Transcript Chapter 1 Ratios and Proportional Reasoning Notes

Lesson 1
Take your pulse for 2 minutes and record
your results.
𝑏𝑒𝑎𝑡𝑠
𝑚𝑖𝑛𝑢𝑡𝑒𝑠
----Beats in 2 minutes = ? ---Number of beats Number of beats
in 1 minute
in 1 minute
-- minutes--
-- minutes--
Use your results to determine the number
1
of beats for minute.
2
Rate:
Definition: a ratio that compares two
different quantities
Example:
160 𝑏𝑒𝑎𝑡𝑠
2 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
Unit Rate:
Definition: when a rate has a denominator
of 1
Example:
80 𝑏𝑒𝑎𝑡𝑠
1 𝑚𝑖𝑛𝑢𝑡𝑒
Rate
Unit Rate
Abbreviation
Name
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑖𝑙𝑒𝑠
1 ℎ𝑜𝑢𝑟
mi/h or mph
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑖𝑙𝑒𝑠
1 𝑔𝑎𝑙𝑙𝑜𝑛
miles per
hour
miles per
gallon
Average
speed
gas mileage
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑜𝑙𝑙𝑎𝑟𝑠
1 𝑝𝑜𝑢𝑛𝑑
price per
pound
dollars/lb
mi/gal or
mpg
unit price
Adrienne biked 24 miles in 4 hours. If she
biked at a constant speed, how many miles
did she ride in one hour?
24 miles in 4 hours =
24 mi  4
4 hr  4
24 miles
4 hour
=
6 miles
1 hour
Adrienne biked 6 miles in 1 hour.
Find 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 hour
Find the unit rate if it costs $2 for eight
juice boxes.
$2 for eight boxes =
$2  8
8 boxes  8
$2
8 boxes
=
$0.25
1 box
The unit price is $0.25 per
juice box.
The prices of 3 different
bags of dog food are
given in the table. Which 40-pound bag:
$49.00  40 ≈ $1.23
size bag has the lowest
price per pound rounded per pound
to the nearest cent?
20-pound bag:
Dog Food Prices
$23.44  20 ≈ $1.17 per
Bag Size
pound
Price ($)
(lb)
40
20
8
49.00
23.44
9.88
8-pound bag:
$9.88  8 ≈ $1.24 per
pound
Tito 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 Sales
Brand
Sales Price
Nutty
12 oz for $2.19
Grandma’s
18 oz for $2.79
Bee’s
28 oz for $4.69
Sav-A-Lot
40 oz for $6.60
Nutty:
$0.18 per oz
Grandma’s:
$0.155
Bee’s:
$0.1675
Sav-A-Lot:
$0.165
Lexi painted 2 faces in 8 minutes at the
Crafts Fair. At this rate, how many faces
can she paint in 40 minutes?
2 faces in 8 minutes =
0.25
1 min
2
8 min
=
28
8 min  8
=
0.25
1 min
x 40 min = 10 faces
Lexi can paint 10 faces in 40 minutes.
Lesson 2
Solve. Write in simplest form.
2
3
x
1
6
1
4

3
8
Definition:
Fractions when a numerator and/or
denominator is also a fraction.
Example:
4
7
12
6
1
5
5
8
2
3
Simplify
1
4
2
.
1
4
2
=
1
4
1
4
 2
x
1
2
=
1
8
Simplify
1
1
2
.
1
1
2
=1
1 x 2
=2
1
2
1
1
3
1
4
Josh can jog
miles in hour. Find his
average speed in miles per hour.
1
1
3 = 11  1
1
3
4
4
4
1
= 
3
4
4
4
= x
3
1
=
16
3
1
=5
3
Josh jogs at an average
1
speed of 5 miles per
3
hour.
Tia is painting her house. She paints
3
4
1
34
2
square feet in hour. At this rate, how
many square feet can she paint each hour?
276
1
=
34 ft
1
3
6
2 = 34 
= 46
3
2
4
ft
4 69
3
=

2
4
Tia can paint 46 square
feet per hour.
69
4
=
x
2
3
a.
Mr. Ito is spreading mulch in his yard.
2
He spreads 4 square yards in 2 hours.
3
How many square yards can he mulch
per hour?
1
2
3
b.
square yards per hour
1
4
2
1
1
2
Aubrey can walk
miles in
hours.
Find her average speed in miles per
hour.
3 miles per hour
On Javier’s soccer team, about
1
33 %
3
players have scored a goal. Write
a fraction in simplest form.
1
33 %
3
=
=
100
3
100
3
x
=
1
3
33
100
 100
1
100
=
1
3
of the
1
33 %
3
as
Lesson 3
Write 32
1
%
8
as a fraction in simplest form.
Customary Units of Measure
Smaller
Larger
12 inches
1 foot
16 ounces
1 pound
8 pints
1 gallon
3 feet
1 yard
5,280 feet
1 mile
Metric Units of Measure
Smaller
Larger
100 cm
1 meter
1,000 grams
1 kilogram
1,000 ml
1 liter
10 mm
1 centimeter
1,000 mg
1 gram
Like a unit rate, a unit ratio has a
denominator of 1.
Example:
12 𝑖𝑛𝑐ℎ𝑒𝑠
1 𝑓𝑜𝑜𝑡
16 𝑜𝑢𝑛𝑐𝑒𝑠
1 𝑝𝑜𝑢𝑛𝑑
100 𝑐𝑚
1 𝑚𝑒𝑡𝑒𝑟
A remote control car travels at a rate of 10
feet per second. How many inches per
second is this?
10 𝑓𝑒𝑒𝑡
10 𝑓𝑒𝑒𝑡
12 𝑖𝑛𝑐ℎ𝑒𝑠
=
𝑥
1 𝑠𝑒𝑐𝑜𝑛𝑑
1 𝑠𝑒𝑐𝑜𝑛𝑑
1 𝑓𝑜𝑜𝑡
Divide out the common units
10 𝑓𝑒𝑒𝑡
12 𝑖𝑛𝑐ℎ𝑒𝑠
=
𝑥
1 𝑠𝑒𝑐𝑜𝑛𝑑
1 𝑓𝑜𝑜𝑡
A remote control car travels at a rate of 10
feet per second. How many inches per
second is this?
Simplify:
10 𝑥 12 𝑖𝑛𝑐ℎ𝑒𝑠
=
1 𝑠𝑒𝑐𝑜𝑛𝑑 𝑥 1
120 𝑖𝑛𝑐ℎ𝑒𝑠
=
1 𝑠𝑒𝑐𝑜𝑛𝑑
So, 10 feet per second equals 120 inches per second.
A swordfish can swim at a rate of 60 miles
per hour. How many feet per hour in this?
60 𝑚𝑖𝑙𝑒𝑠
60 𝑚𝑖𝑙𝑒𝑠 5,280 𝑓𝑒𝑒𝑡
=
𝑥
1 ℎ𝑜𝑢𝑟
1 ℎ𝑜𝑢𝑟
1 𝑚𝑖𝑙𝑒
Divide out the common units
60 𝑚𝑖𝑙𝑒𝑠 5,280 𝑓𝑒𝑒𝑡
=
𝑥
1 ℎ𝑜𝑢𝑟
1 𝑚𝑖𝑙𝑒
A swordfish can swim at a rate of 60 miles
per hour. How many feet per hour in this?
Simplify:
60 𝑥 5,280 𝑓𝑒𝑒𝑡
=
1𝑥1ℎ
316,800 𝑓𝑒𝑒𝑡
=
1 ℎ𝑜𝑢𝑟
Swordfish can swim at a rate of 316,800 feet per
hour.
Marvin walks at a speed of 7 feet per
second. How many feet per hour is this?
7 𝑓𝑡
7 𝑓𝑡
60 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
=
𝑥
𝑥
1𝑠
1𝑠
1 𝑚𝑖𝑛
1 ℎ𝑜𝑢𝑟
Divide out the common units
7 𝑓𝑡
60 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
=
𝑥
𝑥
1𝑠
1 𝑚𝑖𝑛
1 ℎ𝑜𝑢𝑟
Marvin walks at a speed of 7 feet per
second. How many feet per hour is this?
Simplify:
7 𝑥 60 𝑥 60 𝑓𝑒𝑒𝑡
=
1 𝑥 1 𝑥 1 ℎ𝑟
25,200 𝑓𝑒𝑒𝑡
=
1 ℎ𝑜𝑢𝑟
Marvin walks 25,200 feet in 1 hour.
The average speed of one team in a relay
race is about 10 miles per hour. What is
the speed in feet per second?
10 𝑚𝑖 10 𝑚𝑖 5,280 𝑓𝑡
1 ℎ𝑟
1 𝑚𝑖𝑛
=
𝑥
𝑥
𝑥
1 ℎ𝑟
1 ℎ𝑟
1 𝑚𝑖
60 𝑚𝑖𝑛 60 𝑠𝑒𝑐
Divide out the common units
10 𝑚𝑖 5,280 𝑓𝑡
1 ℎ𝑟
1 𝑚𝑖𝑛
=
𝑥
𝑥
𝑥
1 ℎ𝑟
1 𝑚𝑖
60 𝑚𝑖𝑛 60 𝑠𝑒𝑐
The average speed of one team in a relay
race is about 10 miles per hour. What is
the speed in feet per second?
Simplify:
10 𝑥 5,280 𝑥 1 𝑥 1 𝑓𝑡
=
1 𝑥 1 𝑥 60 𝑥 60 𝑠𝑒𝑐
52,800 𝑓𝑒𝑒𝑡
=
3,600 ℎ𝑜𝑢𝑟
The relay teams runs at an average speed of 14.7
feet per seconds
Lesson 4
Fill in the blank.
20 miles/hour = __________feet/min
Proportional – has a constant rate or a unit
rate
Nonproportional – does NOT have a
constant rate or a unit rate
𝑐𝑜𝑠𝑡 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟
16
24
32
=
=
=
𝑜𝑟 $8 𝑝𝑒𝑟 𝑝𝑖𝑧𝑧𝑎
𝑝𝑖𝑧𝑧𝑎𝑠 𝑜𝑟𝑑𝑒𝑟𝑒𝑑
2
3
4
These fractions are equivalent fractions
because they all equal the same value.
Andrew 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 $
TIME (HR)
18
1
36
2
54
3
72
4
Andrew 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
𝐸𝐴𝑅𝑁𝐼𝑁𝐺𝑆 $
18
36
54
72
=
=
=
=
𝑇𝐼𝑀𝐸 (𝐻𝑅)
1
2
3
4
Do they all equal each other?
Yes, the amount Andrew earns is proportional
to the number of hours he works.
At 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.
Uptown 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 $
TICKETS
ORDERED
7+3 = 10
1
2(7) + 3 = 17
2
3(7) + 3 = 24
3
COST $
TICKETS
ORDERED
7+3 = 10
1
2(7) + 3 = 17
2
3(7) + 3 = 24
3
STEP 2: Make equivalent fractions
𝐶𝑂𝑆𝑇 $
10
17
24
=
=
=
𝑇𝐼𝐶𝐾𝐸𝑇𝑆 𝑂𝑅𝐷𝐸𝑅𝐸𝐷
1
2
3
Are these fractions true?
No, these are not equal so the cost and
tickets ordered are not proportional.
You can use the recipe shown to make a fruit
punch. Is the amount of sugar used proportional
to the amount of mix used?
CUPS OF
SUGAR
ENVELOPES OF
MIX
½
1
1
2
1½
3
2
4
𝐶𝑈𝑃𝑆 𝑂𝐹 𝑆𝑈𝐺𝐴𝑅
0.5
1
1.5
2
=
= =
=
𝐸𝑁𝑉𝐸𝐿𝑂𝑃𝐸𝑆 𝑂𝐹 𝑀𝐼𝑋
1
2
3
4
Are the ratios equivalent? Yes, so the sugar and
mix are proportional.
At the beginning of the year, Isabel had $120 in
the bank. Each week, she deposits another $20.
Is her account balance proportional to the
number of weeks of deposits? Use the table
below and explain your reasoning.
TIME (WK)
BALANCE ($)
1
2
3
4
140
160
180
200
No, the balance and the number of weeks are
not proportion because the ratios are not
equal.
The tables shown represent the number of
pages Martin and Gabriel read over time.
Which situation represents a proportional
relationship?
PAGES
MARTIN
READ
TIME
(MIN)
PAGES
GABRIEL
READ
TIME
(MIN)
2
5
3
5
4
10
4
10
6
15
7
15
All of Martin’s ratios equal each other, so
Martin’s table is proportional.
Lesson 5
From a graph:
A proportional relationship is…
1. a straight line
2. a line goes through the origin (0,0)
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.
Step 1: Make a table
The 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.
James 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 money
earned is proportional
to the number of hours
because the line is straight
and goes through the origin.
The 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.
Determine 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.
Which 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.
Lesson 6
Definition: a proportion is an equation
stating that two ratios or rates are
equivalent.
Numbers:
Algebra:
6
3
=
8
4
𝑎
𝑐
= , 𝑤ℎ𝑒𝑟𝑒 𝑏 ≠ 0, 𝑑  0
𝑏
𝑑
After 2 hours, the air temperature had
risen 7F. Write and solve a proportion to
find the amount of time it will take at this
rate for the temperature to rise an
additional 13F.
7
13
=
2
𝑡
7𝑡
26
=
7
7
7t = 2(13)
1t ≈ 3.7
7t = 26
It will take about 3.7 hours
to rise additional 13F.
Solve each proportion.
a.
𝑥
4
=
9
10
x = 3.6
c.
7
3
=
𝑛
21
n = 49
b.
2
34
=
5
𝑦
y = 85
If the ratio of Type O to non-Type O donors at a
blood drive was 37:43, how many donors would be
Type O, out of 300 donors?
Type O
Total Donors
37
37
=
37 + 43
80
37
𝑡
=
80
300
If the ratio of Type O to non-Type O donors at a
blood drive was 37:43, how many donors would be
Type O, out of 300 donors?
37(300) = 80t
11,100 = 80t
11,100
80𝑡
=
80
80
138.75 = t
About 139
donors would
have a blood
Type of 0
The 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 students
Olivia bought 6 containers of yogurt for $7.68.
Write an equation relating the cost c to the
number of yogurts y.
𝑐𝑜𝑠𝑡 $
7.68
=
= $1.28 𝑝𝑒𝑟 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟
𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟𝑠
6
Cost = 1.28y
How much would Olivia pay for 10 yogurts at this
same rate?
1.28(10) = $12.80
Jaycee bought 8 gallons of gas for $31.12.
Write an equation for the cost c to the
number of gallons g.
𝑐𝑜𝑠𝑡 $
31.12
=
= $3.89 𝑝𝑒𝑟 𝑔𝑎𝑙𝑙𝑜𝑛
𝑔𝑎𝑙𝑙𝑜𝑛𝑠
8
Cost = 3.89g
How much would Jaycee pay for 11 gallons
of gas at this rate?
3.89(11) = $42.79
Olivia 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 minutes
Lesson 8
Reggie started a running program to
prepare for track season. Every day for
60 days, he ran a half hour in the morning
and a half hour in the evening. He
averaged 6.5 miles per hour. At this rate,
what is the total number of miles Reggie
ran over the 60 –day period?
The 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.
The 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.
Find the slope.
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
25 − 20
5
=
= =5
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
5−4
1
The slope is 5 seconds for 1 mile.
Graph 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/week
Ronald 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.
The slope =
=
1200 −600
4 −2
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
600
2
= 300
The slope is $300 per week.
Jessica 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
How is rate of change related to slope?
Lesson 9
Determine if this situation is proportional:
Taxi cab passengers are charged $2.50
upon entering the cab. They are then
charged $1.00 for every mile traveled.
Words: 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 proportionality”
The height of the water as a pool is being
filled is shown in the graph. Determine the
rate in inches per minute.
ℎ𝑒𝑖𝑔ℎ𝑡 (𝑦)
𝑡𝑖𝑚𝑒 (𝑥)
𝟐
𝟎.𝟒
𝒐𝒓
𝟓
𝟏
𝟒
𝟎.𝟒
𝒐𝒓
𝟏𝟎
𝟏
𝟔
𝟎.𝟒
𝒐𝒓
𝟏𝟓
𝟏
𝟖
𝟎.𝟒
𝒐𝒓
𝟐𝟎
𝟏
Two minutes after a diver enters the water,
he has descended 52 feet. After 5 minutes,
he has descended 130 feet. At what rate is
the scuba diver descending?
HINT: Time is usually the x
𝑦
52
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = =
𝑜𝑟 26 𝑓𝑡/𝑚𝑖𝑛
𝑥
2
k = 26 feet per minute
The 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.
The 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 = kx
y = 55x
constant = 55
The family traveled 55 miles per hour
Pizzas 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.
Step 2: Is there a direct variation?
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑖𝑧𝑧𝑎𝑠
𝑐𝑜𝑠𝑡
𝟏𝟏
𝒐𝒓 $11
𝟏
𝟐𝟕
𝒐𝒓 $𝟗
𝟑
𝟏𝟗
𝒐𝒓 $𝟗. 𝟓
𝟐
𝟑𝟓
𝒐𝒓 $𝟖. 𝟕𝟓
𝟒
No, the ratios are not constant so there is n
direct variation.
Two 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.
Determine if this linear relationship shows
a direct variation.
Yes, the ratios are the same so this table
shows a direct variation.