Transcript Interactive Chalkboard - Tuslaw Local School District

Lesson 6-1 Ratios and Rates
Lesson 6-2 Using Proportions
Lesson 6-3 Scale Drawings and Models
Lesson 6-4 Fractions, Decimals, and Percents
Lesson 6-5 Using the Percent Proportion
Lesson 6-6 Finding Percents Mentally
Lesson 6-7 Using Percent Equations
Lesson 6-8 Percent of Change
Lesson 6-9 Probability and Predictions
Example 1 Write Ratios as Fractions
Example 2 Write Ratios as Fractions
Example 3 Find Unit Rate
Example 4 Convert Rates
Express the ratio 10 roses out of 12 flowers as
a fraction in simplest form.
Divide the numerator and denominator
by the GCF, 2.
The ratio of roses to flowers is 5 to 6. This means that for
every 6 flowers, 5 of them are roses.
Answer:
Express the ratio 8 golden retrievers out of 12 dogs
as a fraction in simplest form.
Answer:
Express the ratio 21 inches to 2 yards as a fraction
in simplest form.
Convert 2 yards to inches.
Divide the numerator and
denominator by the GCF, 3.
Written in simplest form, the ratio is 7 to 24.
Answer:
Express the ratio 4 feet to 18 inches as a fraction
in simplest form.
Answer:
Shopping A 12-oz bottle of cleaner costs $4.50.
A 16-oz bottle of cleaner costs $6.56. Which costs
less per ounce?
Find and compare the unit rates of the bottles.
Divide the numerator and
denominator by 12 to get a
denominator of 1.
For the 12-oz bottle, the unit rate is
$0.38 per ounce.
Divide the numerator and
denominator by 16 to get a
denominator of 1.
For the 16-oz bottle, the unit rate is $0.41 per ounce.
Answer: The 12-oz bottle has the lower cost per ounce.
Shopping A 6-pack of a soft drink costs $1.50. A
12-pack of a soft drink costs $2.76. Which pack costs
less per can?
Answer: The 12-pack costs less per can.
Animals A snail moved 30 feet in 2 hours. How many
inches per minute did the snail move?
You need to convert
.
to
.
There are 12 inches in 1 foot and 60 minutes in 1 hour.
Write 30 feet in 2 hours as
.
.
Convert feet to inches
and hours to minutes.
.
The reciprocal of
is
.
3
1
6
.
1
Divide the common
factors and units.
2
1
Simplify.
Answer: 30 feet in 2 hours is equivalent to 3 inches
per minute.
Jogging Dave jogs 2 miles in 22 minutes. How many
feet per second is this?
Answer: This is 8 feet per second.
Example 1 Identify Proportions
Example 2 Solve Proportions
Example 3 Use a Proportion to Solve a Problem
Example 4 Convert Measurements
Determine whether the pair of ratios forms
a proportion.
Write a proportion.
Cross products
Simplify.
Answer: So,
.
Determine whether the pair of ratios forms
a proportion.
Write a proportion.
Cross products
Simplify.
Answer: So,
.
Determine whether each pair of ratios forms
a proportion.
a.
Answer:
b.
Answer:
Solve the proportion
.
Cross products
Multiply.
Divide.
Answer: The solution is 21.6.
Solve the proportion
.
Cross products
Multiply.
Divide.
Answer: The solution is 5.
Solve each proportion.
a.
Answer: 4.5
b.
Answer: 7
Architecture An architect builds a model of a
building before the actual building is built. The model
is 8 inches tall and the actual building will be 22 feet
tall. The model is 20 inches wide. Find the width of
the actual building.
Explore
You know the actual height of the building and
the corresponding height of the model. You
need to find the actual width of the building that
corresponds with a model width of 20 inches.
Plan
Write and solve a proportion using ratios that
compare the actual building to the model. Let
w represent the actual width of the building.
Solve
Write a proportion.
Cross products
Multiply.
Divide.
Simplify.
Answer: The actual width of the building is 55 feet.
Examine Check the cross products. Since
and
, the answer is correct.
Planes A model of a jet airplane has a length of
9 inches and a wingspan of 6 inches. Find the
wingspan of the actual plane if the length is 120 feet.
Answer: The wingspan of the actual plane is 80 feet.
Attractions The Circleville Pumpkin Show in
Circleville, Ohio, boasts the world’s largest pumpkin
pie. The pie weighs 350 pounds and is 5 feet in
diameter. Find the diameter of the pie in centimeters
if 1 foot = 30.48 centimeters.
Let x represent the diameter in centimeters.
customary measurement
metric measurement
customary measurement
metric measurement
Cross products
Simplify.
Answer: The diameter of the pie is
152.4 centimeters.
School The gymnasium in a new school building
measures 55 feet in length. Find the length of the
gymnasium in centimeters if 1 ft = 30.48 centimeters.
Answer: The length of the gymnasium is
1676.4 centimeters.
Example 1 Find Actual Measurements
Example 2 Determine the Scale
Example 3 Construct a Scale Drawing
Map A map has a scale of 1 inch = 8 miles. Two
towns are 3.25 inches apart on the map. What is the
actual distance between the two towns?
Let x represent the actual distance between the two
towns. Write and solve a proportion.
map distance
actual distance
map distance
actual distance
Find the cross products.
Simplify.
Answer: The actual distance between the two
towns is 26 miles.
Map A map has a scale of 1 inch = 8 miles. Two
towns are 3.25 inches apart on the map. What is the
scale factor?
To find the scale factor, write the ratio of 1 inch to 8 miles
in simplest form.
Convert 8 miles to inches.
Answer: The scale factor is
measurement on the map is
the actual measurement.
. That is, each
Scale Drawing A scale drawing of a new house has a
scale of 1 inch = 4 feet. The height of the living room
ceiling is 2.75 inches on the scale drawing.
a. What is the actual height of the ceiling?
Answer: 11 feet
b. What is the scale factor?
Answer:
Model Car A model car is 4 inches long. The actual
car is 12 feet long. What is the scale of the model?
Write the ratio of the length of the model to the actual
length of the car. Then solve a proportion in which the
length of the model is 1 inch and the actual length is x feet.
model length
model length
actual length
actual length
Find the cross products.
Simplify.
Divide each side by 4.
Simplify.
Answer: The scale is 1 inch = 3 feet.
Log Cabin A model log cabin is 12 inches high. The
actual log cabin is 42 feet high. What is the scale of
the model?
Answer: The scale is 1 inch = 3.5 feet.
Patio Design Sheila is designing a patio that is
16 feet long and 14 feet wide. Make a scale drawing
of the patio. Use a scale of 0.5 inch = 4 feet.
Step 1
Find the measure of the patio’s length on the
drawing. Let x represent the length.
drawing length
actual length
drawing length
actual length
Find the cross products.
Simplify.
Divide each side by 4.
On the drawing, the length is 2 inches.
Step 2
Find the measure of the patio’s width on the
drawing. Let w represent the width.
drawing width
actual width
drawing width
actual width
Find the cross products.
Simplify.
Divide each side by 4.
Simplify.
On the drawing, the width is 1.75 or
inches.
Step 3
Make the scale drawing.
Use
-inch grid paper. Since 2 inches =
8 squares and
inches = 7 squares, draw
a rectangle that is 8 squares by 7 squares.
Answer:
Gardening Bob is designing a garden that is 18 feet
long and 14 feet wide. Make a scale drawing of the
garden. Use a scale of 0.5 inch = 4 feet.
Answer: The length is
width is
inches = 9 squares and the
inches = 7 squares.
Answer:
Example 1 Percents as Fractions
Example 2 Fractions as Percents
Example 3 Percents as Decimals
Example 4 Decimals as Percents
Example 5 Fractions as Percents
Example 6 Compare Numbers
Express 60% as a fraction in simplest form.
Answer:
Express 104% as a fraction in simplest form.
Answer:
Express 0.3% as a fraction in simplest form.
Multiply by
to eliminate
the decimal in the numerator.
Answer:
Express
as a fraction in simplest form.
The fraction bar
indicates division.
9
Answer:
4
Express each percent as a fraction in simplest form.
a.
Answer:
b.
Answer:
c.
Answer:
d.
Answer:
Express
Answer:
as a percent.
Express
Answer:
as a percent.
Express each fraction as a percent.
a.
Answer: 68%
b.
Answer: 140%
Express 60% as a decimal.
Divide by 100 and remove the %.
Answer:
Express 2% as a decimal.
Divide by 100 and remove the %.
Answer:
Express 658% as a decimal.
Divide by 100 and remove the %.
Answer:
Express 0.4% as a decimal.
Divide by 100 and remove the %.
Answer:
Express each percent as a decimal.
a. 84%
Answer: 0.84
b. 7%
Answer: 0.07
c. 302%
Answer: 3.02
d. 0.9%
Answer: 0.009
Express 0.4 as a percent.
Multiply by 100 and add the %.
Answer:
Express 0.05 as a percent.
Multiply by 100 and add the %.
Answer:
Express 0.0008 as a percent.
Multiply by 100 and add the %.
Answer:
Express 7.3 as a percent.
Multiply by 100 and add the %.
Answer:
Express each decimal as a percent.
a. 0.84
Answer: 84%
b. 0.01
Answer: 1%
c. 0.004
Answer: 0.4%
d. 2.39
Answer: 239%
Express
as a percent. Round to the nearest
tenth percent, if necessary.
Answer:
Express
as a percent. Round to the nearest
tenth percent, if necessary.
Answer:
Express
as a percent. Round to the nearest
tenth percent, if necessary.
Answer:
Express
as a percent. Round to the nearest
tenth percent, if necessary.
Answer:
Express each fraction as a percent. Round to the
nearest tenth percent, if necessary.
a.
Answer: 37.5%
b.
Answer: 41.7%
c.
Answer: 1.3%
d.
Answer: 123.5%
Bakery A baker says that 25% of his customers buy
only bread and
of his customers buy only
cookies. Which group is larger?
Write
as a percent. Then compare.
Answer: Since 40% is greater than 25%, the group that
buys only cookies is larger.
School The school principal states that
of the
students are involved in instrumental music while
42% are involved in vocal music. Which group is
larger?
Answer: The group involved in vocal music is larger.
Example 1 Find the Percent
Example 2 Find the Percent
Example 3 Apply the Percent Proportion
Example 4 Find the Part
Example 5 Apply the Percent Proportion
Example 6 Find the Base
Twenty is what percent of 25?
Twenty is being compared to 25. So, 20 is the part and
25 is the base. Let p represent the percent.
Replace a with 20 and b with 25.
Find the cross products.
Simplify.
Divide each side by 25.
Simplify.
Answer: 20 is 80% of 25.
Twelve is what percent of 40?
Answer: 12 is 30% of 40.
What percent of 8 is 12?
Twelve is being compared to 8. So, 12 is the part and 8
is the base. Let p represent the percent.
Replace a with 12 and b with 8.
Find the cross products.
Simplify.
Divide each side by 8.
Simplify.
Answer: 150% of 8 is 12.
What percent of 20 is 35?
Answer: 175% of 20 is 35.
Tennis From the years 1988 through 1996 Steffi Graf
won the U.S. Open tennis championships five times
and Wimbledon seven times. What percent of both
tournaments combined during those years was Steffi
Graf the women’s champion?
Compare the number of Steffi Graf’s wins, 12, to the total
number of tournaments played, 18. Let a represent the
part, 12, and let b represent the base, 18, in the percent
proportion. Let p represent the percent.
Replace a with 12 and b with 18.
Find the cross products.
Simplify.
Divide each side by 18.
Simplify.
Answer: Steffi Graf won about 66.7% of the tournaments.
Bake Sale At the school bake sale, 23 chocolate
chip cookies, 18 oatmeal raisin cookies, and 7
peanut butter cookies were sold. If the sale started
with a total of 90 cookies, what percent of the
cookies were sold?
Answer: About 53.3% of the cookies were sold.
What number is 8.8% of 20?
The percent is 8.8, and the base is 20. Let a represent
the part.
Replace b with 20
and p with 8.8.
Find the cross products.
Simplify.
Mentally divide each
side by 100.
Answer: 8.8% of 20 is 1.76.
What number is 42.5% of 90?
Answer: 42.5% of 90 is 38.25.
Recycling At Kington Middle School, 600 students
were asked if their families recycle cans and/or
newspapers. 32% said they recycle both; 25% said
they recycle cans; 19% said they recycle
newspapers; 24% said they do not recycle. How
many families recycle both cans and newspapers?
The total number of families is 600. So, 600 is the base.
The percent is 32%.
To find 32% of 600, let b represent the base, 600, and let
p represent the percent, 32%, in the percent proportion.
Let a represent the part.
Replace b with 600
and p with 32.
Find the cross products.
Simplify.
Mentally divide each
side by 100.
Answer: 192 families recycle both cans and newspapers.
Market Research A survey of 500 teenagers found
that 37% prefer a soft drink with dinner; 48% prefer
water; and 15% prefer milk. How many of the
teenagers prefer milk with dinner?
Answer: 75 teenagers prefer milk with dinner.
Seventy is 28% of what number?
The percent is 28% and the part is 70. Let b represent
the base.
Replace a with 70
and p with 28.
Find the cross products.
Simplify.
Divide each side by 28.
Simplify.
Answer: 70 is 28% of 250.
Ninety is 24% of what number?
Answer: 90 is 24% of 375.
Example 1 Find Percent of a Number Mentally
Example 2 Estimate Percents
Example 3 Use Estimation to Solve a Problem
Find 50% of 46 mentally.
Think:
Think:
Answer: 50% of 46 is 23.
.
of 46 is 23.
Find 25% of 88 mentally.
Think:
Think:
Answer: 25% of 88 is 22.
.
of 88 is 22.
Find 70% of 110 mentally.
Think:
Think:
So,
Answer: 70% of 110 is 77.
.
of 110 is 11.
of 110 is 77.
Find the percent of each number mentally.
a. 50% of 82
Answer: 41
b. 25% of 36
Answer: 9
c. 80% of 60
Answer: 48
Estimate 22% of 494.
22% is about 20% or
.
494 is about 500.
of 500 is 100.
Answer: 22% of 494 is about 100.
Estimate 63% of 788.
63% is about 60% or
.
788 is about 800.
of 800 is 480.
Answer: 63% of 788 is about 480.
Estimate
of 1219.
1219 is almost 1200.
1% of 1200 is 12.
Answer:
of 1219 is about
or 3.
Estimate 155% of 38.
155% means about 150 for every 100 or about 15
for every 10.
38 has about 4 tens.
Answer: 155% of 38 is about 60.
a. Estimate 38% of 400.
Answer: 160
b. Estimate 72% of 318.
Answer: 210
c. Estimate
of 2482.
d. Estimate 183% of 93.
Answer: 5
Answer: 162
Money A restaurant bill totals $21.35. You want to
leave a 15% tip. What is a reasonable amount for
the tip?
$21.35 is about $21.
10% of $21 is $2.10.
Move the decimal point
1 place to the left.
5% of $21 is $1.05.
5% is one half of 10%.
So, 15% is about
or $3.15.
Answer: A reasonable amount for the tip
would be $3.
Money A restaurant bill totals $59.05. You want to
leave a 15% tip. What is a reasonable amount for
the tip?
Answer: A reasonable amount for the tip
would be $9.
Example 1 Find the Part
Example 2 Find the Percent
Example 3 Find the Base
Example 4 Find Discount
Example 5 Apply Simple Interest Formula
Find 38% of 22.
Estimate: 40% of 20 is 8.
You know that that the base is 22 and the percent is
38%. Let n represent the part.
Write 38% as the decimal 0.38.
Simplify.
Answer: 38% of 22 is 8.36.
Find 64% of 48.
Answer: 64% of 48 is 30.72.
19 is what percent of 25?
Estimate:
, which is 80%.
You know that the base is 25 and the part is 19.
Let n represent the percent.
Divide each side by 25.
Simplify.
Answer: 19 is 76% of 25.
The answer makes sense
compared to the estimate.
8 is what percent of 25?
Answer: 8 is 32% of 25.
84 is 16% of what number?
Estimate:
80 is 16% of 500.
You know that the part is 84 and the percent is 16%. Let
n represent the base.
Write 16% as the decimal 0.16.
Divide each side by 0.16.
Simplify.
Answer: 84 is 16% of 525.
The answer is
reasonable since it is
close to the estimate.
315 is 42% of what number?
Answer: 315 is 42% of 750.
Jewelry The regular price of a ring is $495. It is
on sale at a 20% discount. What is the sale price
of the ring?
Method 1
First, use the percent
equation to find 20% of 495.
Estimate:
Let d represent the discount.
The base is 495 and the
percent is 20%.
Simplify.
Then, find the sale price.
Subtract the discount from the
original price.
Method 2
A discount of 20% means the ring will cost
or 80% of the original price. Use the percent equation to
find 80% of 495.
Let s represent the sale price.
The base is 495 and
the percent is 80%.
Answer: The sale price of the ring will be $396.
Retail The regular price of a stereo system is $1295.
The system is on sale at a 15% discount. Find the sale
price of the stereo system.
Answer: The sale price of the stereo system is $1100.75.
Banking Suppose you invest $2000 at an annual
interest rate of 4.5%. How long will it take for it to
earn $495?
Write the simple interest formula.
Replace I with 495, p with 2000,
and r with 0.045.
Simplify.
Divide each side by 90.
Simplify.
Answer: It will take 5.5 years to earn $495.
Banking Suppose you invest $3500 at an annual
interest rate of 6.25%. How long will it take for it to
earn $875?
Answer: It will take 4 years to earn $875.
Example 1 Find Percent of Change
Example 2 Find Percent of Increase
Example 3 Find Percent of Increase
Example 4 Find Percent of Decrease
Find the percent of change from 325 to 390.
Step 1
Subtract to find the amount of change.
Step 2
Write a ratio that compares the amount of
change to the original amount. Express the ratio
as a percent.
Substitution.
Write the
decimal as
a percent.
Answer: The percent of change from 325 to 390 is 20%.
Find the percent of change from 84 to 105.
Answer: The percent of change from 84 to 105 is 25%.
Tuition In 1965, when John entered college, the
tuition per year was $7500. In 2000, when his daughter
went to the same school, the tuition was $25,500. Find
the percent of change.
Step 1
Subtract to find the amount of change.
Step 2
Write a ratio that compares the amount of
change to the original tuition. Express the ratio
as a percent.
Substitution.
Write the
decimal as
a percent.
Answer: The percent of change is 240%. In this case,
the percent of change is a percent of increase.
Textbooks In 1990, the price of a textbook was $38. In
2000, the price of the same textbook was $81. Find the
percent of change.
Answer: The percent of change was about 113%. In
this case, the percent of change is a percent
of increase.
Multiple-Choice Test Item
Refer to the table shown. Which county had
the least percent of increase in population from
1990 to 2000?
County
Breckenridge
Bracken
Calloway
Fulton
A Breckinridge
C Calloway
1990
2000
16,312
18,648
7766
8279
30,735
34,177
8271
7752
B Bracken
D Fulton
Read the Test Item
Percent of increase tells how much the population has
increased in relation to 1990.
Solve the Test Item
Use a ratio to find each percent of increase. Then
compare the percents.
• Breckinridge
• Bracken
• Calloway
• Fulton
Eliminate this choice because the population decreased.
Answer: Bracken County had the least percent of
increase in population from 1990 to 2000.
The answer is B.
Multiple-Choice Test Item
The table shows test scores on the first two math
tests of the semester for four students. Which
student had the greatest percent of increase from
test 1 to test 2?
A Holly
B Ben
Student Name
C Sally
D Max
Test 1 Score
Test 2 Score
Holly
Ben
84
93
91
89
Sally
65
79
Max
73
98
Answer: Max had the greatest percent of increase
from test 1 to test 2. The answer is D.
Clothing A $110 sweater is on sale for $88. What is
the percent of change?
Step 1
Subtract to find the amount of change.
Step 2
Compare the amount of change to the
original price.
Substitution.
Write the
decimal as
a percent.
Answer: The percent of change is –20%. In this
case, the percent of change is a percent
of decrease.
Shoes A $145 pair of tennis shoes is on sale for $105.
What is the percent of change?
Answer: The percent of change is about –27.6%.
Example 1 Find Probability
Example 2 Find Probability
Example 3 Find Experimental Probability
Example 4 Make a Prediction
Suppose a number cube is rolled. What is the
probability of rolling a 4 or a 5?
There are 2 numbers on the number cube that are
a 4 or a 5: 4 and 5.
There are 6 possible outcomes: 1, 2, 3, 4, 5, and 6.
Answer: The probability of rolling a 4 or a 5 is
Suppose a number cube is rolled. What is the
probability of rolling a number that is divisible by 3?
Answer: The probability of rolling a number that is
divisible by 3 is
Suppose that two number cubes are rolled. Find the
probability of rolling two identical numbers.
Make a table showing the sample space when rolling two
number cubes.
1
2
3
4
5
6
1
(1, 1)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
2
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
3
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
4
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
5
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
6
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
There are 6 outcomes in which the two numbers
are identical.
Answer:
This means there is a
chance of rolling two identical numbers.
Suppose that two number cubes are rolled. Find the
probability of rolling two numbers whose sum is 8.
Answer: There is a
chance of
rolling two numbers whose sum is 8.
Use the table below to determine the experimental
probability of landing on heads for this experiment.
Answer: The experimental probability of landing on
heads in this case is
Basketball Brian is shooting baskets with a
basketball. He makes 13 shots and misses 9 shots.
Determine the experimental probability of Brian
making a shot.
Answer: The experimental probability of making
a shot in this case is
Sports Miss Newman surveyed her class to see
which sport they preferred watching. 44% preferred
football, 28% basketball, 20% soccer, and 8% tennis.
Out of 560 students in the entire school, how many
would you expect to say they prefer watching
basketball?
The total number of students is 560. So, 560 is the base.
The percent is 28%.
To find 28% of 560, let b represent the base, 560, and let
p represent the percent, 28%, in the percent proportion.
Let a represent the part.
part
percent
base
Simplify.
Mentally divide
each side by 100.
Answer: You can expect about 157 students to say
they prefer watching basketball.
Colors The students in an art class were surveyed
about their favorite color. 32% preferred blue, 29%
preferred red, 23% preferred yellow, and 16%
preferred green. Out of 450 students in the entire
school, how many would you expect to say they
prefer red?
Answer: You can expect about 131 students to say
they prefer red.