ratio, rates, & proportions. ratios a ratio is a comparison of two numbers. o example: tamara...
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Ratio, Rates, & Proportions
Ratios• A ratio is a comparison of two
numbers.o 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 don’t
always have identical meanings to other fractions***
• There are two different types of ratioso .Part-to-Part Ratios
Example: Tamara’s dogs to cats is 2 to 8 or 1 to 4
o .Part –to-Whole Ratios Example: Tamara’s dogs to total
pets is 2 to 10 or 1 to 5
Try the following Apples to bananas
• There are 12 apples and 3 bananas
• So the ratio of apples to bananas is
4:1, 4 to 1, OR
Oranges to apples
• There are 15 oranges and 12 apples
• So the ratio of oranges to apples is
5:4, 5 to 4, OR
1
4
4
5
1. Roses to iris
3. Daisies to roses
2. Iris to daisies
4. 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 4
58 to 829 to 9
16 to 188 to 9
24 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
9
29Explain why the ratio
9
is not written as a mixed number:
Comparing Ratios• Compare ratios by writing in simplest
formoAre these ratios equivalent?250 Kit Kats to 4 M&M’s and 500 Kit Kats to 8 M&M’s
o 12 out of 20 doctors agree and 12 out of 30 doctors agree
o The ratio of students in Ms. B’s classes that had HW was 8 to 2 and 80% of Mrs. Long’s class had HW
=5.624
250 5.62
8
500
5
3
20
12 ≠
5
2
30
12
10
8=
100
80
Rates• A rate is a ratio that compares two
different quantities or measurements.
• Rates can be simplified
• Rates us the words per and foro Example: Driving 55 miles per hour
o Example: 3 tickets for $1
Unit Rates• A unit rate is a rate per one unit.
In unit rate the denominator is always one.o 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.o Example: Taylor can type 215 words in
5 min How many words can he type per minute?
Who is the faster typist? How much faster? Taylor is 2wpm faster than Miguel
wpm435
215
1. Film costs $7.50 for 3 rolls
3. 90 students and 5 teachers
5. **Snowfall of 12 ¾ inches in 4 ½ hours.
7. Ian drove 30 miles in 0.5 hours
2. Drive 288 miles on 16 gallons of gas.
4. Earn $49 for 40 hours of work
6. Use 5 ½ quarts of water for every 2 lbs of chicken
8. Sarah drove 5 miles in 20 minutes
Try the Following
$2.50 per roll
15 mph
2 ¾ quarts per lb
$1.23 per hour
18 mpg
18 students per teacher
2 5/6 per hour
60 mph
Complex Unit Rates• Suppose a boat travels 30 miles in 2
hourso How do you write this rate?
• Suppose a boat travels 12 miles in 2/3 hourso How do you write this as a rate?
o How do you write this as a division problem?
o Determine the unit rate: 18mph
mphhours
miles15
2
30
3
212 18
2
312
hours
miles
3/2
12
• Suppose a boat travels 8 ¾ miles in 5/8 hours.oHow do you write this as a rate?
oHow do you write this as a division problem?
oDetermine the unit rate: 14 mph
hours
miles
8/5
4/83
8
5
4
38 14
5
8
4
35
• 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)
1. 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?
3. Doug entered a canoe race. He rowed miles in hour. What is
his average speed?
3. 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?
4. 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 yard
3000 ft per hour
7mph
8 costumes
4
32
2
122
13
2
1
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 you
• Compare 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 box
Price per box
CM - $0.86 per box SF - $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 o Andrew earns $18 per hour for mowing lawns. Is the amount he
earns proportional to the number of hours he spends mowing?• Make a table to show these amounts
• For each number of hours worked, write the relationship of the amount he earned and hour as a ratio in simplest form.
• Are all the rates equivalent?
Earnings ($) 18 36 54 72Time (h) 1 2 3 4
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.
o Uptown Tickets charges $7 per baseball game ticket plus $2 processing fee per order. Is the cost of an order proportional to the number of tickets ordered?
• Make a table to show these amounts
• For each number of tickets, write the relationship of the cost of the and the number of tickets ordered.
• Are all the rates equivalent?
Cost ($) 10 17 24 31Tickets Ordered 1 2 3 4
The rates are not equivalent. This means the total cost of the tickets is not proportional to the number of tickets sold.
o 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 ½
o 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 ½ 1 1 ½ 2Envelopes of Mix 1 2 3 4
Solving Proportions
• A proportion is two equivalent ratios
• When solving proportions we must first ask ourselves – “What are we comparingA lemonade recipe calls for ½ 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
27
20
17
10and
24
9
8
3and
4
2
1
2/1
• 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.o Example: Determine if the two ratios are proportional:
12
8
9
6and
8
3
10
4and
15
10
7
2and
b.
c.
Yes 72 = 72 No 32 = 30 No 30 = 70
o Example: Determine the value of x:
o 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?”
o 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
36
10
6
x = 60
$8.25 per hour
3 poundsx
1470
5
2450
pounds
bills
hours
$
3025
25.206 x $247.50
Coordinate Plane Review
The Coordinate Plane
y-axis
Origin
x-axis
Quadrant IQuadrant II
Quadrant III Quadrant 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.o Example: Determine if the two ratios are proportional:
12
8
9
6and
8
3
10
4and
15
10
7
2and
Yes 72 = 72 No 32 = 30 No 30 = 70
Ordered Pairs• Ordered Pair: is a pair of numbers that
can be used to locate a point on a coordinate plane.– Example: (3, 2)
y - coordinate
x - coordinate
●
●
III
III IV
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)
●
●
●
●
II
I
IV
II
Steps for Graphing
1. Draw and label the x and y axis – don’t forget your arrows
2. Make a table of values to represent the problem. Be sure to include the values: -1, 0, 1, and 2
3. Graph your order pairs- you need at least 3 points to make a line
4. Draw a line through the points – don’t forget your arrows
5. 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.
y
x
Number of Minutes 1 2 3
Number of Feet 6 12 18
Yes – it is a straight line
through the origin
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.
y
x
Number of Minutes 1 2 3
Number of Feet 4 8 13
No – it is not a straight line and it
doesn’t go through the origin
Slope
• .Slope is the rate of change between any two points on a line
• The 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 line
• Slope is the or
Positive Slope
• The line goes up 3 (rise) and over 1 (run).
• Slope = 3
Negative Slope
1
3
run
rise
1
3
__
__
xinchange
yinchange
• The line goes down 2 (rise) and over 1 (run).
• Slope = -2
1
2
run
rise
1
2
__
__
xinchange
yinchange
Tell whether the slope is positive or negative . Then find the slope
• Negative• Slope = -1
1
1
run
rise
• Positive• Slope = 4/3
3
4
run
rise
Use the given slope and point to graph each line
Use the given slope and point to graph each line
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
Tell whether each graph shows a constant or variable rate of change
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)