Rates

RatesLesson 1No calculators for this unit1Real-World Link----Beats in 2 minutes = ? ---- Number of beatsin 1 minuteNumber of beatsin 1 minute-- minutes-- -- minutes-- Find a Unit RateCommon Unit RatesExample 1Got it? 1Find each unit rate. Round to the nearest hundredth if necessary.

a. $300 for 6 hours

$50 per hour

b. 220 miles on 8 gallons

27.5 miles per hour6

Example 2Example 3The prices of 3 different bags of dog food are given in the table. Which size bag has the lowest price per pound rounded to the nearest cent?Dog Food PricesBag Size (lb)Price ($)4049.002023.4489.8840-pound bag: $49.00 40 $1.23 per pound

20-pound bag: $23.44 20 $1.17 per pound

8-pound bag: $9.88 8 $1.24 per poundGot it? 3Tito wants to buy some peanut butter to donate to the local food pantry. Tito wants to buy as much peanut butter as possible. Which brand should he buy?Peanut Butter SalesBrandSales PriceNutty12 oz for $2.19Grandmas18 oz for $2.79Bees28 oz for $4.69Sav-A-Lot40 oz for $6.60Nutty: $0.18 per oz

Grandmas: $0.155

Bees: $0.1675

Sav-A-Lot:$0.165Example 4Complex Fractions and Unit RatesLesson 2Complex FractionsExample 1Example 2Example 3Example 4Tia can paint 46 square feet per hour. Got it?Pick one problems and solve. 17Example 5Convert Unit RatesLesson 3Commonly Used MeasurementsCustomary Units of MeasureSmallerLarger12 inches1 foot16 ounces1 pound8 pints1 gallon3 feet1 yard5,280 feet1 mileMetric Units of MeasureSmallerLarger100 cm1 meter1,000 grams1 kilogram1,000 ml1 liter10 mm1 centimeter1,000 mg1 gramUnit RatioExample 1A remote control car travels at a rate of 10 feet per second. How many inches per second is this?Example 1A remote control car travels at a rate of 10 feet per second. How many inches per second is this?Example 2A swordfish can swim at a rate of 60 miles per hour. How many feet per hour in this?

Example 2A swordfish can swim at a rate of 60 miles per hour. How many feet per hour in this?

Example 3Marvin walks at a speed of 7 feet per second. How many feet per hour is this?Example 3Marvin walks at a speed of 7 feet per second. How many feet per hour is this?Example 4The average speed of one team in a relay race is about 10 miles per hour. What is the speed in feet per second?Example 4The average speed of one team in a relay race is about 10 miles per hour. What is the speed in feet per second?Proportional and Nonproportional RelationshipsLesson 4Proportional vs. NonproportionalExample 1Andrew earns $18 per hour for mowing lawns. Is the amount of money he earns proportional to the number of hours he spends mowing? Explain.

Step 1: Make a table

EARNINGS $18365472TIME (HR)1234Example 1Andrew earns $18 per hour for mowing lawns. Is the amount of money he earns proportional to the number of hours he spends mowing? Explain.

Step 2: Make equivalent fractions

Do they all equal each other? Yes, the amount Andrew earns is proportional to the number of hours he works.

Got it? 1At Lakeview Middle School, there are 2 homeroom teachers assigned to every 48 students. Is the number of students at this school proportional to the number of teachers? Explain your reasoning.

The ratio is proportional since the ratio is 24 students to every teacher. Example 2Uptown Tickets charges $7 per baseball game plus a $3 processing fee to order. Is the cost of an order proportional to the number of tickets ordered? Explain.

STEP 1: Make a table.COST $7+3 = 102(7) + 3 = 173(7) + 3 = 24TICKETS ORDERED123Example 2STEP 2: Make equivalent fractions

Are these fractions true? No, these are not equal so the cost and tickets ordered are not proportional. COST $7+3 = 102(7) + 3 = 173(7) + 3 = 24TICKETS ORDERED123Example 3You can use the recipe shown to make a fruit punch. Is the amount of sugar used proportional to the amount of mix used?

Are the ratios equivalent? Yes, so the sugar and mix are proportional.

CUPS OF SUGARENVELOPES OF MIX1121 324Got it? 2 & 3At the beginning of the year, Isabel had $120 in the bank. Each week, she deposits another $20. Is her account balance proportional to the number of weeks of deposits? Use the table below and explain your reasoning.

No, the balance and the number of weeks are not proportion because the ratios are not equal. TIME (WK)1234BALANCE ($)140160180200Example 4The tables shown represent the number of pages Martin and Gabriel read over time. Which situation represents a proportional relationship?

All of Martins ratios equal each other, so Martins table is proportional. PAGES MARTIN READTIME(MIN)25410615PAGES GABRIEL READTIME(MIN)35410715MID-CHAPTER CHECKGraphing Proportional RelationshipsLesson 5Identifying Proportional RelationshipsFrom a graph:

A proportional relationship is

1. a straight line

2. a line goes through the origin (0,0)Example 1The slowest mammal on Earth is the tree sloth. It moves at a speed of 6 feet per minute. Determine whether the number of feet the sloth moves is proportional to the number of minutes it moves by graphing. Explain.

Step 1: Make a table

Example 1The slowest mammal on Earth is the tree sloth. It moves at a speed of 6 feet per minute. Determine whither the number of feet the sloth moves is proportional to the number of minutes it moves by graphing. Explain.

Step 2: Graph the ordered pairs

The line passes through the origin and the line is straight,so, this situation is proportional.

Got it? 1James earned $5 an hour babysitting. Determine whether the amount of money James earns babysitting is proportional to the number of hours he babysits by graphing. Explain.

The amount of moneyearned is proportionalto the number of hours because the line is straightand goes through the origin.

Example 2The cost of renting video games from Games Inc. is shown in the table. Does this represent a proportional relationship? Explain.

No, even though the line is straight, it does not go through the origin. Got it? 2Determine is the number of calories and the number of minutes is proportional based on the table below.

No, even though the line goes through the origin, it is not a straight line. Example 3Which batting cage represents a proportional relationships between the number of pitches and the cost? Explain.

Fun Center shows a proportional relationship because it goes through the origin. Solve Proportional RelationshipsLesson 6Write and Solve ProportionsExample 1After 2 hours, the air temperature had risen 7F. Write and solve a proportion to find the amount of time it will take at this rate for the temperature to rise an additional 13F.It will take about 3.7 hours to rise additional 13F. Got it? 1x = 3.6y = 85n = 49Example 2Example 2About 139 donors would have a bloodType of 0Got it? 2The ratio of 7th grade students to 8th grade students in a soccer league is 17:23. If there are 200 students in all, how many are in the 7th grade?

85 studentsExample 3 Using Unit RateExample 4Got it? 3 & 4Olivia typed 2 pages in 15 minutes. Write an equation relating the number of minutes m to the number of pages p typed. How long will it take her to type 10 pages at this rate?

M = 7.5p

75 minutes or 1 hour 15 minutesSlopeLesson 8Slope

Example 1The table below shows the relationship between the number of seconds y it takes to hear thunder after a lightning strike and the miles x you are from the lightning.

Graph the data.

Example 1

Got it? 1Graph the data about plant height for a science fair project. Then find the slope and explain what it represents.

Slope = 1.5; the plant grows 1.5 cm/weekExample 2Ronald opened a savings account. Each week he deposits $300. Draw a graph of the account balance versus time. Find the numerical value of the slope and interpret it in words.

Got it? 2Jessica has a balance of $45 on her cell phone account. She adds $10 each week for the next four weeks. In the work zone, graph the account balance versus time. Find the numerical value of the slope and interpret it in words.

The slope = $10/week

Jessica deposits $10 per week Direct VariationLesson 9Direct VariationWords: a line that has a constant k and goes through the origin

Symbols: y = kx, where k is a number (positive or negative)

Example: y = 3x

3 or k is called constant of variation or constant of proportionalityExample 1

Got it? 1Example 2The equation y = 10x represents the amount of money y Julio earns for x hours he works. Identify the constant of proportionality. Explain what it means in this situation.

Constant of proportionality = k

y = kx

y = 10x

$10 is the constant and it means that Julio earns $10 an hour.

Got it? 2The distance y traveled in miles by the Chang family in x hours is represented by the equation y = 55x. Identify the constant of proportionality. Explain what it represents.

y = kxy = 55x

constant = 55The family traveled 55 miles per hourExample 3 Determining Direct VariationPizzas cost $8 each plus a $3 delivery charge. Show the cost of 1, 2, 3, and 4 pizzas. Is there a direct variation?

Step 1: Make a table.

Example 3 Determining Direct VariationGot it? 3Two pounds of cheese cost $8.40. Show the cost for 1, 2, 3, and 4 on a table. Is this an example of direct variation?

Yes, the constant rate is $4.20 per pound.Example 4Determine if this linear relationship shows a direct variation.

Yes, the ratios are the same so this table shows a direct variation.