A UNIT RATE Step 1: Write the rate as a fraction. Step 2: Make the denominator equal to 1 by dividing the quantity in the numerator and denominator by the quantity in the denominator. Example 1: Given the rate 200 miles to 8 gallons, determine the unit rate. 200 miles 8 gallons 200 miles 8 = 8 gallons 8 = 25 miles 1 gallon Write the rate as a fraction. Divide the numerator and denominator by the denominator. Simplify. The unit rate is 25 miles per gallon. Practice 1: Determine the unit rate for driving 288 miles on 8 gallons of gasoline. Watch It: Answer: 36 miles per gallon Example 2: A typist can type 240 words in 4 minutes. Find the unit rate. 240 words 4 minutes 240 words 4 = 4 minutes 4 = 60 words 1 minute Write the rate as a fraction. Divide the numerator and denominator by the denominator. Simplify. The unit rate is 60 words per minute. 435

12 CCBC Math 081 Unit Rates Section pages Practice 2: A typist can type 480 words in 6 minutes. Determine the unit rate. Watch It: Answer: 80 words per minute Example 3: Find the unit rate for driving 135 miles on 4 gallons of gasoline. 135 miles 4 gallons 135 miles 4 = 4 gallons miles = 1 gallon Write the rate as a fraction. Divide the numerator and denominator by the denominator. Simplify. The unit rate is miles per gallon. Practice 3: Determine the unit rate for driving 850 miles on 20 gallons of gasoline. Watch It: Answer: 42.5 miles per gallon Calling all shoppers! One very useful unit rate is called a unit price. It gives the price of a single part. For instance, it could be the price of one pound or the price of one liter or the price of one quart, or the price of one tile. Example 4: Mina bought 2.5 kilograms of brown rice for $7.50. Determine the unit price. $ kilograms $ = 2.5 kilograms 2.5 $3 = 1 kilogram Write the rate as a fraction. Divide the numerator and denominator by the denominator. Simplify. The unit rate is $3.00 per kilogram. Practice 4: George bought 2 kilograms of brown rice for $3.50. Determine the unit rate or unit price (of brown rice per kilogram). Watch It: Answer: $1.75 per kilogram 436

13 CCBC Math 081 Unit Rates Section pages Example 5: The price of 8 pounds of apples is $10. What is the unit price? $10 8 pounds $10 8 = 8 pounds 8 = $ pound Write the rate as a fraction. Divide the numerator and denominator by the denominator. Simplify. The unit rate is $1.25 per pound. Practice 5: The price of 4 pounds of apples is $8. What is the unit rate (price of apples per one pound)? Watch It: Answer: $2 per pound Example 6: A 20-ounce box of cereal costs $3.40. What is the unit price? $ ounces $ = 20 ounces 20 = $ ounce Write the rate as a fraction. Divide the numerator and denominator by the denominator. Simplify. The unit rate is $0.17 per ounce (or 17 per ounce). Practice 6: A 16-ounce box of cereal costs $2.40. What is the unit price? Watch It: Answer: $0.15 per ounce Sometimes we will need to convert one of the units before we can determine the unit rate. This may happen in a problem if the rate given does not have the same units as the unit rate that we are asked to find. For instance, suppose we are given the rate 15 miles to 20 minutes. But we are asked to find the unit rate in miles per hour. To solve a problem like this we must first get the units to match, then we can proceed to find the unit rate. 437

14 CCBC Math 081 Unit Rates Section pages Example 7: Given the rate 15 miles to 20 minutes, determine the unit rate in miles per hour. Change minutes to hours: 20 minutes 1 hour 20 hour 20 hour minutes hour 1 60 minutes The given rate was 15 miles to 20 minutes, but now we can express this as 15 miles to 1 3 hour. 15 miles 1 hour 3 1 = 15 3 Write the rate as a fraction. Divide the numerator by the denominator. = = = 45 1 Change division to multiplication of the reciprocal. Multiply. The unit rate is 45 miles per hour or 45 mph. Practice 7: Given the rate 35 miles in 40 minutes, determine the unit rate in miles per hour. Watch It: Answer: 52.5 miles per hour Watch All: 438

15 CCBC Math 081 Unit Rates Section pages 6.3 Unit Rate Exercises Determine the unit rate calories per 4 servings of pie miles in 11 hours miles per 12 gallons 4. $15.20 per 3 hours grams per 17 milliliters pages in 6 hours meters per 8 seconds 8. $8.25 per 3 boxes 9. $45.35 per 5 hours calories per 15 servings Solve each problem. 11. An international call costs $9.35 for 32 minutes. What is the cost per minute? 12. At a warehouse store, 10 cans of soup cost $8.22. What is the price per can of soup? 13. A secretary can type 2435 words in 20 minutes. What is the secretary s unit rate in words per minute? 14. A 12 ounce can of beans costs $0.88. What is the unit price per ounce? 15. A car traveled 120 miles in 2 hours. What was the unit rate in miles per hour? 16. Ann babysat for 6 hours and was paid $36. What is her pay per hour? 17. Four cookies have 125 calories. What is the number of calories per cookie? 18. Sue bought 5 apples for $1.50. What is the price per apple? 19. Bill got his phone bill and was charged $4.25 for a 20 minute call. How much was he charged per minute? 20. Caleb rode his bicycle 5 miles in 30 minutes. What is the rate of miles per hour? 439

16 CCBC Math 081 Unit Rates Section pages 6.3 Unit Rates Exercises Answers calories per serving miles per hour miles per gallon 4. $5.07 per hour (rounded to the nearest cent) 5. 2 grams per millimeter pages per hour meters per second 8. $2.75 per box 9. $9.07 per hour calories per serving 11. $0.29 per minute (rounded to the nearest cent) 12. $0.82 per can (rounded to the nearest cent) words per minute 14. $0.07 per ounce (rounded to the nearest cent) miles per hour 16. $6 per hour calories per cookie 18. $0.30 per apple 19. $0.21 per minute (rounded to the nearest cent) miles per hour 440

17 CCBC Math 081 Mid-Chapter 6 Review Section 6.1 to 6.3 Write each ratio in two other ways. CHAPTER 6 Mid-Chapter Review 1. 5 to to : :11 Write each ratio or rate in simplest form to to to : : pounds to 300 pounds male students to 24 female students cars to 45 trucks words per 4 minutes meters per 15 seconds calories per 4 bags 20. A group consists of 27 females and 12 males. What is the ratio of females to males? 21. Juliet paid $6 for lunch and $36 for dinner while traveling for work. What is the ratio of the cost of lunch to the cost of dinner? 22. A cabinet at school has 18 boxes of yellow chalk and 21 boxes of white chalk. What is the ratio of boxes of white chalk to all the boxes of chalk in the cabinet? Find the unit rate miles per 8 gallons of gas 24. $54.40 for 16 pounds 25. $49.56 for 14 gallons 26. $2496 for 15 people books for 16 people gallons for 20 people miles in 5 hours calories for six cupcakes 441

18 CCBC Math 081 Mid-Chapter 6 Review Section 6.1 to 6.3 M i d - C h a p t e r 6 R e v i e w A n s w e r s 1. 5:8 ; :3 ; to 15 ; to 11 ; :53 ; 18 to :100 ; 99 to to to to : : pounds 6 pounds 3 male students 2 female students cars 3 trucks 241 words 2 minutes 11 meters 3 seconds 65 calories 4 bags 9 females 4 males $1 lunch $6 dinner 7 boxes white chalk 13 boxes of chalk miles per gallon 24. $3.40 per pound 25. $3.54 per gallon 26. $ per person books per person gallons per person miles per hour calories per cupcake 442

19 CCBC Math 081 Proportions Section pages 6.4 Proportions In this section you will study a special type of equation called a proportion. Think of taking two fractions and inserting an equal sign between them. That s a proportion! PROPORTION Definition Math Statement Example A proportion is a statement that two ratios (fractions) are equal. a c b d Read a is to b as c is to d Means (Middle Terms) Extremes (Outer Terms) Read 2 is to 6 as 1 is to 3 Means: 6 and 1 Extremes: 2 and 3 Proportion True or False Proportions are either true or false. For instance, 4 1 True if we reduce we get 1 ; the two fractions are equal False if we reduce we do not get 4 ; the fractions are not equal. 9 Sometimes it is difficult to determine whether a proportion is true or false by reducing and comparing the fractions. For instance, it would be difficult with the proportion For that reason, there is another method that can be used to determine if a proportion is true or false. The method is shown in the following box. 443

20 CCBC Math 081 Proportions Section pages PROPERTY OF PROPORTIONS A proportion is true if and only if the product of the extremes equals the product of the means. a c is true only if a d b c b d This can be stated more simply in the following way: A proportion is true only if its cross products are equal. a b = c d is true only if ad bc Example 1: Is 5 15 a true statement? =? Multiply diagonally to get the cross products. Perform the multiplications to see if the products are equal Since the cross products are equal, the proportion is true. Practice 1: Is 7 28 a true statement? Answer: no Watch It: Example 2: Is a true statement? 7 5 = Multiply diagonally to get the cross products.? (1.2)(5) (7)(0.9) Perform the multiplications to see if the products are equal Cross products are not equal, so the proportion is not true. 444

21 CCBC Math 081 Proportions Section pages Practice 2: Is a true statement? Answer: yes 3 17 Watch It: A proportion contains four numbers. But in some problems, only three of the numbers are given and a variable represents the fourth. Our goal will be to determine the value of the variable that makes the proportion true. We will solve for the missing number using cross products and algebra as explained below. SOLVING PROPORTIONS To solve a proportion for a missing number (represented by a variable): 1. Cross multiply (multiply diagonally) and set the cross products equal to each other. 2. Perform the multiplication. 3. Divide both sides of the equation by the coefficient of the variable. 4. Simplify. Example 3: Solve for n: n This is a proportion. So, cross multiply (multiply diagonally). 51 n Set the cross products equal to each other. 17 n n 5100 Simplify by performing the multiplication on both sides. 17n 5100 Divide both sides by the coefficient of the variable n 300 Simplify by performing the division on both sides. Practice 3: Solve for n: 3 9 Answer: 15 5 n Watch It: 445

22 CCBC Math 081 Proportions Section pages Example 4: Solve for x: x Practice 4: Solve for x: x 1.5 This is a proportion. So, cross multiply (multiply diagonally) (2.5)( x) (7.5)(1.5) Set the cross products equal to each other. 2.5x Simplify by performing the multiplication on both sides. 2.5x Divide both sides by the coefficient of the variable x 4.5 Simplify by performing the division on both sides. x 3.5 Answer: Watch It: Example 5: Solve for n: 3 5 n This is a proportion. So, cross multiply (multiply diagonally). n 4 Set the cross products equal to each other n 12 5n Simplify by performing the multiplication on both sides. 12 5n Divide both sides by the coefficient of the variable = n OR 2.4 n Simplify by performing the division on both sides. 5 Practice 5: Solve for x: 7-2 Answer: x 9 Watch It: 446

23 CCBC Math 081 Proportions Section pages Example 6: Solve for n: 1 2 n n This is a proportion. So, cross multiply (multiply diagonally) n Set the cross products equal to each other n Simplify by performing the multiplication on both sides. 9 3 n 10 5 Divide both sides by the coefficient of the variable n Rewrite the left side of the equation so it is easier to solve Simplify the right side of the equation. 9 5 n Multiply by the reciprocal n Divide out common factors = n OR. n Simplify. Practice 6: Solve for n: 1 3 n Watch It: Answer: 5 3 Watch All: 447

24 CCBC Math 081 Proportions Section pages 6.4 Proportion Exercises Determine whether the following proportions are true or false? Solve the following proportions. For problems involving decimals, give the answer in decimal form. For problems involving fractions, give the answer in fractional form n 5 25 n n n 2 n n n n n n n n n n n n

25 CCBC Math 081 Proportions Section pages 6.4 Proportions Exercises Answers 1. False 11. n True 12. n 5 3. True 13. n False 14. n n n or n n or n n n n or n or n n or n 63 7 or

26 CCBC Math 081 Applications Section pages 6.5 Applications You are in the grocery store about to pick up some cans of tuna. There is a 6-ounce can for $1.20 and a 10-ounce can for $1.60. Of course you want the best deal. Which do you buy? Hmmm... I believe we have a ratio problem to solve! In this first type of application problem we need to compare two ratios using unit rates. APPLICATION: COMPARING UNIT RATES 1. For each item, write a ratio (rate) as a fraction. 2. Get the unit rate for each fraction by dividing the numerator by the denominator. 3. Compare the unit rates. Example 1: A 6-ounce can of tuna sells for $1.20 and a 10-ounce can sells for $1.60. Which is the better deal? Rate = Unit Rate = Cost # of ounces Cost 1 ounce $ ounces $ ounces $0.20 per ounce 20 per ounce $ ounces $ ounces $0.16 per ounce 16 per ounce The 10-ounce can of tuna is the better deal because its unit price is lower. Practice 1: A 20-ounce box of cereal costs $4.20 and a 16-ounce box of the same cereal costs $3.60. Which is the better deal? Watch It: Answer: 20-ounce box for $4.20 You are still in the grocery store and you are ready to purchase eggs. You are going to make cupcakes for a family reunion. Your recipe, that makes 18 cupcakes, requires 3 eggs. But you decide to make 4 dozen cupcakes this is going to be a large reunion. How many eggs will you need? Looks like we have a proportion problem that needs to be solved! In this type of application, we are dealing with quantities that are proportional. The number of cupcakes you bake is proportional to the number of eggs you use. For instance, if we wanted to double the number of cupcakes and make 36, then we would have to double the number of eggs and use 6. If we wanted to triple the number of cupcakes and make 54, then we would have to triple the number of eggs and use 9. The numbers are not so easy to calculate mentally for 4 dozen, or 48 cupcakes. So we will use algebra. Review the procedure that follows. 450

27 CCBC Math 081 Applications Section pages APPLICATION: SOLVING PROPORTIONS 1. Assign a variable to the unknown quantity. 2. Set up a ratio (fraction) in words to specify the quantities being compared. Use this setup to complete the next two steps. 3. Write a ratio (fraction) using the given values. 4. Write another ratio (fraction) using the unknown quantity (the variable). 5. Write a proportion by setting the ratios equal to each other. 6. Solve the proportion to isolate the variable. Example 2: A recipe that makes 18 cupcakes requires 3 eggs. How many eggs are needed to make 4 dozen, or 48, cupcakes? 1. Variable: assign variable to the unknown quantity n = number of eggs to make 4 dozen cupcakes 2. Ratio in Words: quantities being compared 3. Ratio of Given Values: set up like last step 4. Ratio with Variable: set up like last step # of Cupcakes # of Eggs 18 cupcakes 3 eggs 48 cupcakes n eggs Note that in all three ratios above, cupcakes were in the numerator of the fraction and eggs in the denominator. It is very important to be consistent when you set up your fractions. 5. Proportion: set ratios equal n 6. Solve Proportion: cross multiply 18n 144 divide by coefficient of variable You will need 8 eggs to bake 48 cupcakes. 18n simplify n 8 451

28 CCBC Math 081 Applications Section pages Practice 2: To make a cup of hot cocoa, Bob mixes 3 teaspoons of cocoa powder with 2 cups of milk. How much cocoa powder would be needed to mix with 12 cups of milk? Watch It: Answer: 18 teaspoons Example 3: Water is being pumped out of a basement at a rate of 140 gallons per hour. How many hours will it take to pump 2030 gallons of water out of the basement? 1. Variable: assign variable to the unknown quantity n = number of hours to pump water out of basement 2. Ratio in Words: quantities being compared 3. Ratio of Given Values: set up like last step 4. Ratio with Variable: set up like last step gallons of Water # of Hours 140 gallons 1 hour 2030 gallons n hours Note that in all three ratios above, the number of gallons of water was in the numerator and the number of hours was in the denominator. It is very important to be consistent with this. 5. Proportion: set ratios equal n 6. Solve Proportion: cross multiply 140n 2030 divide by coefficient of variable 140n simplify n 14.5 The answer represents number of hours, so remember to include those units in your answer. It will take 14.5 hours to pump 2030 gallons of water from the basement. Practice 3: Water is pumped into a pool at a rate of 120 gallons per hour. How many hours will it take to pump 3000 gallons of water into the pool? Watch It: Answer: 25 hours 452

29 CCBC Math 081 Applications Section pages Example 4: You need to combine 98 grams of sulfuric acid and 70 grams of sodium hydroxide to produce sodium sulfate (a kind of chemical salt). How many grams of sulfuric acid would need to combine with 20 grams of sodium hydroxide to produce sodium sulfate? 1. Variable: assign variable to the unknown quantity n = grams of sulfuric acid to combine with 20 grams of sodium hydroxide 2. Ratio in Words: quantities being compared 3. Ratio of Given Values: set up like last step grams of Sulfuric Acid grams of Sodium Hydroxide 98 grams Sulfuric Acid 70 grams Sodium Hydroxide 4. Ratio with Variable: set up like last step n grams Sulfuric Acid 20 grams Sodium Hydroxide Note that in all three ratios above, Sulfuric Acid was in the numerator and Sodium Hydroxide was in the denominator. It is very important to be consistent with this. 98 n 5. Proportion: set ratios equal Solve Proportion: cross multiply n divide by coefficient of variable n simplify 28 n The answer represents the number of grams, so remember to include those units in your answer. We found that 28 grams of Sulfuric Acid are needed to combine with 20 grams of Sodium Hydroxide. Practice 4: You need to combine 98 grams of sulfuric acid (H2SO4) and 80 grams of sodium hydroxide (NaOH) to produce sodium sulfate (a kind of chemical salt). How many grams of sulfuric acid (H2SO4) would you need to combine with 40 grams of sodium hydroxide (NaOH) to produce sodium sulfate? Watch It: Answer: 49 grams 453

30 CCBC Math 081 Applications Section pages Example 5: You know that there are 56 milligrams of cholesterol in much cholesterol is there in 8 ounces of trout? ounces of trout. How 1. Variable: assign variable to the unknown quantity n = mg of cholesterol 2. Ratio in Words: quantities being compared mg of Cholesterol ounces of Trout 3. Ratio of Given Values: set up like last step 4. Ratio with Variable: set up like last step 56 mg of Cholesterol 1 3 ounces of Trout 2 n mg of Cholesterol 8 ounces of Trout Note that in all three ratios above, cholesterol was in the numerator and trout in the denominator. It is very important to be consistent with this. 5. Proportion: set ratios equal 56 n Solve Proportion: write mixed number as an improper fraction cross multiply divide by coefficient of variable simplify multiply by reciprocal 56 n n 7 n n n cancel n simplify 128 n The answer represents the number of milligrams, so remember to include those units in your answer. There are 128 mg of cholesterol in 8 ounces of trout. 454

31 CCBC Math 081 Applications Section pages Practice 5: A single tablet of One-A-Day Vitamin for men contains 75 milligrams of Vitamin 1 C. How many milligrams of Vitamin C are in 2 tablets of One-A-Day 3 Vitamin? Watch It: Answer: 175 milligrams Watch All: 455

32 CCBC Math 081 Applications Section pages 6.5 Application Exercises Translate each into a proportion and solve the problem. 1. Harry gets 23 mi per gallon of gasoline in his truck. If he has 4 gallons of gasoline in his truck, how far can he go? 2. On a map, 3 in. represents 8 mi. How many in. will represent a distance of 24 mi? 3. On a map, 2 cm represents 3 km. How many km are represented by 15 cm? 4. A nurse has to give a patient a dose of medication. The dosage says 3 ml of medication for a 150 lb. person. If the person weighs 200 lbs., how many ml of medication is the person to receive? 5. 4 mg of a drug are to be given for every 10 kg of body weight. Find the size of a person requiring 25 mg of the drug. 6. A nurse has to give a child a dose of Tylenol. The dosage says 2 tsp of medication for a 50 lb person. If the person weighs 75 lbs, how many tsp of medication will the person receive? 7. A 2-lb box of sugar costs $1.20 and a 5-lb bag of sugar costs $2.75. What is the better deal? 8. A baker can make 72 cookies using 4 c of flour. How many cups of flour are needed to make 288 cookies? 9. Three oz of a chemical are needed to treat 25 oz of water. How many oz of the chemical are needed to prepare 100 oz of water? 10. To make 4 moles of water, 2 moles of oxygen gas are needed. How many moles of water can you make with 21 moles of oxygen gas? 456

33 CCBC Math 081 Applications Section pages 11. A child care advertises that they have a ratio of 2 care givers for every 9 children. If there are 6 care givers, how many children are at the child care? 12. A college has a ratio of 2 male students for every 3 female students. If there are 5322 male students, how many female students attend the college? 13. An office assistant can type 525 words in 5 minutes. At this rate, how many words can the office assistant type in 20 minutes? 14. A baseball player gets 36 hits in 90 times at bat. At this rate, how many hits should he get in 250 times at bat? 15. There are 45 mg of cholesterol in 2 oz of egg substitute. How many mg of cholesterol are there in 3 oz of egg substitute? 16. There are 18 g of fat in a 4 oz steak. How many grams of fat are in a 6 oz steak? 17. At the grocery store, a 16 oz bag of rice is $ 2.34 and a 24 oz bag of rice is $3.42. What is the better buy? 18. At a warehouse store you can purchase 60 cans of soda for $8.95. At a regular grocery store, you can purchase 12 cans of soda for $1.85. What is the better deal? 19. If 5 snakes can eat 3 mice, how many mice will 30 snakes eat? 20. A recipe calls for ¼ tsp of salt for every ½ cup of flour. How much salt should be used for 5 cups of flour? 457

34 CCBC Math 081 Applications Section pages 6.5 Application Exercise Answers mi 2. 9 in km 4. 4 ml kg 6. 3 tsp 7. 5-lb bag c oz of chemical moles of water children females words hits mg g oz bag cans mice tsp 458

35 CCBC Math 081 Chapter 6 Summary CHAPTER 6 SUMMARY Ratios, Rates, & Proportions Section 6.1 Section 6.2 Section 6.3 Section 6.4 Section 6.5 Ratio: a comparison of two quantities Write the ratio of 1 cm to 18 cm in 3 ways. (1) 1 to 18 (2) 1:18 (3) 18 1 Always reduce to lowest terms. Never write as a whole number or mixed number. There are 8 blue cars and 20 green cars on the lot. a. Write the ratio of blue cars to green cars. b. Write the ratio of green cars to the total number of cars. Blue Cars Green Cars Green Cars Total Cars Rate: a ratio that compares two quantities that have different units The patient weighs 150 pounds and receives 45 mg of medicine. What is the rate of mg to 45 mg 45 mg 15 3 mg pounds for this medication? 150 pounds 150 pounds pounds Unit Rate: a rate in which the denominator is 1 What is the unit rate for driving 126 miles on 4 gallons of gas? * Divide numerator and denominator by the denominator: Proportion: a statement that two ratios (fractions) are equal; 126 miles 126 miles miles 4 gallons 4 gallons 4 1 gallon a c only if ad bc b d Is a true statement? Solve for n: n? Get cross products: (3. 6)(9. 5) (8)(4) Get cross products: 4n n 105 The proportion is not true: Solve for variable. 4n n Applications: On a map, 2 cm represents 9 km. How many cm would represent 36 km? 1. Assign Variable: n = number of centimeters 2. Write Ratio in Words: Centimeters Kilometers 3. Write Ratio of Given Values: 2 cm 9 km 4. Write Ratio with Variable: n cm 36 km 5. Write Proportion: 2 cm n cm 9 km 36 km 6. Solve Proportion:. 2 n n 72 9n 72 9n n On the map, 8 cm will represent 36 km. 459

36 CCBC Math 081 CHAPTER 6 Chapter Review Simplify the ratio to :45 Write each in simplest fraction form grams to 15 milliliters miles to 15 gallons pages in 30 hours calories in 6 servings Determine the unit rate calories per 4 servings miles in 4 hours 10. $345 for 3 months of insurance pages in 8 hours Determine which purchase is the better deal boxes for $3.69 or 3 boxes for $ cans for $5.99 or 10 cans for $ $600 for 120 gallons or $450 for 60 gallons surf boards for $ or $93.75 for 15 surf boards towels for $47.88 or 8 towels for $31.60 Determine if the proportion is true or false Solve the following proportions n n x w 7 15 n x n m 5 460

37 CCBC Math 081 Chapter 6 Review Translate each into a proportion and solve the problem. 29. Tony gets 9 miles per gallon of gasoline in his truck. If he has 11 gallons of gasoline in his truck, how far can he go? 30. On a map, 2 inches represent 7 miles. How many inches will represent a distance of 91 miles? 31. A recipe calls for 1 teaspoon of vanilla for every 2 teaspoons of baking powder. How 2 much vanilla will be used with 7 teaspoons of baking powder? 32. Brian drives 360 miles on an 18 gallon tank of gasoline. If the car is showing 35 miles until empty, how many gallons of gas are in the car? 33. Jennifer sells five pieces of her homemade jewelry, earning $31.50 profit. How many pieces does she need to sell in order to earn $ profit? 34. John and Joe bought 10 dozen steamed crabs for $85. If Bill purchased 3 dozen crabs from the same company at the same rate, how much does he pay? 35. Jaime had to write a twenty page paper for school. If it took her seven hours to write four pages, how long will it take her to write the entire paper? Mixed Review 36. Convert 64 fluid ounces to quarts. 37. Convert hm to cm. 38. Compute Write 1 as a decimal with three decimal places Compute Compute Mary was training for a marathon. Over a week, she tracked the distance she ran each day to be 18.5 miles, 20.5 miles, 19 miles, 21.5 miles, 22 miles, and 21.5 miles. What are the mean, median, and mode for the distances she ran that week? 43. Determine the area of a circle with radius of 12 inches. Use

38 CCBC Math 081 Chapter 6 Review Determine the circumference of a circle with diameter of 84 cm. Use Distribute 3(8x 5). 46. Solve x x 47. Solve Solve 6x2x Solve 2 x Three-tenths of a number plus 6.2 is 3.8. Translate into a math equation and solve the equation to determine the number. 462

39 CCBC Math 081 Chapter 6 Review C h a p t e r 6 R e v i e w A n s w e r s to : grams 5 milliliters miles 3 gallons pages 3 hours 62 calories 3 servings calories per serving miles per hour 10. $115 per month pages per hour boxes for $ cans for $ $600 for 120 gallons 15. $93.75 for 15 surf boards towels for $ False 18. True 19. True 20. False 21. n = n = x = w = n x = n m miles inches tsp of vanilla gallons or 1 gallons pieces of jewelry 34. $ hours quarts cm Mean = 20.5 miles Median = 21 miles Mode = 21.5 miles in cm x x = x = x = x

40 CCBC Math 081 Unit 3 Review Chapters 5 and 6 CHAPTERS 5 & 6 Unit Three Review 1. Simplify 7x 9x Simplify -4y + 9x + 3x 11y + 8y 6x 3. Simplify 3 x 5 x Evaluate 7x 3y if x = 4 and y = Evaluate 3 x if x = Distribute 4(p 6) 7. Distribute 4 (7 h 35) 7 8. Fill in the blanks: An example of the property of is 9 + y = y Rewrite 4 (y c) using the associative property of multiplication 10. Determine whether v = 8 is a solution of 7 = v Solve 9 + x = Solve y 7.4 = Solve 5x = 55 z 14. Solve Solve 5y + 4 = (2x + 3) 10x = 4 464

41 CCBC Math 081 Unit 3 Review Chapters 5 and The sum of 6 and three times a number is 11. Determine the number. 18. Two fifths of a number is six sevenths. Determine the number. 19. Write the following ratio using a colon and as a fraction. 6 to Simplify the ratio: 16: A shelter has 27 cats and 21 dogs. What is the ratio of dogs to cats? 22. Write the ratio for $75 per 12 hours in simplest fraction form. 23. Determine the unit rate. 84 grams of sugar per 14 ounces of cereal. 24. A jogger can run 17.5 miles in 4 hours. What is the unit rate in miles per hour? 25. Determine whether the following proportion is true or false Solve the proportion. Give the answer in decimal form n 27. Solve the proportion. Give the answer in fraction form. 3 4 n Translate into a proportion and solve the problem. 15 mg of a drug are to be given every 5 hours. How much of the drug should be given in 24 hours? 29. Translate into a proportion and solve the problem. There are 16 salmon in 50 cubic tons of water in a lake. In how many cubic tons of water would you need to find 72 salmon? 30. Translate into a proportion and solve the problem. 2.5 cups of mix make 14 pancakes. How much mix is needed to make 35 pancakes? 465

42 CCBC Math 081 Unit 3 Review Chapters 5 and 6 U n i t T h r e e R e v i e w A n s w e r s 1. 2x y 6x x p h Commutative; addition 9. (4 y) c 10. No 11. x = y = x = z = y x = x = 11; x ; :17, : dogs 9 cats $25 4 hours g of sugar/oz miles/hour 25. False 26. n n mg cubic tons of water cups 466