50 No matter which way you write it, the way you say it is 1 to 50.

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1 RATIO & PROPORTION Sec 1. Defining Ratio & Proportion A RATIO is a comparison between two quantities. We use ratios everyay; one Pepsi costs 50 cents escribes a ratio. On a map, the legen might tell us one inch is equivalent to 50 miles or we might notice one han has five fingers. Those are all examples of comparisons ratios. A ratio can be written three ifferent ways. If we wante to show the comparison of one inch representing 50 miles on a map, we coul write that as; 1 to 50 or using a colon 1 : 50 or Using a fraction 1 50 No matter which way you write it, the way you say it is 1 to 50. Because we are going to learn to solve problems, it s easier to write the ratios using fractional notation. If we looke at the ratio of one inch representing 50 miles on a map, 1 50, we might etermine 2 inches represents 100 miles, 3 inches represents 150 miles by using equivalent fractions. That just seems to make sense. Look at that from a mathematical stanpoint, it appears that we might also be able to simplify ratios to Does represent the same comparison as 1 50? The answer is yes an if we looke at other ratios, we woul see that reucing/simplifying ratios oes not effect those comparisons. Now, we have some goo news. We not only iscovere how to write ratios, we also learne they can be simplifie. But before we move on, let me make a point. While ratios can be written like a fraction an simplifie like a fraction, a ratio is NOT a fraction! A fraction is part of a unit. A ratio represents a comparison. For instance; Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 1

2 = 5 6 as a fraction = 2 5 as a ratio What??? On the first example, you foun the common enominator, mae equal fractions, ae the numerators an brought own the enominator because you were etermining how much you ha for a whole unit fractions. In the secon example, let s say you got 1 hit out of 3 times at bat last week an 1 hit out of 2 times at bat this week. That results in 2 hits out of 5 times at bat. What we want clear here is a ratio is a comparison between hits an times at bat. We can not get one-thir a hit. We either get a hit or we on t. Now back to the example using a map. We notice that 3, 3 inches represents miles, coul be reuce to 1 50 meaning 1 inch represents 50 miles. Mathematically, by setting the ratios equal, we coul write 1 50 = Simplifying Ratios You simplify ratios the same way you simplify fractions. That is, you fin the GCF an simplify. So if 7 bars of cany cost $1.40, we write that using fractional notation an simplify. 7 bars $1.40 = 1 bar $0.20 From our previously learne math working with fractions, nothing has change. Simplifying ratios is one exactly the same way you woul simplify fractions. Since we notice the equality, that leas us to a new efinition: A PROPORTION is a statement of equality between 2 ratios. Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 2

3 Looking at a proportion like 1 = 3, we might see some other relationships that exist if 2 6 we take time an manipulate the numbers. For instance, what woul happen if we tippe both ratios up-sie own? 2 1 an 6 3, notice they are also equal, so 2 1 = 6 3 Noticing that is interesting, because it leas us to something we might o in our aily lives, checking prices. In math, we call it unit ratios. Example 1. Express each ratio as a fraction. a. 3 to 4 b. 8 to 5 c. 9:13. 15:7 Example 2. Express each ratio in simplest form. a. 12 to 10 b. 24:36 c Example 3. A certain math test has 50 questions. The first 10 are true false an the rest are matching. Fin: a. The ratio of true-false questions to matching questions. b. The ratio of true-false questions to the total number of questions. c. The ration of the total number of questions to matching questions. Sec. 2 Unit Ratios Unit ratios are ratios where the number written on the bottom escribes ONE unit, hence the wor unit. So, in the previous example if 7 bars of cany cost $1.40, we wrote that using fractional notation an simplifie. 7 bars $1.40 = 1 bar $0.20 An that was great. But we just iscovere that we can tip ratios upsie own an maintain equalities. Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 3

4 That is $ bars = $ bar is the same as the first proportion we wrote. An, that allows us to write the ratio as a unit ratio, the enominator being ONE. The cost is $0.20 per cany bar. Usually, we write the ratio the way it was given to us, an simplify it. To fin unit pricing, the best approach is to etermine how you want to write the ratio first by asking yourself how you want to compare the quantities so you will have the unit written in the enominator. There are times the ratios will not simplify in a way that results in ONE being in the enominator. Example If 24 cases of peanuts cost $16, how much woul one case cost? Writing that as a proportion, I want to know the cost of one case, so I am going to write the ratio with cases in the enominator. $16 24 cases = $2 3 cases Simplifying that ratio i not result in a UNIT ratio. In other wors, it i not tell me the price of ONE case. Note, that when working with unit ratios, ollars is written in the numerator. A better way to hanle unit rates is to simply ivie the numerator by the enominator. In other wors, once we ecie how to write the ratio to get the unit we are looking for in the enominator just ivie. So, in that problem, the unit ratio woul be 16/24 or 2/3 simplifie, then ivie or 3 2 Either way, the answer is $ or $0.67 per case. Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 4

5 Sec 3. Properties Of Proportion Now, let s go back an look at these proportions. We foun that we coul write proportions upsie own an maintain the equality. 1 = How about writing the original proportion sieways, will we get another equality? 1 3 an 2 6, notice they are equal also, so 1 3 = 2 6 So, that works out nicely. If we continue looking at the original proportion 1 2 = 3, we might also notice we coul 6 cross multiply an retain an equality. In other wors 1x6 = 2x3. This iea of manipulating numbers is pretty interesting stuff, on t you think? Makes you woner whether tipping ratios up-sie own, writing them sieways or cross multiplying only works for our original proportion? Well, to make that etermination, we woul have to play with some more proportions. Try some, if our observation hols up, we ll be able to generalize what we saw. Let s try these observations with the proportion 2 3 = 4 6 Can I tip them upsie own an still retain an equality? In other wors, oes 3 2 = 6 4? How about writing them sieways, oes 2 4 = 3 6 How about cross-multiplying in the original proportion, oes 2x6 The answer to all three questions is yes. = 3x4? Since everything seems to be working, we will generalize our observations using letters instea of numbers. Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 5

6 If a b = c, then 1. b a = c 2. a c = b 3. a = bc Those 3 observations are referre to as Properties of Proportions. Those properties can an will be use to help us solve problems. While we mae these properties by observation, we coul also have shown them to be true by using the Properties of Real Numbers. In other wors, if the original proportion is true, a b = c is true, I coul multiply both sies of the equation by the common enominator, b, that woul have resulte in property 3 above (cross-multiplying). Then, I coul have further manipulate to get the relationships in properties 1 an 2. In other wors, we coul prove these relationships exist for all proportions. Sec 4. Solving Problems Using Proportions To solve problems, most people use either equivalent fractions or cross multiplying to solve proportions. Example If a turtle travels 5 inches every 10 secons, how far will it travel in 50 secons? What we are going to o is set up a proportion. The way we ll o this is to ientify the comparison we are making. In this case we are saying 5 inches every 10 secons. Therefore, an this is very important, we are going to set up our proportion by saying inches is to secons. On one sie we have 5 escribing inches to secons. On the other sie we have to 10 again use the same comparison, inches to secons. We on t know the inches, so we ll call it n. Where will the 50 go in the ratio, top or bottom? Bottom, because it escribes secons goo eal. So now we have = n 50 Now, we can fin n by equivalent fractions or we coul use property 3 an cross multiply. Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 6

7 5 10 = n 50 10n = 5x 50 10n = 250 n = 25 The turtle will travel 25 inches in 50 secons It makes it easier to unerstan initially to write the same comparisons on both sies of the equal signs. In other wors, if we ha a ratio on one sie comparing inches to secons, then we write inches to secons on the other sie. If we compare the number of boys to girls on one sie, we woul have to write the same comparison on the other sie, boys to girls. We coul also write it as girls to boys on one sie as long as we wrote girls to boys on the other sie. The first Property of Proportion, tipping the ratios upsie own, permits this to happen. Solve these problems by setting up a proportion. 1. If there were 7 males for every 12 females at the ance, how many females were there if there were 21 males at the ance? Ask yourself; is there a ratio, a comparison in that problem? What s being compare? 2. Davi rea 40 pages of a book in 5 minutes. How many pages will he rea in 80 minutes if he reas at a constant rate? 3. On a map, one inch represents 150 miles. If Las Vegas an Reno are five inches apart on the map, what is the actual istance between them? 4. Bob ha 21 problems correct on a math test that ha a total of 25 questions, what percent grae i he earn? (In other wors, how many questions woul we expect him to get correct if there were 100 questions on the test?) 5. If there shoul be three calculators for every 4 stuents in an elementary school, how many calculators shoul be in a classroom that has 44 stuents? If a new school is scheule to open with 600 stuents, how many calculators shoul be orere? 6. If your car can go 350 miles on 20 gallons of gas, at that rate, how much gas woul you have to purchase to take a cross country trip that was 3000 miles long? Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 7

8 Sec 5. Direct Variation; y = kx We can also write proportions algebraically. In fact, y = kx is how a irect variation is efine. Here s the goo news, that s pretty much the same thing we have been oing. If I solve y = kx for k, I woul have k = y/x. In other wors, I woul have a ratio. Setting two ratios equal, I have a proportion. It turns out the k is calle the constant of variation. Variation, because these are proportions or variations, an it is a constant because as we sai before, the ratios are always equal, the same number, a constant. Earlier, we efine a ratio as a comparison between two numbers. That is, there are three boys for every 4 girls, resulting in a ratio of boys to girls is 3 : 4. Someone else may have escribe that relationship as girl to boys or 4 :3. Using functional notation, y = mx + b, where b = 0 an we let m = k, we have y = kx. k is the slope or the constant of variation. Our experiences inicate that x is the inepenent variable an y is the epenent variable. In other wors, the value of y varies, is epenent, with the value of x. Why is that important? In the boy girl ratio, we coul solve problems in terms of the ratio of boys to girls or girls to boys. The constant of variation, woul be ifferent epening upon how you chose to set up the ratio. In other wors, it is not clear which woul be the inepenent vs epenent variable. In algebra, we want that better efine so proportions can be seen as a function. So, we will say y varies irectly as x. So x is the inepenent variable an y is the epenent variable an is written as y = kx. That results in y/x = k, the constant of variation. Problem number 6 above was one by setting the ratios equal, then using equivalent fractions or cross multiplying to solve for the amount of gas. In that problem, we really we knew the ratios were equal, a constant of variation existe, but we in t care what it was. Other than setting the ratios equal, we in t care an really i not nee to know what that constant was. Let s o that same problem using y = kx. To solve using algebra, we first must fin the value of k, the constant of variation. Your car can travel 350 miles on 20 gallons of gas. Using y = kx, the miles (y) you can travel epens on the amount of gas (x) you have in the car. So, how many miles per gallon oes your car get that woul be the constant of variation. We coul fin that 2 ways 350/20, setting up the ratio as we i before or we coul substitute those values into y = kx. 350 = k (20) 350/20 = k Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 8

9 17.5 = k That means you get 17.5 miles per gallon. Now to fin out how much gas you woul nee to travel 3000 miles, you substitute the values of k an y into y = kx 3000 = 17.5 x = x When you rea y = kx mathematically, we say y varies irectly as x. Also, while we call k the constant of variation, it is also the slope of a line, the rate of change normally written an recognize in algebra by using m. Sec 6. Information Not Given in terms of Ratio The ratio an proportions problems we have one up to this point have expresse a ratio, then given you more information in terms of the ratio previously expresse. In other wors, if the ratio expresse was male to female, then more information was given to you in terms of males or females an you set up the proportion. Piece of cake, right? Well, what happens if you were given a ratio, like males to females, but then the aitional information you receive was not given in the terms of the original ratio? Maybe the aitional information tol you how many stuents were in a class altogether? You wouln t be able to set up a proportion base on what we know now. But, on t you love it when someboy says but? It normally means more is to come. So hol on to your chair to contain the excitement, we get to learn more by seeing patterns evelop. Up to this point, we know if 2 fractions are equal, then I can tip them upsie own, write them sieways an cross multiply an we will continue to have an equality. If a b = c,then 1. b a = c 2. a c = b an 3. Remember, the way we evelope those properties was by looking at equal fractions an looking for patterns. If we continue to look at equal fractions, we might come up with even more patterns. Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 9

10 Let s look, we originally sai that 1 3 = 2. Now if we continue to play with these 6 proportions, then looke to see if the same things we notice with these hel up for other equalities, then we might be able to make some generalizations. For instance, if I were to keep the numerator 1 on the left sie, an a the numerator an enominator together to make a new enominator on the left sie, woul I still have an equality if I i the same thing to the right sie. Let s peek. We have 1 3 = 2, keeping the same numerators, then aing the numerator an 6 enominator together for a new enominator, we get 1 2 an. Are they equal? Does = 2 8? Oh boy, the answer is yes. Don t you woner who stays up at night to play with patterns like this? If we looke at other equal fractions, we woul fin this seems to be true. So what o is generalize this. To show this works, we general using variables, by aing one to both sies of the original proportion. If a b = c, then a b + b b = c + a + b b = c + Other patterns we might see by looking at the fraction 1 3 = 2 6 inclue aing the numerators an writing those over the sum of the enominators that woul be equal to either of the original fractions. In other wors, = 1 3 = 2 6 An of course, since this also seems to work with a number of ifferent equal fractions, we again make a generalization. If a a+c = c, then = a = c b b+ b Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 10

11 So now we have iscovere two more patterns for a total of 5 Properties of Proportion. Now, going back to the problem we were escribing earlier that gave a ratio, then gave aitional information that was not in terms of the ratio, we ll be able to manipulate the properties of proportion to solve aitional problems. Say yes to math, this is really great stuff. Example If there are 3 boys for every 7 girls at school, how many boys atten the school if the total stuent enrollment consists of 440 stuents? The first comparison given is boys to girls. Knowing this, we woul like to set up a proportion that looks like this: boys girls = boys girls just as we have one before. The problem that we encounter is while we can put the 3 an 7 on the left sie to represent boys an girls, we have to ask ourselves, where oes 440 go? It oes not represent just boys or just girls, so we can t put it in either position on the right sie. 440 represent the total number of boys an girls. Remembering what we just i, see there was a reason for looking for more patterns, we notice if we have a proportion like b g = b g, then b b + g = b b + g woul be true. From our problem, we now can see b + g woul represent the total of the boys an girls. Filling in this proportion, we have = b = b 440 Solving; 10b = 3x b = 1320 b = 132 There woul be 132 boys; to fin the number of girls we coul subtract 132 from 440. Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 11

12 The point being, if you were given a problem being escribe by a ratio, then aitional information was given to you not using the escriptors in the original ratio, you coul manipulate the information using one of the properties of proportion. Sec 7. Using Equations to Solve Proportions To be quite frank, that is not the way I woul usually attack that sort of problem. What I woul prefer oing is setting up the problem algebraically. Let s step back an see how ratios work. Remember, we sai you coul reuce ratios. In 3 1 other wors, if I ha the ratio of, I coul reuce it to Visually, the way you reuce is by iviing out a common factor. To reuce coul rewrite it as 1x3 1. Diviing out the 3 s, we woul have 50 x , we Now going back to the previous example, we ha 3 boys for every 7 girls, with a total enrollment of 440. Doing this algebraically, we still have the same ratio, boys to girls. But, again, we realize the aitional information is not given in terms of boys or girls. We just i this problem by playing with the properties of proportion. Using algebra, the ratio of boys to girls, b g 3 is. 7 Does that mean we have exactly three boys an seven girls? No, that ratio comes from reucing the actual number in the boys to girls ratio. Since we on t know the common factor we woul have ivie out a common factor, we ll call it X. Unbelievable concept. Using the ratio; b g 3 or, we now know we have 3X boys an 7X girls. So the ratio of b 7 g looks like this; 3X 7 X. But notice the sum of 3X an 7X woul be the total number of stuents The total number of stuents is 440, therefore we have Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 12

13 boys + girls = 440 3X + 7 X = X = 440 X = 44 The ratio of boys to girls 3X 7 X, that means there are 3X boys or 3 (44) which is 132 boys. That manipulation allowe us to solve a proportion problem where more information was not given in terms of the first ratio. Example If the ratios of the sies of a triangle are 4:5:6 an the perimeter is 75 inches, how long is each sie? Immeiately notice we are given ratios in terms of sies an more information is given in terms of perimeter. To me, that suggests oing this problem algebraically. The sies therefore are 4x, 5x, an 6x. The perimeter is the sum of the sies of a polygon, in this case 75 inches. So, 4x + 5x + 6x = 75 15x = 75 x = 5 So the sies are 20, 25, an 30 inches. From my stanpoint, there are two types of Ratio & Proportion problems. Type 1 Type II A ratio is given, an then more information is given in terms of the escriptors of the first ratio. For this type of problem, you set the two ratios equal an solve by equivalent fractions or cross multiplying A ratio is given, an then more information is given to you that is not using the escriptors in the original ratio. This type of problem shoul be one algebraically as we i in the last example. Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 13

14 The 5 Properties of Proportion if a b = c,then 1. b a = c (upsie own) 2. a c = b (sieways) 3. a = bc (cross multiply) 4. a +b b = c + (sum of num an en over eb) 5. a+c a c = = b+ b (a num, a en) As you try some of these problems, first etermine if they are Type I or Type II, then use the appropriate problem solving strategy. 1. Cross Multiplying Proofs Properties of Proportion If a b = c, then I can multiply both sies by the common enominator b( a b ) = b( c ), ivie cout common factors a = bc or a = bc Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 14

15 2. Upsie own if a b = c, then from the previous proof, we know a= bc, since we want the a an c in the enominator, we ivie both sies by ac a ac = bc ivie out common factors ac c = b a or b a = by the Symmetric Property c 3. Sieways if a b = c, then from the previous proof, we know a= bc, since we want the c an in the enominator, a c = bc c a c = b we ivie both sies by c ivie out common factors 4. Sum of Numerator an Denominator over Denominator if a b = c, a 1 to both sies in terms of enominators a b + b b = c + a + b b = c + Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 15

16 5. Sum of Numerators an Denominators if a b = c, to get a sum in the problem, I nee to a something that will result in the esire sums a b = c, which results in a=bc a + c = bc+ c, aing c to both sies (a+c)=c(b+), factoring a+ c b+ = c, iving both sies by (b+) a+ c b+ = c = a b, since c = a b Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 16

17 RATIO AND PROPORTION 1. Express each ratio as a fraction. a. 3 to 4 b. 8 to 5 c. 9:13. 15:7 2. Express each ratio in simplest form. a. 12 to 10 b. 24:36 c A certain math test has 50 questions. The first 10 are true false an the rest are matching. Fin: a. The ratio of true-false questions to matching questions. b. The ratio of true-false questions to the total number of questions. c. The ration of the total number of questions to matching questions. 4. A 30 poun moonling weighs 180 pouns on the earth. How much oes a 300 poun Earthling weigh on the moon? 5. The ratio of the sies of a certain triangle is 2:7:8. If the longest sie of the triangle is 40 cm, how long are the other two sies? 6. If a quarterback completes 20 out of 45 passes in his first game, how many passes o you expect him to complete in his secon game if he only throws 18 passes? 7. On a trip across the country Joe use 20 gallons of gas to go 300 miles. At this rate, how much gas must he use to go 3500 miles? 8. On a map 3 inches represents 10 miles. How many miles o 16 inches represent? 9. If our class is representative of the university an there are 2 males for every 12 females. How many men atten the university if the female population totals 15,000? 10. The ratio of length to with of a rectangle is 8:3. Fin the imensions of the rectangle if the perimeter is 88m. Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 17

18 RATIO & PROPORTION Stuents shoul be able to: Problem set Ientify, write, an simplify ratios Set up a proportion from a wor problem Solve problems using proportions Solve problems using unit pricing 1. Of 300 stuents in the cafeteria, 140 ha lunch. Write the ratio of the stuents in the cafeteria to the stuents that ha lunch. 2. A math test has 50 questions. The first ten are True-False an the rest are matching, fin the ratio of True-False questions to matching questions. The ratio of True-False to the total number of questions. 3. Write a proportion to represent the conitional if there are 5 boys for every 7 girls in math class, how many boys are there if there are 35 girls? 4. A baseball team won 8 games an lost 3. What is the ratio of wins to games playe? 5. Use the chart: Grae Distribution A 3 B 5 C 7 D 4 F 1 What is the ratio of A s to the class? What is the ratio of B s to D s? 6. A 30 poun moon ling weighs 180 pouns on earth. How much oes a 300 poun Earthling weigh on the moon? 7. A quarterback completes 12 out of 20 passes in his first game, how many passes woul you expect him to complete in his secon game if he only throws 15 passes? 8. On a trip across country, Any use 20 gallons of gas to travel 300 miles. At this rate, how much gas must he use to go 2000 miles? 9. On a map, if one inch represents 25 miles, how many miles oes 7 inches represent? Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 18

19 10. At Intercee Inc., 7 out of 10 people receive raises. If they employ 350 people, how many got raises? 11. Snow is falling at a rate of one inch every 3.5 hours. At this rate how long will it take for 6.5 inches of snow to fall? 12. The ratio of with to length in a rectangle is 3:8. If the length of the rectangle is 20 inches, what is the with? 13. There were 5 boys for every 3 girls in the class. Out of the 56 stuents in the class, how many were girls? 14. Fre pai $72 last week in gas for his car. If he use 56 gallons of gas, how much i he pay per gallon? 15. A store sells bars of soap at 4 for $3.56 or 3 for $2.97. What is the unit price of the soap that is the better buy? 16. Bob rove 320 miles in 5 hours. At this rate, how far coul he travel in three hours? 17. 1½ cups of sugar will serve 6 people, how many people will 2 ½ cups serve? 18. A 15 poun ham is enough to serve 20 people. How much ham o you nee to serve 50 people? 19. The ratio length to with of a rectangle is 8:3. Fin the imensions of the rectangle if the perimeter is 88 inches. 20. The ratio of men to women in a class is 2:7. Of the 36 stuents in the class, how many were women? MORE PROBLEMS 1) On last week s math test Carol ha 21 correct out of 25 problems. What percent grae i she receive? 2) Bob receive an 85% on his history exam. If there were 20 questions, how many i he have correct? 3) John receives a 5% commission on his sales. If he receive $30. How much i he sell? 4) Bob receive an 84% on his history exam. If there were 50 questions, how many i he have wrong? Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 19

20 5) If you receive 20% off on a pair of pants that cost $25, how much woul you pay? 6) Jessie earns $25 per week. If 8% is eucte for social security, what is the amount Jessie receives? 7) A raio costs $20; Harol buys it for $16. What percent off i he receive? 8) A quarterback complete 18 of 25 passes in a football game. What percent of the passes i he complete? 9) Mrs. Freeze bought a watch for $45, she was charge 10% feeral tax an 4% sales tax. How much i she have to pay altogether? SOLVING PROBLEMS 1) Jack sol his river for $17.85, making a profit of $5.95. The profit is what percent of the selling price? 2) Twenty-one percent of Maria s salary goes to her car payment. If she earns $400, how much oes she have to pay for her car? 3) Mirana s ball club won 5 of 6 games. What percent of the games i his team lose? 4) At 4 th Avenue there are 630 stuents. If 70 were absent Monay, what percent is this? 5) Last year I earne $5000, this year I earne $5500. What percent increase is this? 6) At 4 th Avenue 60 stuents receive a 1 in math. This was 20% of the stuents taking math in the 8 th grae. How many math stuents in the 8 th grae? 7) In the Falcon s Nest ice cream cost 20. If the school pays 15, what percent markup is it? 8. The pep club was ecrease from 15 members to 12 members. What was the percent of ecrease? 9) $15 is what percent more than $10? Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 20

21 Ratio & Proportion Name Date Definitions 1. Ratio 2. Proportion 3. Unit ratio 4. Express the ratio 5 to 7 as a fraction. 5. Express each ratio in simplest form. a. 6 to 8 b. 24 to Bob s club has 50 members. Ten are male an the remainers are females. Fin: a. The ratio of males to females. b. The ratio of males to the total number of members in the club. c. The ratio of the total number of people in the club to the number of females. 7. If there are 12 inches in one foot, how many inches are there in five feet? 8. The legen on a map inicates one inch equals 35 miles. If the istance travele on the map is four inches, how many miles woul have to be travele? 9. If a basketball player makes 25 out of 45 baskets in his first game, how many baskets woul you expect him to make in the secon game if he attempte 18? Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 21

22 10. On a trip across country Bob use 25 gallons of gas to travel 300 miles. At this rate, how much gas will he use to travel 3000 miles? 11. If three pieces of cany cost 10, how much will 16 pieces cost? 12. If the ratio of boys to girls in art class is 5 to 7, how many girls are there in class if there are 60 stuents enrolle? 13. The ratio of length to with of a rectangle is 8 : 3. Fin the imensions of the rectangle if the perimeter is 44 inches. 14. Davi reas 40 pages of a book in 50 minutes. How many pages shoul he be able to rea in 80 minutes? 15. Davi reas 40 pages of a book in 50 minutes. How many pages shoul he be able to rea in one minute? 16. Bob saves $16 in 16 ays. His sister Sarah saves $49 in 7 weeks, are these rates equivalent. Explain your answer. 17. A canle is 30 inches long. After burning 12 minutes, the canle is 25 inches long. How long woul it take the whole canle to burn? 18. Write 5 Properties of Proportion an give an example of each. 19. If a b = c, show mathematically how a = bc Ratio & Proportion Copyright Hanlonmath 2003 Hanlonmath.com 22