Integrated Algebra Ratios, Proportions and Rates $ =

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1 % ½ ⅘ Integrated Algebra Ratios, Proportions and Rates $ = ⅓

2 Ratio A ratio is a comparison of two quantities. Ex: A teacher graded 180 bonus quizzes during the school year. The number of quizzes receiving A's, B's, and C's were in the ratio of 5 : 3 : 1, respectively. Find the number of bonus quizzes that received a grade of A for the school year.

3 1. In a senior class, there are b boys and g girls. Express the ratio of the number of boys to the total number in the class. Ratio Practice 6. The measures of the angles of a triangle are in the ratio of 2 : 3 : 4. Find the number of degrees in the smallest angle of the triangle. 2. A soccer team played 25 games and won 17. a. What is the ratio of the number of wins to the number of loses? 7. Hannah took a trip to visit her cousin. She drove 120 miles to reach her cousin's house and the same distance back home. b. What is the ratio of the number of games played to the number of games won? 3. The Yankees won 125 games, the Red Sox won 97 games, and the Mets won 86 games. What is the ratio of wins of the Yankees to the Red Sox to the Mets? a. It took her 1.2 hours to get halfway to her cousin's house. What was her average speed, in miles per hour, for the first 1.2 hours of her trip? 4. Two numbers are in a ratio of 5 : 3. Their sum is 80. Find the largest number. b. Hannah's average speed for the remainder of the trip to her cousin's house was 40 miles per hour. How long, in hours, did it take her to drive the remaining distance? 5. Mr. Smith and Mr. Kelly are business partners. They agreed to divide the profits in the ratio of 3 : 2. The profit amounted to $24,000. How much did each person receive? c. Traveling home along the same route, Hannah drove at an average rate of 55 miles per hour. After 2 hours her car broke down. How many miles was she from home?

4 Proportion A proportion is a comparison of ratios. Ex: The length of a stadium is 100 yards and its width is 75 yards. If 1 inch represents 25 yards, what would be the dimensions of the stadium drawn on a sheet of paper?

5 1. Are the followingtrue proportions? a. Proportion Practice 5. If 4 tickets to a show cost $9.00, find the cost of 14 tickets. b. 2. Solve for x: 6. A house which is assessed for $10,000 pays $300 in taxes. What should the tax be on a house assessed at $15,500? 3. Solve for x: 7. The length of the thruway is 600 miles. If 0.5 inch represents 50 miles, what is the length of the thruway on the map? 4. Solve for x: 8. If on a scale drawing 48 feet are represented by 12 inches, then a scale of 1/4 inch represents how many feet?

6 Direct Variation When two variable quantities have a constant (unchanged) ratio, their relationship is called a direct variation. It is said that one variable "varies directly" as the other. The constant ratio is called the constant of variation. The formula for direct variation is y = kx, where k is the constant of variation."y varies directly as x". Solving for k: Ex: If mvaries directly as yand m is 6 when y is 36, find the constant of variation. 1. "avaries directly as b". If a = 3 whenb = 24, find b when a = 10. Direct Variation Practice 3. There are about 200 calories in 50 grams of Swiss cheese. Willie ate 70 grams of this cheese. About how many calories were in the cheese that he ate if the number of calories varies directly as the weight of the cheese. 2. In the following chart, does one variable vary directly with the other? M N One variable (A) varies directly as the other (C). Find the missing numbers x and y. Write the formula which relates the variables. A 1 2 x C 3 y 15

7 Percents Percents are used to describe parts of a whole base amount. When one of the parts of the relationship is unknown, we can solve an algebraic equation for the unknown quantity. Ex 1: Find 2.5% of 600. Ex 2: If 120 million roses were sold on Valentine's Day, and 75% of the roses were red, how many red roses were sold on Valentine's Day? Percent Practice is 15% of what number? 4. 3 is what percent of 12? 2. A real estate company pays commissions to their sales people for selling property. A well known company paid 6% commissions last year to their sales staff, totaling $480,000. What was the dollar value of the real estate sold by the company that year? 5. Juan missed 6 out of 92 questions on a test. To the nearest percent, what percent of the questions did he solve correctly? 3. In a magazine, 30 pages of the 80 pages are devoted to sports. What percent of the magazine is devoted to sports? 6. After Mary lost 20% of her investment, she had $2000 left. How much did she invest originally?

8 Percent of Increase, Decrease, Discount Use the percent to find the amount the original number is changing by ADD raise, appreciate, tax, tip SUBTRACT depreciate, discount Ex 1: You normally work at the mall over the spring break for $450. This year the boss tells you that you will be receiving a 7% raise. How much of an increase will you be receiving? How much will you be earning in total this year? Ex 2: An I-phone is on sale for 20% off the original price. If the original price is $270, what is the sale price? Ex 3: The enrollment at a local elementary school had 560 students last year. This year the enrollment has decreased by 48 students. What is the percent of decrease to the nearest tenth of a percent? Percent Increase, Decrease, Discount Practice 1. Find the percent of change, rounded to the nearest whole percent. Describe the change as an increase or decrease. $6.25/h to $6.75/h 4. The circulation of a newsletter decreased from 6500 to What was the percentage decrease in circulation? 2. Joshua's normal body temperature is 98.5ºF. Due to a cold, his temperature went up 3ºF. To the nearest percent, what was the percent of increase in his body temperature? 5. Walter is a waiter at the Towne Diner. He earns a daily wage of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday? 3. Vicki paid $11.25 for a pair of pants that usually sells for $25. What percent discount did she receive? 6. Rashawn bought a CD that cost $18.99 and paid $20.51, including sales tax. What was the rate of the sales tax?

9 Rate A rate is a ratio that compares two different kinds of numbers, such as miles per hour, or inches per minute. A unit rate compares a quantity to its unit of measure. A rate expresses how long it takes to do something. Solving a problem dealing with rate usually involves solving a proportion. Ex 1: How long, in minutes, did it take the bug to cover 350 inches at a rate of 50 inches per minute? Ex 2:The bug drives his matchbox cruiser to his friend s house traveling at the rate of 50 inches per minute. He then walks back to his home at the rate of 10 inches per minute. If the round trip took 9 minutes, how far is it from the bug's home to his friend's house? 1. A cell phone can receive 120 messages per minute. At this rate, how many messages can the phone receive in 150 seconds? Rate Practice 3. Joseph typed a 1,200-word essay in 25 minutes. At this rate, determine how many words he can type in 45 minutes. 2. Nicole s aerobics class exercises to fastpaced music. If the rate of the music is 120 beats per minute, how many beats would there be in a class that is 0.75 hour long? 4. A car uses one gallon of gasoline for every 20 miles it travels. If a gallon of gasoline costs $3.98, how much will the gas cost, to the nearest dollar, to travel 180 miles? 5. Tom drove 290 miles from his college to home and used 23.2 gallons of gasoline. His sister, Ann, drove 225 miles from her college to home and used 15 gallons of gasoline. Whose vehicle had better gas mileage? Justify your answer.

10 Conversions Sometimes you need to convert from one unit of measure to another similar unit. Ex: Roberta needs ribbon for a craft project. The ribbon sells for $3.75 per yard. Find the cost, in dollars, for 48 inches of the ribbon.

11 1. Peter walked 8,900 feet from home to school. Conversion Practice 4. A soda container holds gallons of soda. How many ounces of soda does this container hold? How far, to the nearest tenth of a mile, did he walk? 2. Elizabeth is baking chocolate chip cookies. A single batch uses teaspoon of vanilla. If Elizabeth is mixing the ingredients for five batches at the same time, how many tablespoons of vanilla will she use? 5. A jogger ran at a rate of 5.4 miles per hour. Find the jogger's exact rate, in feet per minute. 3. If the speed of sound is 344 meters per second, what is the approximate speed of sound, in meters per hour? 6. Andy is 6 feet tall. If 1 inch equals 2.54 centimeters, how tall is Andy, to the nearest centimeter?

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