Proportional Relationships

Transcription

1 Proportional Relationships The graph should be linear. For a directl proportional relationship, the graph will alwas be a straight line through the origin. NAME: CLASS: TEACHER: Ms. Schmidt _

2 Computing Unit Rate Classwork Da 1 Vocabular Ratio - Unit Rate Eample 1: How fast is our class? Trial Number of Papers Passed Time (in seconds) Ratio of Number of Papers Passed to Time Rate Unit Rate A unit rate is a rate per one given unit; such as 34 miles per 1 gallon Find the unit rate of each: 1) 216 meters in 8 seconds how man meters for 1 second? 2) Epress the ratio of $10 for 8 fish as a unit rate (1 fish). 3) $2,702 for 28 people- how much mone for 1 person? Epress each ratio as a fraction in simplest form: 4) 27 rooms to 48 windows 5) 3 gallons to 15 quarts

3 Computing Unit Rate Classwork Da 1 Which is the better price? 6) Si cand bars for $2.62 or eight bars for $3.40? 7) Paint brushes that sell in a packet of one dozen for $6.46 or eighteen for $9.90? 8) If ou spend $11.13 for 8 gallons of gasoline, how much would ou spend on 14 gallons? TRY THESE: 1) Mr. K needs help solving this problem. A hot dog truck sells 9 hot dogs for $ a) Find the unit rate b) If he wants to bu 3 hot dogs for Mrs. Trantino, how much will it cost? 2) Write in simplest form: 13 diamonds to 52 cards 3) Which is the better price? 32 ounces for $3.84 or 40 ounces for $4.40 More Practice: Find the unit rate of each: 1) The Seneca s Student Government sold $75 worth of tickets for a talent show in 3 hours. How man tickets did the sell in one hour? $ $ hours hours 2) At Si Flags, 1,473 people entered the park in 3 hours. How man people entered the park in 1 hour? people hours people hours 3) A wedding a Villa Lombardi s cost $9,750 for 150 people. How much does Villa Lombardi s charge per guest? 4) April showers bring Ma flowers! If 3 inches of rain fell in 5 hours, how man inches fell per hour? 5) You can bu 4 apples at Stop and Shop for $0.96. You can bu 6 of the same apples at Pathmark for $1.50. Which store has the better bu?

4 Computing Unit Rate Classwork Da 1 6) If a runner ran 102 meters in 12 seconds, how man meters did he/she run per second? 7) Ticketmaster sold 1200 tickets to the Mets-Yankees game in 3 hours. How man tickets were sold is one hour? Which is the better bargain? Find the unit price for each and compare them. 8) Pens: $4.50 for 3 pens or $3.20 for 20 pens 9) Pencils: 16 for $8.32 or 35 for $ ) Luc went awa on vacation for 10 das and when she came home she had 280 s. How man s did she get per da? 11) Derek just got a new I-Phone and downloaded 348 songs in 6 hours. How man songs did he download per hour? 12) Ran and his brother are comparing the prices of two brands of cereal. Frosted Flakes costs $2.25 for a 15- ounce bo. Luck charms costs $3.90 for a 30-ounce bo. Which brand is more epensive and b how much per-ounce? 13) Gas mileage is the average number of miles ou can drive a car per gallon of gasoline. A test of a new car resulted in 2,250 miles being driven using 125 gallons of gas. Find the new car s gas mileage. 14) The table shows the prices that Mrs. Dragotta paid at 3 different gas stations. Complete the table to determine which gas station had the better price per gallon. Gas Station Gallons Price Price per Gallon (Show work here) Hess 15 $43.50 Coastal 10 $29.40 Amoco 12 $35.88

5 Proportional Relationships Classwork Da 2 Vocabular Constant of Proportionalit - The value of the ratio of quantities in a proportional relationship. This value is also equivalent to the unit rate. Understand what the phrase proportional to means. A ver common misconception is that two variables are directl proportional to if one increases as the other increases. Two variables are said to be directl proportional if, and onl if, their ratio is a constant for all values of each variable. Therefore when one variable is divided b the other, the answer is alwas a constant. The have the same unit rate. Directl Proportional NOT Directl Proportional Independent Variable (Domain ) Dependent Variable (Range ) Look at the tables and determine if the quantities given are in a proportional relationship. ****In order to test for proportional relationships, quantities must have equivalent ratios. **** Compare each ratio to see if the are equivalent. Is there a constant rule? If es, it is proportional. ****Are the cross products equal? Eample 1) Eample 2) Hour $ Proportional? Yes or No Hour Miles Proportional? Yes or No Using a ratio to identif a unit rate-practice 1) Gas Mileage 2) Cooking Times Miles Gallons of gas used Weight of Turke(lb) Cooking Time (hour) Proportional es/no Proportional es/no Unit Rate(miles per gallon) = Unit Rate(pounds per hour) = 3) Paint Coverage 4) Grapes per pound Amount of Paint (gallons) Area Covered (square feet) 1/2 2,000 3/4 3, , ,000 Grapes (pound lb) Cost (per lb) 5 $ $3.60 1/4 $1.20 Proportional es/no Proportional es/no

6 Proportional Relationships Classwork Da 2 Find the Unit Rate and Missing Value. 1) Babsitting Pa-Salar per hour 2) Dog Biscuits Hours (h) Pa (p) $11 $55 Biscuits (lb) Price $1.65 $5.50 $9.90 Unit Rate in words Unit Rate($ per hour) Unit Rate in words Unit Rate (Cost per biscuit) 3) Teting Prices 4) Calories burned for 130 lb. woman running 5 mph # of tets Pa (p) $150 $225 $18.75 Length of workout (hours) Calories burned ,416 Unit Rate in words Unit Rate (cost per tet) Unit Rate in words Unit Rate (calories per hour) Etra Problems: Determine whether each table forms a proportional relationship. (SHOW ALL WORK) *Remember the table must have equivalent ratio. 1) ) ) Proportional? es or no Proportional? es or no Proportional? Yes or No 4) ) ) Proportional? es or no Proportional? es or no Proportional? es or no

7 Identifing Proportional Relationships Classwork Da 3 Vocabular Unit Rate Constant of Proportionalit = c or = k Constant rate of change (slope) Origin *Proportional relationships can be represented on a coordinate plane. A graph of ever proportional relationship will be a straight line that includes the origin, the point (0,0). Can ou draw this? 1) The graph shows the relationship between the time it takes a turtle to walk and its distance. How far does it travel at 0 hours? miles How far does it travel at 1 hour? miles How far does it travel at 2 hours? miles How far does it travel at 3 hours? miles How far will it travel in 6 hours? miles What does the point (2, 2) mean? What is the unit rate (1,r)? Since this graph goes through the origin and the unit rate is constant it is a proportional relationship. Distance (miles) Turtle Speed Time (hr) Determine whether each graph is proportional. Sod Sales Mowing Lawns 2) 3) Total Cost ($) Profit ($) Area (sq. ft) Yes or No- Justif Lawns Yes or No- Justif

8 Identifing Proportional Relationships Classwork Da 3 Miles 4) Road Trip Hour a) Is the graph showing a proportional relationship? b) Speculate what might have happened during the 3 rd and 4 th hour of the trip. c) What is the average speed from hour 1 to hour 3? d) What is the average speed for the entire trip? 5) (0,0), (1,2), (2,4), (3,6) Proportional? Yes or No (Direct Variation) 6) (0,4), (1,6), (2,8), (3,10) Proportional? Yes or No (Direct Variation) Justif Justif Justif 7) The graph shows distances traveled for a bike-a-thon. Use the information displaed in the graph to find out how man miles the participant rides in 11 hour. Triccle-a-thon 8) A student tring to save the Holtsville Ecolog site was getting signatures on a petition at a rate of 30 signatures a da. At this rate, how man signatures will he have in 1 week? Petition 360 Miles Signatures 180 What is the unit rate? (miles per hour) Hour Das

9 Identifing Proportional Relationships Classwork Da 3 9) The graph shows our wages for mowing lawns during the summer. How man lawns will ou mow if ou earned $390? Wage ($) Mowing Lawns Lawns 10) 11) Wh? Wh? Proportional? es or no Proportional? es or no Proportional? Yes or No Proportional? Yes or No Unit Rate 12) Isaiah sold cand bars to help raise mone for his scouting troop. The table shows the amount of cand he sold to the mone he received. Is the amount of cand bars sold proportional to the mone Isaiah received? How do ou know? Eample 1: From a Table to Graph

10 Identifing Proportional Relationships Homework Da 3 1) Complete the table below. 2) Using the graph, answer the following questions. Yogurt Costs Amount of Price Yogurt (c) ($) Show Work for #1 here a) What is the unit rate? b) How man inches will be 18 ards? Number of Inches YARDS AND INCHES Number of Yards ) During Jose s phsical education class toda, students visited activit stations. Net to each station was a chart depicting how man calories (on average) would be burned b completing the activit. Calories burned while Jumping Rope a) Is the number of Calories burned proportional to time? How do ou know? b) If Jose jumped rope for 6.5 minutes, how man calories would he epect to burn? 4) Multipl: ) What is as an improper fraction? 6) Order from least to greatest 2 1, 7%, ) Evaluate: 8) What is a solution 1 3 for = and = 27 of 0? 3 4

11 Unit Rate as the Constant of Proportionalit in an Equation Classwork Da 4 Find the Constant and Write the Formula = c or = k When the ratios of two quantities are alwas the same, the quantities are proportional. The value of the ratio is called the constant of proportionalit(k or c). This value is also equivalent to the unit rate. k is the constant of proportionalit. The height is half the width. h =.5w is the constant of proportionalit. h = w Constant of Proportionalit is the same as unit rate (slope). Vocabular Constant- Coefficient - Variable- Identif the constant (Hint-circle the word after the word per because that is our (input).) Find the constant of proportionalit for each table/graph and write the equation of the direct variation. 1) ards of cloth per blanket 2) pa per hour 3) Yards () Blankets (b) Constant of proportionalit (c) = Hours (h) Pa (p) $11 $55 $88 Constant of proportionalit (c) = Equation Equation 4) The graph to the right shows the distance (in ft.) ran b a Jaguar. a) What does the point (5, 280) represent in the contet of the situation? b) What does the point (3, 168) represent in the contet of the situation? c) Is the distance run b the Jaguar proportional to the time? Eplain wh or wh not. Distance (ft) Constant of proportionalit (c) = Equation Jaguar s Run d) Write an equation to represent the distance ran b the Jaguar. Eplain or model our reasoning.

12 Unit Rate as the Constant of Proportionalit in an Equation Classwork Da 4 When two values are directl proportional, the value of the output () divided b the input () will alwas have the same value. This value will be the coefficient of the input,. Find the constant of proportionalit (unit rate) in each of the relationships that follow: 1) = 3 2) = 1 3 3) = 4) = c = c = c = c = If ou are given a table or a graph, ou can find the constant of proportionalit (slope) b dividing the output,, b the input,. ( ) Find the constant of proportionalit in the chart and graph below. Net, write the equation for the situation ) Find the constant of proportionalit. Write the equation that satisfies this table. = c, Then write our equation as 2 8 = ) Find the constant of proportionalit. Write the equation that satisfies this graph. Constant/slope Equation 3) Find the constant of proportionalit. Write the equation that satisfies this table. Before ou begin, which value do ou think is the output? Hours (h) Pa (p) $16 $80 $192 $320 4) Find the constant of proportionalit. Write the equation that satisfies this graph. Constant/slope Equation

13 Unit Rate as the Constant of Proportionalit in an Equation Classwork Da 4 5) Find the constant of proportionalit. Write the equation that satisfies this table = k, Then write our equation as = 6) Find the constant of proportionalit. Write the equation that satisfies this graph. Constant/slope Equation 7) Find the constant of proportionalit. Write the equation that satisfies this table. Before ou begin, which value do ou think is the output? 8) Find the constant of proportionalit. Write the equation that satisfies this graph. Constant/slope Hours (h) # of rooms painted (p) Equation 9) If the constant of proportionalit is 3.5, what is the equation? 10) A truck driver has travelled 350 miles in 5 hours. Write an equation that represents his distance travelled per hour. 11) The cost of a certain vegetable is 0.59 per pound. Write an equation to represent this situation, using c to represent the cost and p, for pounds. 12) The new data plan offers 2MB of data for $30. Write an equation to represent this situation, using c to represent the cost and d, for data.

14 Unit Rate as the Constant of Proportionalit in an Equation Homework Da 4 K or c = constant of proportionalit Find the Constant (unit rate/slope) and Write the Formula = k or = c 1) wages per da 2) price per pound 3) pounds per bag Das (d) Wages (w) $51.25 $ $ Constant of proportionalit (k) = Pounds Price $7.96 $9.95 $11.94 Constant of proportionalit (k) = Bags (b) Dog Food (lb) (d) Constant of proportionalit (k) = Equation Equation Equation Fruit Price per Pound 4) 5) Dance Tickets Price (p) Profit (p) Oranges (f) Constant of proportionalit (k) = Equation Tickets (t) Constant of proportionalit (k) = Equation 6) A baker can produce 120 cookies for ever 3 hours. What is the constant of proportionalit? What is the equation that represents this situation?

15 7) The following table shows the amount of cand and price paid. a) Is the cost of cand proportional to the amount of cand? b) Write an equation to illustrate the relationship between the amount of cand and the cost. c) Using the equation, predict how much it will cost for 12 pounds of cand? d) What is the maimum amount of cand ou can bu with $60? 8) Plot the following points on a coordinate grid. (2,2), (4,4), (6,6), (8,8) Find the constant of proportionalit What is the equation? 9) Plot the following points on a coordinate grid. (3,1), (6,2), (9,3) Find the constant of proportionalit What is the equation? 10) Create a real-life question that has a constant of proportionalit that is a whole number. Be sure to write the equation and eplain what it means. 11) Create a real-life question that has a constant of proportionalit that is a fraction. Be sure to write the equation and eplain what it means.

16 Write the Constant of Proportionalit as a Table Classwork Da 5 Write the Equation (=c) from a table Complete the following tables. Using the table of values, write the equation on the line. 1) 2) 3) X Y X Y X Y c 4) 5) 6) X Y X Y X Y ) How is #5 different from all the other tables? Can ou figure out the rule?

17 Write the Constant of Proportionalit as a Table Classwork Da 5 8) Given the equation of a line, = 4, 9) Given the equation of a line, = 2 + 1, complete the following table. complete the following table ) Fill in the blanks to the right. Not a direct Variation (not a constant of proportionalit) Walking Constant/slope (c) Distance (miles) Equation Time (hr) 11) Write an equation to find the price for 12) Write an equation to find the amount for an an amount of minutes. amount of das Minutes (m) Price (p) in $ 100 $ $50 1,000 $100 1,500 $150 Dollars (hundreds) Art Sales Das Constant/slope (c) Equation Constant/slope (c) Equation

18 Write the Constant of Proportionalit as a Table Homework Da 5 c = c c = constant of proportionalit Write the linear equation that gives the rule for this table. Write our answer as an equation with first, followed b an equals sign. 1) 2) 3) 4) Constant Constant Constant Constant Equation Equation Equation Equation 5) Write the equation for the relationship shown in the graph. Use whole numbers: Review- Show work on separate paper. Don t be afraid of all the words. 6) Rand is planning to drive from New Jerse to Florida. Rand recorded the distance traveled and the total number of gallons used ever time he stopped for gas. Assume miles driven are proportional to Gallons Consumed in order to complete the table.

19 Write the Constant of Proportionalit as a Table Homework Da 5 7) Andrea is a street artist in New Orleans. She draws caricatures (cartoon-like portraits of tourists). People have their portrait drawn and then come back later to pick it up from her. The graph below shows the relationship between the number of portraits she draws and the amount of time in hours needed to draw the portraits. a) Write several ordered pairs from the graph and eplain what each coordinate pair means in the contet of this graph. b) Write an equation that would relate the number of portraits drawn to the time spent drawing the portraits. c) Determine the constant of proportionalit and eplain what it means in this situation. 8) The graph below shows the amount of time a person can shower with a certain amount of water. a) Can ou determine b looking at the graph whether the length of the shower is proportional to the number of gallons of water? Eplain how ou know. Shower Water Usage b) How long can a person shower with 15 gallons of water and with 60 gallons of water? c) What are the coordinates of point A? Describe point A in the contet of the problem. d) Can ou use the graph to identif the unit rate? e) Plot the unit rate on the graph. Is the point on the line of this relationship? f) Write the equation to represent the relationship between the number of gallons used and the length of a shower.

20 Using Unit Rate as a Scale Factor Classwork Da 6 Eplain what a point (,) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0,0) and (1,r) where r is the unit rate. Do Now - Write an equation that will model the proportional relationship given in each real world situation. 1) There are 3 cans that store 9 tennis balls. Consider the number of balls per can. a. Find the constant of proportionalit for this situation. Write in words b. Write an equation to represent the relationship. 2) In 25 minutes Li can run 10 laps around the track. Consider the number of laps she can run per minute. a. Find the constant of proportionalit in this situation. b. Write an equation to represent the relationship. Eamples 1) On average, Susan downloads 60 songs per month. An online music vendor sells package prices for songs that can be downloaded on to personal digital devices. The graph below shows the package prices for the most popular promotions. Susan wants to know if she should bu her music from this compan or pa a flat fee of $58.00 for the month offered b another compan. Which is the better bu? 2) Ms. Robinson decided to make juice to serve along with the pizza at the Student Government part. The directions said to mi 2 scoops of powdered drink mi with a half a gallon of water to make each pitcher of juice. One of Ms. Robinson s students said she will mi 8 scoops with 2 gallons of water to get 4 pitchers. How can ou use the concept of proportion to decide whether the student is correct?

21 Using Unit Rate as a Scale Factor Classwork Da 6 3) A student is making trail mi. A) Create a graph, using the coordinate plane below, to determine if the quantities of nuts and fruit are proportional for each serving size listed in the table. Label the aes and title our graph. Serving Size Cups of Nuts () Cups of fruits () B) If the quantities are proportional, what is the constant of proportionalit (slope/unit rate) that defines the relationship? C) Eplain how the constant of proportionalit was determined and how it relates to both the table and graph. D) What does the point (0, 0) mean in regards to the situation? E) What does the point (1, 2) mean in regards to the situation? 4) If Kala can walk 2 miles is ½ an hour, what would be her unit rate per hour? 5) A car on the epresswa is travelling 15 miles in.25 hours, what speed is the car travelling per hour?

22 Using Unit Rate as a Scale Factor Classwork Da 6 6) The graph below represents the cost of a pack of gum. The unit rate is represented as $ each pack. Represent the relationship using b completing the table and writing an equation of the line. Cost in dollars (d) Number of packs (g) Number of Packs (g) Cost in Dollars (d) Equation A) Eplain what each point on the graph means. B) How much will 20 packs of gum cost? C) Eplain what the point (0,0) means. D) Eplain what the point (1, 0.75) means. Lesson Summar: The points (0,0) and (1, r), where r is the unit rate, will alwas fall on the line representing two quantities that are proportional to each other. he graph. These two points ma not alwas be given as part of the set of data for a given real-world or mathematical situation, but the will alwas fall on the line that passes through the given data points. tit.

23 Using Unit Rate as a Scale Factor Homework Da 6 1) The graph to the right shows the distance (in ft.) ran b an athlete in training. a) What does the point (5, 280) represent in the contet of the situation? b) What does the point (3, 168) represent in the contet of the situation? c) Is the distance run b the athlete proportional to the time? Eplain wh or wh not. Distance Athlete Running Time d) Write an equation to represent the distance run b the athlete. Eplain or model our reasoning. 2) The following graph represents the total cost of renting a car. The cost of renting a car is a fied amount each da regardless of how man miles the car is driven. It does not matter how man miles ou drive; ou just pa an amount per da. Car Rental Cost a) What does the ordered pair (4, 250) represent? b) What would be the cost to rent the car for a week? Eplain or model our reasoning. c) What is the unit rate and what does it mean? 3) The following table shows the amount of broccoli and price paid. Amount of Broccoli (pounds) Cost (Dollars) a) Is the cost of broccoli proportional to the amount of broccoli? b) Write an equation to illustrate the relationship between the amount of broccoli and the cost. c) Using the equation, predict how much it will cost for 12 pounds of broccoli? d) What is the maimum amount of broccoli ou can bu with $11.25?

24 Interpreting Graphs with Proportional Relationship Classwork Da 7 Remember back on da 1 of this lesson, ou were told ou would be able to tell if something formed a proportional relationship, what the constant of proportionalit is, and how to graph and write an equation. 1) Haile works for Cake Boss making brownies all da. She can bake 6 batches of brownies in 3 hours. d) a) Find the constant of proportionalit. b) Fill in the table below: Hours(h) Batches(b) c) Write an equation to represent this situation. d) Graph this situation in the graph on the right. Be sure to label our aes for batches and for hours. Be sure to title our graph. 2) Looking to the right we have a sample question from the state. Travel Last summer, a famil took a trip to a beach that was about 200 miles awa from their home. The graph to the right shows the distance driven, in miles, and the time, in hours, taken for the trip. Show all work. What was their average speed from hour 1 to hour 4? a) 25 miles per hour b) miles per hour Miles Hours c) miles per hour d) 100 miles per hour

25 3) Spencer rides his biccle for 10 hours. He can bike 25 miles in 2 hours. a) Find the constant of proportionalit. b) Fill in the table below: Hours Distance c) Write an equation to represent this situation d) Graph this situation in the graph on the right. Be sure to label our aes with miles and for hours. Be sure to title our graph. 4) At NASA, a rocket was test fired. The graph to the right shows the distance risen and fallen, in miles, and the time, in minutes, taken for the trip. Show all work. a) What was the rockets average speed from minute 0 to minute 3? b) What happened between minute 3 through minute 6? c) During what minutes did the rocket descend? Miles Height Minutes d) What was the rockets average rate of descent? 5) What does the points (0,0) and (1, r) represent on a graph? 6) Define the constant of proportionalit in our own words.

26 Interpreting Graphs with Proportional Relationship Homework Da 7 1) 15 Basement Flooding 12 Number of Inches (i) Number of Hours (h) 1) Eplain what the point (2, 6) means in reference to the graph. 2) Eplain what (0, 0) means. 3) Eplain what (1, r) means where r is the unit rate. 4) This summer, Maggie would like to start saving mone. Maggie is planning on working all 10 weeks of the summer. She saves $20 ever two weeks. a) Find the constant of proportionalit. b) Fill in the table below: Weeks Savings c) Write an equation to represent this situation d) Graph this situation in the graph on the right. Be sure to label our aes with weeks and savings. Be sure to title our graph.

27 5) Fill in the blanks: Weeks Savings 35 6) A bo scout convention takes a road trip. There are 282 people going and onl 47 cars. How man people will need to fit in each car? 7) One da ou download 4 songs for $5. Write an equation that uses the constant of proportionalit to describe the relationships between s songs and the cost in d dollars. 8) Last month the electric bill was $50.64 for 450 kilowatt-hours of electricit. At that rate, what would be the cost for 240 kilowatt-hours? 9) Make up our own proportional relationship. *Create a table *Create a graph *State the unit rate * Write situation in words *Write an equation to represent the constant of proportionalit. Eplain our situation in words. Table Graph (label aes and title) Unit Rate/Constant/Slope Equation

28 Unit 3 Proportional Relationships Performance Task 1) Ale spent the summer helping out at his famil s business. He was hoping to earn enough mone to bu a new $220 gaming sstem b the end of the summer. Halfwa through the summer, after working for 4 weeks, he had earned $112. Ale wonders, If I continue to work and earn mone at this rate, will I have enough mone to bu the gaming sstem b the end of the summer? To check his assumption, he decided to make a table. He entered his total mone earned at the end of week 1 and his total mone earned at the end of Week 4. 2) Carli s class built some solar-powered robots. The raced the robots in the parking lot of the school. The graphs below show the distance d, in meters, that each of three robots traveled after t seconds. a) Each graph has a point labeled. What does the point tell ou about how far that robot has traveled? b) Carli said that the ratio between the number of seconds each robot travels and the number of meters it has traveled is constant. Is she correct? Eplain. c) How fast is each robot traveling? How can ou see this in the graph?

29 3) Al s Produce Stand sells 7 ears of corn for $1.50. Barbara s Produce stand sells 13 ears of corn for $2.75. Write two equations, one for each produce stand that models the relationship between the number of ears of corn sold and the cost. Then use each equation to help complete the tables below. 4) During their last workout, Izz ran 2 ¼ miles in 15 minutes and her friend Julia ran 3 ¾ miles in 25 minutes. Each girl thought she were the faster runner. Based on their last run, which girl is correct? Show all work. 5) Championship T-shirts sell for $22 each. a. What point(s) MUST be on the graph for the quantities to be proportional to each other? b. What does the ordered pair (5, 110) represent in the contet of this problem? c. How man T-shirts were sold if ou spent a total of $88?