Download Proportions – Blue Problems number of women

Proportions – Blue Problems
If somebody says that men and women are in a 3-to-8 ratio, it means that
number of men
3
=
number of women 8
There are many ways this equation could be true. For instance, there could be 3 men and 8 women.
Or there could be 6 men and 16 women because
6 (3) (2) 3
=
=
16 (8) (2) 8
The number of men must be a multiply of 3, and the number of women must be the same multiple
8. This means that there is a number x such that
3x = number of men,
8x = number of women.
The number x is a common factor of the number of men and the number of women. Suppose that
in addition to knowing the ratio 3:8 (a short way of writing the ratio 3 to 8), you also know that
there is a total of 407 people in the group. How many of each would there be?
Since 3x is the number of men and 8x is the number of women, you can write
3x + 8x = 407
11x = 407
x = 37
So there are (3)(37), or 111, men and (8)(37), or 296, women.
Example 1
Fish Problem. Naturalists estimate that there are 3000 fish in Lake Muchimuck. Some are perch
and the rest are bass. They drag a fishing net through the lake and catch 24 perch and 21 bass.
a. What is the ratio of perch to bass caught? Express the answer in lowest terms.
b. Assuming that the entire lake has this ration, how many of each kind of fish were there
(including those caught)?
Answer:
number of perch 24 8
a.
=
=
number of bass 21 7
Ratio is 8 : 7
Think these reasons
b. Let x = common factor.
8x = Number of perch.
7x = Number of bass
Define a variable.
Write expressions
numbers of fish
8x + 7x = 3000
Total
number
of fish
is
for
Write an equation
3000.
15x = 3000
x = 200
.. . 8x = 1600,
7x = 1400
1600 perch, 1400 bass.
Solve the equation.
Evaluate the expressions
Answer the question
Check: 1600 + 1400 = 3000, and 1600 = 8 , which checks.
1400
7
the
Practice Problems:
1. Ratio Problem. The ratio of two integers is 13 : 6. The smaller integer is 54. Find
the larger integer.
2. The ratio of two integers is 9 : 7. Their sum is 1024. Find the two integers.
3. Grandchildren Problem. Mae Berry has 18 grandchildren on her son’s side and 12
grandchildren on her daughter’s side.
a.
What is the ratio of these numbers, in lowest terms?
b. The elder Berry divides 7200 acres of land into two tracts, whose areas are in
this ratio. How many acres are in each tract?
4. Price-to-Earnings Ratio Problem. The stock of a certain company is priced at
$18.20 per share. The company’s earnings for one year amounted to $2.80 per
share.
a.
Find the price-to-earnings ratio. Express it both as a ratio of two integers in
lowest terms and as a ratio __ : 1 (as it appears in newspapers), where the
second number is 1 and the first number is not necessarily an integer.
b. If the company earned $3,360,000 in that year, what was the total value of its
stock?
5. Basketball Tickets Problem. You are called upon to estimate how many of the
12,000 people who attend a professional basketball game are women. From a small
sample, you determine that the ratio of men to women is about 3 : 2. About how
many of the people attending are women?
6. Milk Stock Problem. Suppose that you work in a supermarket. The milk display case
can hold a total of 160 one-gallon containers of milk. Sales figures show that the
cheaper generic milk outsells the national brand milk by 7 : 3. How many containers
of each kind should you put in the case in order to have this ratio?
7. “Average” American. I recently read an article where the writer stated that Bill
Gates is worth about $38,000,000,000! This is a pretty difficult number to really
understand, so to help us out, the writer said that if Bill Gates buys a $250,000
Lamborghini sports car it is similar to the “average” American spending 65 cents
for a pack of gum. How much is the “average” American worth?
8. Tom ran 1.5 km in 7.5 minutes.
a.
Write a proportion to determine how long it will take him to run a 5 km race at
this same rate.
b. What is his rate in meters per minute?
c.
What is his rate in minutes per km?
9. Plastic bags are recycled at the local grocery store. For every 5 bags that are
recycled by customers, 3 new bags can be made. How many new bags can be made if
800 bags are returned?
10. You bought a car with 14-gallon tank that is advertised to get 25 miles per gallon.
You begin a trip on a full tank of gas and after 300 miles you are on empty. Was the
advertisement accurate for this trip? Explain.
11. A 12-foot board is cut into three pieces whose
lengths are in the ratio 3:1:2. How many inches
are in the length of the shortest piece?
12. Homer paid $3 for 6 donuts. How many dollars would Homer pay for 3 dozen
donuts?
13. x + 3 = 5 _____
x -5 2
14.
2y - 1 3
= _____
4y - 3 7
15.
3x - 2 = x + 4 _____
2
4
16. Sharn has a photograph which is 15 cm long and 10 cm wide. She wants to have it
enlarged so that the new photograph will be 42 cm long. How wide should the
enlarged photograph be?
17. A model robot is 40 cm tall and the real robot is 92 cm tall.
a.
If the real robot has his name printed and the letters are 10 cm tall, how tall
will the letters appear on the model robot?
b. If the model robot’s arm length is 16 cm, how long is the real robot’s arm?
18. A British television program shows homes that are for sale. One home is listed as
costing 500,000 pounds, or $850,000. The ratio of pounds to dollars used in this
TV program can be written in the form “1 pound : x dollars.” What is the value of x?
Express your answer as a decimal to the nearest tenth.
19. Eighteen acres of land sold for $27,766.80. At the same rate, what is the cost of
six acres of land? Express your answer to the nearest whole dollar.
20. Of 75 pairs of jeans, 7 have flaws. Estimate how many of 24,000 pairs of jeans are
flawed.
21. Eight of the 32 students in your math class have a cold. The school population is
450. A student estimates that 112 students in the school have a cold.
a.
Why is your math class not representative of the population?
b. Describe a survey plan you could use to better estimate the number of
students who have a cold.
Proportions – Blue Solutions
1. 117
2. 576 and 448
3.
a.
3:2
4. a.
13 : 2
6.5 : 1
b.
4320 acres and 2880 acres
b.
$21,840,000
5. 4800 women
6. 112 cartons generic, 48 cartons national
7. According to the date given, the “average” American is worth $98,800.
8. a.
25 min
b.
200 meter per min
c.
5 minutes per km
9. There are 800 ÷ 5 = 160 groups of bags returned by customers, so 160 x 3 = 480 bags can be made.
10. No; The car averaged 21.4 miles per gallon.
11. The three pieces are in ratio 3:1:2. Since 3 + 1 + 2 = 6, then a 6-foot board would be cut into pieces
that were 3, 1, and 2 feet, respectively. Since the board you are given is 12 feet, or twice as long, the
piece should each be twice as long, and the smallest piece will be 2 feet.
Since the question asks for the number of inches in the smallest piece, the 2 feet must be converted
to inches: 2 feet x 12 inches/foot = 24 inches.
12. Paying $3 for 6 donuts is the same as paying $6 for one dozen donuts. There are 12 donuts in one
dozen, so if one dozen donuts cost $6, then 3 dozen cost 3 x 6 = $18.
13. 10 1
14. -1
15. 10
3
16. 28 cm wide
17. a.
4.3 cm
b.
36.8 cm
18. We need to simplify the ratio 500,000 pounds : 850,000 dollars. A first step might be to divide each
number by 10,000, which gives us 50 pounds : 85 dollars. A next step might be to divide each number
by 5, which gives us 10 pounds : 17 dollars. Finally, dividing both numbers by 10, we get 1 pound : 1.7
dollars, so x = 1.7.
19. Six acres is one-third of 18 acres, so the cost of six acres is $27,766.80 ÷ 3 = $9255.60 or $9256 to
the nearest whole dollar.
20. 2,240
21. a.
b.
The cold virus may be passed from student to student in that class so that more of them have
colds than in the total school population.
Various answers
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In
some cases, sources were not known.
Problems
Bibliography Information
7
The Math Forum @ Drexel
(http://mathforum.org/)
9 – 12, 18 - 19
Math Counts
(http://mathcounts.org)
1-6
Algebra I: Expressions,
Equations, and
Applications (Hardcover)~
Paul A. Foerster, AddisonWesley Publishing
Company, Menlo Park, CA,
1999
20 - 21
Davison, David M.
Prentice Hall Pre-Algebra
Tools for a Changing
World. Needham, Mass:
Prentice Hall, 2001. Print.