Ratio and Proportion Handout This handout will explain how to express simple ratios and solve proportion problems. After completion of the worksheet you should be able to: A. Set up a ratio with like units B. Set up a ratio with unlike units C. Determine if two ratios are proportional D. Solve for the missing number in a proportion E. Solve word problems Ratios • Definition: A ratio is a comparison between two numbers. A ratio statement can be written three ways: 3 , 3 to 2, 3:2 2 You want to bet on a horse race at the track and the odds are 2 to 1; this is a ratio that can be written 2 to 1, 2:1, 2 1 You bake cookies and the recipe calls for 4 parts (cups) flour to 2 parts (cups) sugar. The comparison of flour to sugar is a ratio: 4 to 2, 4:2, 4 . 2 Problem Set I: (answers to problem sets begin on page 10) Express the following comparisons as ratios Suppose a class has 14 redheads, 8 brunettes, and 6 blondes a. What is the ratio of redheads to brunettes? b. What is the ratio of redheads to blondes? c. What is the ratio of blondes to brunettes? What is ratio of blondes to total students? d. Ratios and Proportions Handout Revised @ 2009 MLC page 1 of 10 Problem Set II Express the following as ratios in fraction form and reduce a. 3 to 12 b. 25 to 5 c. 6 to 30 d. 100 to 10 e. 42 to 4 f. 7 to 30 Finding common units: Ratios should be written in the same units or measure whenever 4 cups 4 quart rather than . This makes possible i.e. 2 cups 2 cups comparisons easier and accurate. Note the problem “3 hours to 60 minutes” These units (hours and minutes) are not alike. You must convert one to the other’s unit so that you have minutes to minutes or hours to hours. It is easier to convert the bigger unit (hours) to the smaller unit (minutes). Use dimensional analysis or proportions to make the conversion. Dimensional Analysis method: 3 hours = _______ minutes 3 hours 60 min 180 × = = 180 minutes 1 1 hour 1 Proportional method: 60 minutes x min therefore x = 180 minutes = 1 hour 3 hours Note: Select the above method you like best and use it for all conversions. Now you know that 3 hours is the same as 180 minutes so you can substitute 180 minutes in the ratio and have 180 minutes then reduce to 3 minutes 60 min 1 min Ratios and Proportions Handout Revised @ 2009 MLC page 2 of 10 Summary: 1. State problem 3 hours to 60 minutes as a ratio 2. Analyze Change one of the unlike units 3. Convert bigger unit to smaller 3 hours to 180 minutes 4. Substitute the converted 180 minutes to 60 minutes 180 = 18 = 3 number into the problem and 60 6 1 write as a fraction Example: Compare 2 quarters to 3 pennies. When comparing money, it is frequently easier to convert to pennies. Therefore, 2 quarters equal 50 pennies. Substitute 50 pennies for the 2 quarters. Now write as a ratio 50 to 3: 50 . 3 25 pennies 1 How about comparing a quarter to a dollar? = 100 pennies 4 How would you write the ratio “a dollar to a quarter?” 100 pennies 4 Remember, the first number goes on top: = 25 pennies 1 Example: Compare 4 yards to 3 feet. First analyze. Change (convert) 4 yards to equivalent in feet. Do either dimensional analysis or proportions to make the conversion. Dimensional analysis: 4 yards 3 feet 12 × = = 12feet 1 1 yard 1 Proportions: 3 feet = x feet therefore x = 12 feet 1 yard 4 yards Problems Set III Express each of the following ratios in fractional form then simplify. 5¢ to $2 1. 12 feet to 2 yards 2. 30 minutes to 2 hours 3. 5 days to 1 year 4. 1 dime to 1 quarter 5. Comparing unlike units (rates) Ratios and Proportions Handout Revised @ 2009 MLC page 3 of 10 Sometimes measurable quantities of unlike are compared. These cannot be converted to a common unit because there is no equivalent for them. Example: 80¢ for 2 lbs. of bananas ¢ or lbs. measure two different quantities, money and weight 80¢ = 40¢ (40¢ per pound) 2 lbs. 1 lb. Example: 200 miles on 8 gallons of gas Ratio = 200 miles: 8 gallons = 200 miles = 25 mi. (25 miles per gallon) 8 gallons 1 gal. Example: 200 miles: 240 minutes 200 miles In comparing distance to time, the answer is always given 240 minutes in miles per hour (mph). Therefore, time must be converted to hours. 200 miles = 50 mi. or 50 mph 4 hours 1 hr. Problem Set IV Express the following rates in fractional form and reduce to lowest terms. 40¢ : 5 lbs 1. 2. 60 benches for 180 people 100 miles to 120 minutes (in miles per hour) 3. 84 miles on 2 gallons of gas 4. Proportions Ratios and Proportions Handout Revised @ 2009 MLC page 4 of 10 • Definition: A proportion is a mathematical sentence that states that two ratios are equal. It is two ratios joined by an equal sign. The units do not have to be the same. b) 5 lbs. = 7 lbs. Some examples: a) 2 = 6 3 9 $2.00 $2.80 36 3 inches 55 1 inches 4 8 c) = 2 blouses 3 blouses These can be written as 2:3::6:9; 5 lbs.:$2.00::7 lbs.:$2.80; 36 3 inches : 2 blouses :: 55 1 : 3 blouses 4 8 Notice that when you cross-multiply the diagonal numbers, you get the same answers. This means the proportion is true. Look at the first example a) 2 = 6 3 9 Cross multiply: 2 x 9 = 18; 3 x 6 = 18 Look at the second example b) 5 lbs. = 7 lbs. $2.00 $2.80 5 x $2.80 = $14.00; 7 x $2.00 = $14.00 36 3 inches 55 1 inches 4 8 = Look at the third example c) 2 blouses 3 blouses 3 x 36 3 = 441 2 x 55 1 = 441 4 4 8 4 If the products of this cross multiplication are equal, then the ratios are a true proportion. This method can be used to see if you have done your math correctly in the following section. Solving Proportion Problems Ratios and Proportions Handout Revised @ 2009 MLC page 5 of 10 Sometimes you will be given two equivalent ratios but one number will be missing. You must find the number that goes where the x is placed. If your answer is correct then the cross multiplications will be equal. You can find the missing number by using a formula called “Lonely Mate.” This formula consists of two steps: 1) Cross multiply the two diagonal numbers 2) Divide that answer by the remaining number (“lonely mate”) multiply 3 x 24 = 72 Example: 3 = x 8 24 divide 72 by 8 X=9 Example: Example: 3=9 2 x 4 5= 5 x 6 2 multiply 2 x 9 = 18 divide 18 by 3 X=6 multiply 4 × 6 = 24 5 1 5 divide 24 ÷ 2 5 5 X = 12 Example: 5:25::x:150 Problem Set V write in fraction form first: 5 =x 25 150 cross multiply 5 x 150 = 750 divide by 25 x = 30 Solve for the given variable Ratios and Proportions Handout Revised @ 2009 MLC page 6 of 10 1. 3 =x 10 50 2. 5 3. 12 2 = 80 x 6= 5 x 80 4. 5. 6. 1 =x 3 1.8 1 1 4= 2 31 2 x x:81::27:2.43 Word Problems In real life situations you will use ratios and proportions to solve problems. The hard part will be the set up of the equation. Example: Sandra wants to give a party for 60 people. She has a punch recipe that makes 2 gallons of punch and serves 15 people. How many gallons of punch should she make for her party? 1) Set up a ratio from the recipe: gallons of punch to the 2 gallons number of people. 15 people 2) Set up a ratio of gallons to the people coming to the x gallons party. (Place the gallons on top) 60 people 2 gallons x gallons 3) Set up a proportion = 15 people 60 people *Like units should be across from each other 4) Solve using the “lonely mate” formula 2 gallons x gallons = 15 people 60 people 2 X 60 = 120 120 ÷ 15 = 8 This means she should make 8 gallons of punch! Ratios and Proportions Handout Revised @ 2009 MLC page 7 of 10 Example: Orlando stuffs envelopes for extra money. He makes a quarter for every dozen he stuffs. How many envelopes will he have to stuff to make $10.00? 1. Set up a ratio between the amount of money he makes and the dozen $.25 envelopes he stuffs 12 envelopes 2. Set up a ratio between the amount of money he will make and the $10 number of envelopes he will have to stuff x envelopes 3. Set up a proportion between the two ratios $.25 $10 = 12 envelopes x envelopes 4. Solve using “lonely mate” 10 x 12 = 120 a. 120 ÷ .25 = 480 b. This means that he will have to stuff 480 envelopes! Problem Set VI Ratios and Proportions Handout Revised @ 2009 MLC page 8 of 10 1. A premature infant is gaining 2.5 ounces a day. At that rate, how many days will it take for him to gain 8 ounces? 2. A copy machine can duplicate 2400 copies in one hour. How many copies can it make per minute? 3. A pants factory pays its seamstresses by their production output. If the company paid a seamstress 5¢ for every 4 pockets she sewed on, how many pockets would she have to sew on to receive $60.00? 4. A five-kilogram sack of Beefy-Bones dog food weighs 11¼ lbs. How much would a 1-kilogram sack weigh? 5. A patient is receiving 1 liter of IV fluids every 8 hours. At that rate, how much will he receive in 3 hours? Answers to Problem Sets Ratios and Proportions Handout Revised @ 2009 MLC page 9 of 10 Problem Set I (Note: the first number goes on top of the fraction bar) a. b. c. d. 14 or 14 to 8 or 14:8 8 14 or 14 to 6 or 14:6 6 6 or 6 to 8 or 6:8 8 6 or 6 to 28 or 6:28 28 Problem Set II 3 =1 a. 12 4 25 = 5 b. 5 1 6 =1 c. 30 5 100 = 10 d. 10 1 42 = 21 e. 4 2 7 f. 30 Problem Set III 1. 5¢ = 5 = 1 $2.00 200 40 2. 12 ft. = 12 ft. = 2 2 yds. 6 ft. 1 3. 30 min. = 30 min. = 1 2 hrs. 120 min. 4 4. 5 days 1 5 days = = 1 year 365 days 73 1 dime = 10 = 2 1 quarter 25 5 5. Problem Set IV 1. 2. 3. 4. 40¢ : 5 lbs = 40¢ = 8¢ 5 lbs. 1 lb. 60 benches for 180 people = 60 = 1 bench 180 3 people 100 miles : 120 minutes = 100 miles = 50 miles 2 hours 1 hour 84 miles on 2 gallons = 84 miles 42 miles = 2 gallons 1 gallon Problem Set V 1. x = 15 4. x = .6 2. x = 192 5. x = 7 3. x = 5⅓ or 5.33 6. x = 900 Problem Set VI 1. x = 3.2 days 2. x = 40 copies 3. x = 4800 pockets 4. x = 2.25 lbs. 5. x = .375 L Ratios and Proportions Handout Revised @ 2009 MLC page 10 of 10