%
½
⅘
Integrated Algebra
Ratios, Proportions and Rates
$
=
⅓
Ratio
A ratio is a comparison of two quantities.
Ex: A teacher graded 180 bonus quizzes during the school year. The number of quizzes
receiving A's, B's, and C's were in the ratio of 5 : 3 : 1, respectively. Find the number of
bonus quizzes that received a grade of A for the school year.
Ratio Practice
1. In a senior class, there are b boys and g
girls. Express the ratio of the number of boys
to the total number in the class.
6. The measures of the angles of a triangle
are in the ratio of 2 : 3 : 4. Find the number of
degrees in the smallest angle of the triangle.
2. A soccer team played 25 games and won
17.
7.
Hannah took a trip to visit her cousin. She
drove 120 miles to reach her cousin's house
and the same distance back home.
a. What is the ratio of the number of wins to
the number of loses?
b. What is the ratio of the number of games
played to the number of games won?
3. The Yankees won 125 games, the Red
Sox won 97 games, and the Mets won 86
games. What is the ratio of wins of the
Yankees to the Red Sox to the Mets?
4. Two numbers are in a ratio of 5 : 3. Their
sum is 80. Find the largest number.
5. Mr. Smith and Mr. Kelly are business
partners. They agreed to divide the profits in
the ratio of 3 : 2. The profit amounted to
$24,000. How much did each person
receive?
a. It took her 1.2 hours to get halfway to her
cousin's house. What was her average
speed, in miles per hour, for the first 1.2
hours of her trip?
b. Hannah's average speed for the
remainder of the trip to her cousin's house
was 40 miles per hour. How long, in hours,
did it take her to drive the remaining
distance?
c. Traveling home along the same route,
Hannah drove at an average rate of 55
miles per hour. After 2 hours her car broke
down. How many miles was she from
home?
Proportion
A proportion is a comparison of ratios.
Ex: The length of a stadium is 100 yards and its width is 75 yards. If 1 inch represents 25
yards, what would be the dimensions of the stadium drawn on a sheet of paper?
Proportion Practice
1. Are the followingtrue proportions?
5. If 4 tickets to a show cost $9.00, find the
cost of 14 tickets.
a.
b.
2. Solve for x:
6. A house which is assessed for $10,000
pays $300 in taxes. What should the tax be
on a house assessed at $15,500?
3. Solve for x:
7. The length of the thruway is 600 miles. If
0.5 inch represents 50 miles, what is the
length of the thruway on the map?
4. Solve for x:
8. If on a scale drawing 48 feet are
represented by 12 inches, then a scale of 1/4
inch represents how many feet?
Direct Variation
When two variable quantities have a constant (unchanged) ratio, their relationship is called a
direct variation. It is said that one variable "varies directly" as the other.
The constant ratio is called the constant of variation.
The formula for direct variation is y = kx, where k is the constant of variation."y varies directly
as x". Solving for k:
Ex: If mvaries directly as yand m is 6 when y is 36, find the constant of variation.
Direct Variation Practice
1. "avaries directly as b". If a = 3 whenb = 24,
find b when a = 10.
3. There are about 200 calories in 50 grams of
Swiss cheese. Willie ate 70 grams of this
cheese. About how many calories were in the
cheese that he ate if the number of calories
varies directly as the weight of the cheese.
2. In the following chart, does one variable
vary directly with the other?
4. One variable (A) varies directly as the
other (C). Find the missing numbers x and
y. Write the formula which relates the
variables.
M
N
3
6
4
8
5
10
6
12
7
14
A
C
1
3
2
y
x
15
Percents
Percents are used to describe parts of a whole base amount. When one of the parts of
the relationship is unknown, we can solve an algebraic equation for the unknown
quantity.
Ex 1: Find 2.5% of 600.
Ex 2: If 120 million roses were sold on Valentine's Day, and 75% of the roses were red, how
many red roses were sold on Valentine's Day?
Percent Practice
1.
30 is 15% of what number?
4. 3 is what percent of 12?
2. A real estate company pays commissions to
their sales people for selling property. A well
known company paid 6% commissions last year to
their sales staff, totaling $480,000. What was the
dollar value of the real estate sold by the company
that year?
5. Juan missed 6 out of 92 questions on a
test. To the nearest percent, what percent of
the questions did he solve correctly?
3. In a magazine, 30 pages of the 80 pages
are devoted to sports. What percent of the
magazine is devoted to sports?
6. After Mary lost 20% of her investment, she
had $2000 left. How much did she invest
originally?
Percent of Increase, Decrease, Discount
Use the percent to find the amount the original number is changing by
ADD
raise, appreciate, tax, tip
SUBTRACT
depreciate, discount
Ex 1: You normally work at the mall over the spring break for $450. This year the boss tells you
that you will be receiving a 7% raise. How much of an increase will you be receiving? How
much will you be earning in total this year?
Ex 2: An I-phone is on sale for 20% off the original price. If the original price is $270, what is
the sale price?
Ex 3: The enrollment at a local elementary school had 560 students last year. This year the
enrollment has decreased by 48 students. What is the percent of decrease to the nearest tenth
of a percent?
Percent Increase, Decrease, Discount Practice
1. Find the percent of change, rounded to the
nearest whole percent. Describe the change
as an increase or decrease.
$6.25/h to $6.75/h
4. The circulation of a newsletter decreased
from 6500 to 3575. What was the percentage
decrease in circulation?
2. Joshua's normal body temperature is
98.5ºF. Due to a cold, his temperature went
up 3ºF. To the nearest percent, what was the
percent of increase in his body temperature?
5. Walter is a waiter at the Towne Diner. He
earns a daily wage of $50, plus tips that are
equal to 15% of the total cost of the dinners
he serves. What was the total cost of the
dinners he served if he earned $170 on
Tuesday?
3. Vicki paid $11.25 for a pair of pants that
usually sells for $25. What percent discount
did she receive?
6. Rashawn bought a CD that cost $18.99 and
paid $20.51, including sales tax. What was
the rate of the sales tax?
Rate
A rate is a ratio that compares two different kinds of numbers, such as miles per hour, or
inches per minute. A unit rate compares a quantity to its unit of measure.
A rate expresses how long it takes to do something.
Solving a problem dealing with rate usually involves solving a proportion.
Ex 1: How long, in minutes, did it take the bug to cover 350 inches at a rate of 50 inches per
minute?
Ex 2:The bug drives his matchbox cruiser to his friend’s house traveling at the rate of 50
inches per minute. He then walks back to his home at the rate of 10 inches per minute. If the
round trip took 9 minutes, how far is it from the bug's home to his friend's house?
Rate Practice
1. A cell phone can receive 120 messages
per minute. At this rate, how many messages
can the phone receive in 150 seconds?
3. Joseph typed a 1,200-word essay in 25
minutes. At this rate, determine how many
words he can type in 45 minutes.
2. Nicole’s aerobics class exercises to fastpaced music. If the rate of the music is 120
beats per minute, how many beats would
there be in a class that is 0.75 hour long?
4. A car uses one gallon of gasoline for every
20 miles it travels. If a gallon of gasoline
costs $3.98, how much will the gas cost, to
the nearest dollar, to travel 180 miles?
5. Tom drove 290 miles from his college to home and used 23.2 gallons of gasoline. His
sister, Ann, drove 225 miles from her college to home and used 15 gallons of gasoline. Whose
vehicle had better gas mileage? Justify your answer.
Conversions
Sometimes you need to convert from one unit of measure to another similar unit.
Ex: Roberta needs ribbon for a craft project. The ribbon sells for $3.75 per yard. Find the
cost, in dollars, for 48 inches of the ribbon.
Conversion Practice
1. Peter walked 8,900 feet from home to
school.
4. A soda container holds gallons of soda. How many ounces of soda does this container hold?
How far, to the nearest tenth of a mile, did he
walk?
2. Elizabeth is baking chocolate chip cookies.
A single batch uses
teaspoon of vanilla. If
5. A jogger ran at a rate of 5.4 miles per hour.
Find the jogger's exact rate, in feet per minute.
Elizabeth is mixing the ingredients for five
batches at the same time, how many
tablespoons of vanilla will she use?
3. If the speed of sound is 344 meters per
second, what is the approximate speed of
sound, in meters per hour?
6. Andy is 6 feet tall. If 1 inch equals 2.54
centimeters, how tall is Andy, to the nearest
centimeter?