ratio, rates, & proportions

Ratio, Rates, & Proportions

RatiosA ratio is a comparison of two numbers.

Example: Tamara has 2 dogs and 8 fish. The ratio of dogs to fish can be written in three different ways.

*** Be careful with the fraction ratios they dont always have identical meanings to other fractions***

There are two different types of ratios.Part-to-Part RatiosExample: Tamaras dogs to cats is 2 to 8 or 1 to 4

.Part to-Whole RatiosExample: Tamaras dogs to total pets is 2 to 10 or 1 to 5

Try the following Apples to bananasThere are 12 apples and 3 bananas

So the ratio of apples to bananas is

4:1, 4 to 1, OR Oranges to applesThere are 15 oranges and 12 apples

So the ratio of oranges to apples is

5:4, 5 to 4, OR

Roses to iris

Daisies to roses Iris to daisies

all flowers to roses

A large bouquet of flowers is made up of 18 roses, 16 daisies, and 24 irises. Write each ratio in all three forms in simplest form. Identify which ratios are part-to-part and which ratios are part-to-whole.18 to 243 to 458 to 829 to 916 to 188 to 924 to 163 to 2

The ratio is comparing two number. If you changed it to a mixed number it would no longer be a comparison. This is why fraction ratios are tricky

Explain why the ratio 9

is not written as a mixed number:

Comparing RatiosCompare ratios by writing in simplest formAre these ratios equivalent?250 Kit Kats to 4 M&Ms and 500 Kit Kats to 8 M&Ms

12 out of 20 doctors agree and 12 out of 30 doctors agree

The ratio of students in Ms. Bs classes that had HW was 8 to 2 and 80% of Mrs. Longs class had HW

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RatesA rate is a ratio that compares two different quantities or measurements.

Rates can be simplified

Rates us the words per and for

Example: Driving 55 miles per hour

Example: 3 tickets for $1

Unit RatesA unit rate is a rate per one unit. In unit rate the denominator is always one.

Example: Miguel types 180 words in 4 min. How many words can he type per minute?

= or words per minute

rate unit rate word form

45 45 . 1

Unit rates make it easier to make comparisons.Example: Taylor can type 215 words in 5 minHow many words can he type per minute?

Who is the faster typist? How much faster? Taylor is 2wpm faster than Miguel

Film costs $7.50 for 3 rolls

90 students and 5 teachers

**Snowfall of 12 inches in 4 hours.

Ian drove 30 miles in 0.5 hoursDrive 288 miles on 16 gallons of gas.

Earn $49 for 40 hours of work

Use 5 quarts of water for every 2 lbs of chicken

Sarah drove 5 miles in 20 minutes

Try the Following$2.50 per roll15 mph2 quarts per lb$1.23 per hour18 mpg18 students per teacher2 5/6 per hour60 mph

Complex Unit Rates

Suppose a boat travels 30 miles in 2 hoursHow do you write this rate?

Suppose a boat travels 12 miles in 2/3 hoursHow do you write this as a rate?

How do you write this as a division problem?

Determine the unit rate: 18mph

Suppose a boat travels 8 miles in 5/8 hours.How do you write this as a rate?

How do you write this as a division problem?

Determine the unit rate: 14 mph

.Complex fractions are fractions that have fractions within them. They are either in the numerator, denominator, or both.

Divide complex fractions by multiplying (keep, change, change)

Mary is making pillows for her Life Skills class. She bought yards of fabric. Her total cost was $16. What was the cost per yard?

Doug entered a canoe race. He rowed miles in hour. What is

his average speed? Mrs. Robare is making costumes for the school play. Each costume requires 0.75 yards of fabric. She bought 6 yards of fabric. How many costumes can Mrs. Robare make?

A lawn company advertises that they can spread 7,500 square feet of grass seed in hours. Find the number of square feet of grass seed that can be spread in an hour.

TRY the FOLLOWINGWrite each rate. Then determine the unit rate and write in both fraction and word form$5.82 per yard3000 ft per hour7mph8 costumes

Comparing Unit RatesDario has two options for buying boxes of pasta. At CornerMarket he can buy seven boxes of pasta for $6. At SuperFoodz he can buy six boxes of pasta for $5.He divided 7 by 6 and got 1.166666667 at CornerMarket. He then divided 6 by 7 and got 0.85714286. He was confused. What do these numbers tell him about the price of boxes of pasta at CornerMarket?

Decide which makes more sense to youCompare the two stores prices. Which store offers the better deal?

1.166667 is the number of boxes you can get for $10.85714286 is the price per boxPrice per boxCM - $0.86 per boxSF - $0.83 per box

Proportions

Two quantities are proportional if they have a constant ratio or unit rate.You can determine proportionality by comparing ratios Andrew earns $18 per hour for mowing lawns. Is the amount he earns proportional to the number of hours he spends mowing?Make a table to show these amounts

For each number of hours worked, write the relationship of the amount he earned and hour as a ratio in simplest form.Are all the rates equivalent?

Since each rate simplifies to 18, they are all equivalent. This means the amount of money Andrew earns is proportional to the number of hours he spends mowing.

Earnings ($)18365472Time (h)1234

Uptown Tickets charges $7 per baseball game ticket plus $2 processing fee per order. Is the cost of an order proportional to the number of tickets ordered?Make a table to show these amounts

For each number of tickets, write the relationship of the cost of the and the number of tickets ordered.

Are all the rates equivalent?

The rates are not equivalent. This means the total cost of the tickets is not proportional to the number of tickets sold.

Cost ($)10172431Tickets Ordered1234

Use the recipe to make fruit punch. Is the amount of sugar used proportional to the amount of mix used? Explain.

Yes, they all reduce to

In July, a paleontologist found 368 fossils at a dig. In August, she found about 14 fossils per day.Is the number of fossils the paleontologist found in August proportional to the number of days she spent looking for fossils that month?No, July average 11.87 fossils per day

Cups of Sugar 11 2Envelopes of Mix1234

Solving Proportions

A proportion is two equivalent ratiosWhen solving proportions we must first ask ourselves What are we comparingA lemonade recipe calls for cup of mix for every quart of water. If Jeff wanted to make a gallon of lemonade, is 2 cups of mix proportional for this recipe?YES

Determine if the following ratios are proportional?No, 340 270 Yes, 72 = 72

Proportionality can also be determined between two ratios by simplifying or comparing their cross products

If they reduce to the same ratio, or their cross products are the same, then they are proportional

You can also solve proportions for a missing variable by cross multiplying.Example: Determine if the two ratios are proportional:

b. c. Yes 72 = 72No 32 = 30No 30 = 70

Example: Determine the value of x:

Example: A stack of 2,450 one-dollar bills weighs five pounds. How much do 1,470 one-dollar bills weigh?Set up a proportion ask ourselves what are we comparing?

Example: Whitney earns $206.25 for 25 hours of work. At this rate, how much will Whitney earn for 30 hours of work?

How much does Whitney earn per hour?

x = 60 $8.25 per hour3 pounds$247.50

Coordinate Plane Review

The Coordinate Planey-axisOrigin x-axisQuadrant IQuadrant IIQuadrant IIIQuadrant IV

Ordered Pair: is a pair of numbers that can be used to locate a point on a coordinate plane

Graph the

You can also solve proportions for a missing variable by cross multiplying.Example: Determine if the two ratios are proportional:

Yes 72 = 72No 32 = 30No 30 = 70

Ordered PairsOrdered Pair: is a pair of numbers that can be used to locate a point on a coordinate plane.Example: (3, 2)y - coordinatex - coordinateIIIIIIIV

Graph the following ordered pairs on the coordinate plan and state the quadrants the points are located in(3, 2)

(-5, 4)

(6, -4)

(-7, 7) IIIIVII

Steps for Graphing

Draw and label the x and y axis dont forget your arrows Make a table of values to represent the problem. Be sure to include the values: -1, 0, 1, and 2 Graph your order pairs- you need at least 3 points to make a line Draw a line through the points dont forget your arrows If the line is straight and goes through the origin, then the quantities are proportional

Example: The 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 your reasoning. yxYes it is a straight line through the origin

Number of Minutes123Number of Feet61218

Example: The table below shows the number of calories an athlete burned per minute of exercise. Determine whether the number of calories burned is proportional to the number of minutes by graphing. Explain your reasoning.yxNo it is not a straight line and it doesnt go through the origin

Number of Minutes123Number of Feet4813

.Slope is the rate of change between any two points on a lineThe sign of the slope tells you whether the line is positive or negative.You can find slope of a line by comparing any two points on that lineSlope is the or

Positive Slope

The line goes up 3 (rise) and over 1 (run).

Slope = 3

Negative Slope

The line goes down 2 (rise) and over 1 (run).

Slope = -2

Tell whether the slope is positive or negative . Then find the slope

NegativeSlope = -1

PositiveSlope = 4/3

Use the given slope and point to graph each line

Rate of Change (Slope)

.Rate of change (slope) describes how one quantity changes in relation to another.For graphs, the rate of change (slope) is constant ( a straight line)

Tell whether each graph shows a constant or variable rate of change

Proportional Relationships

A proportional relationship between two quantities is one in which the two quantities vary directly with one another (change the same way). This is called a direct variation.For graphs, the rate of change (slope) is constant ( a straight line)

ratio, rates, & proportions

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