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New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
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Algebra I
Open Ended Application
Problems
2012-08-14
www.njctl.org
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Table of Contents
Number Problems
Age Problems
Geometry Problems
Percent Problems
Mixture Problems
Uniform Motion Problems
Work Problems
Proportionality Problems
Click on the topic to
go to that section
Problem Solving Strategies
Most problems can be solved by incorporating one or more
strategies.
Work backwards
Make a table, chart or diagram
Solve a simpler or similar problem
Guess and check
Look for a pattern
Eliminate possibilities
Draw a picture
Plan for Solving a
Word Problem
Plan: Read the problem several times.
What do you know?
What do you need to find?
Eliminate any unnecessary information.
Set Up: Define a variable.
Making a chart or drawing a picture may be helpful.
Write an open sentence.
Solve the open sentence.
Plan for Solving a
Word Problem
Check:
Reread the problem.
Did you answer the question?
Did you state your answer clearly with the
appropriate units?
Is your answer consistent with the information
given in the problem?
Number Problems
Return to
Table of
Contents
Vocabulary
Consecutive integers are obtained when you count by ones
from any given integer.
Examples
1, 2, 3, 4, 5
-6, -5, -4, -3
x, x + 1, x + 2
x - 1, x, x + 1
Vocabulary
Even integers are integers that are multiples of two.
Consecutive even integers are obtained when you count by
twos from any given even integer.
Examples
2, 4, 6
-6, -4, -2
x, x + 2, x + 4
x - 2, x, x + 2
Vocabulary
Odd integers are integers that are not even.
Consecutive odd integers are obtained when you count by
twos from any given odd integer.
Examples
3, 5, 7
-9, -7, -5
x, x + 2, x + 4
x - 2, x, x + 2
Example 1
One number is 10 greater than another. If the lesser
number is subtracted from three times the greater
number, the difference is 42. Find the numbers.
Plan:
2 numbers - one is 10 greater than the other
Set Up:
n + 10 = greater number
n = lesser number
Write an open sentence:
3(n + 10) - n = 42
Solve:
3(n + 10) - n = 42
3n + 30 - n = 42
2n + 30 = 42
-30 -30
2n = 12
2
2
n=6
So n + 10 = 6 + 10 = 16
The two numbers are 6 and 16.
Check:
Reread the problem. Does your answer make
sense?
Example 2
One number is 12 greater than another. If the sum of
the two numbers is 88, find the numbers.
Plan:
2 numbers - one is 12 greater than the other
Set Up:
n + 12 = one number
n = second number
Write an open sentence:
(n + 12) + n = 88
X
Solve:
(n + 12) + n = 88
2n + 12 = 88
2n + 12 = 88
-12 -12
2n = 76
2
2
n = 38
So n + 12 = 38 + 12 = 50
The two numbers are 38 and 50.
Check:
Reread the problem. Does your answer make
sense?
X
Example 3
Find three consecutive odd integers whose sum
is 183.
Plan:
Find three consecutive odd integers
Set Up:
n = 1st consecutive odd integer
n + 2 = 2nd consecutive odd integer
n + 4 = 3rd consecutive odd integer
Write an open sentence:
n + n + 2 + n + 4 = 183
X
Solve:
n + n + 2 + n + 4 = 183
3n + 6 = 183
-6
-6
3n = 177
3
3
n = 59
So n + 2 = 61 and n + 4 = 63
The three consecutive odd integers are 59, 61 and 63.
Check:
sense?
Reread the problem. Does your answer make
X
Practice 1
One number is 70 greater than a second number. If the
lesser number is subtracted from twice the greater
number, the difference is 174. Find the numbers.
Plan:
Set Up:
Write an open sentence:
2(n + 70) - n = 174
Solve:
The two numbers are 34 and 104.
Check Your Solution:
Practice 2
Find three consecutive integers whose sum is -315.
Plan:
Set Up:
Write an open sentence:
x + x + 1 + x + 2 = -315
Solve:
The three numbers are -106, -105 and -104.
Check Your Solution:
Practice 3
Find three consecutive even integers such that the sum
of the least integer and the greatest integer is -180.
Plan:
Set Up:
Write an open sentence:
Solve:
The three consecutive odd integers are -92,
-90 and -88.
Check Your Solution:
1
Find the largest of four consecutive integers
whose sum is 130.
2
The lengths of the sides of a triangle are
consecutive odd integers. The perimeter is 27
meters. Find the length of the smallest side.
3
Find two consecutive even integers whose sum
is 148.
4
Sam has 6 more than twice as many newspaper
customers as when he started selling
newspapers. If he currently has 98 customers,
how many did he have when he started?
5
There are fifty coins in a jar that contains only
dimes and quarters. The number of dimes in the
jar is 2 less than three times the number of
quarters. How many dimes are in the jar?
Age Problems
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Table of
Contents
Problem Solving Strategy
Sometimes using a chart to organize the information given in
a word problem can be helpful.
You can use this strategy to solve age problems.
Example 1
Jake is 12 years older than his dog. Next year he will
be four times as old as his dog will be. How old is
Jake now?
X
Plan: Find Jake's age now.
Set Up:
Age Now
Age Next Year
Dog
x
x+1
Jake
x + 12
(x + 12) + 1
Write an open sentence:
Next year, Jake will be four times as old as his dog.
(x + 12) + 1 = 4(x + 1)
Solve: (x + 12) + 1 = 4(x + 1)
x + 13 = 4x + 4
-x
-x
13 = 3x + 4
-4
-4
9 = 3x
3 3
3=x
Jake's dog is 3 years old now and 3 + 12 = 15. Therefore, Jake is 15
years old now.
Check: Reread the problem. Does your answer make sense?
X
Example 2
Erica is now four years older than her sister Alicia. In ten
years, Erica will be twice Alicia’s present age. Find the
age of each girl now.
X
Plan: Find the age of each girl.
Set Up:
Age Now
Age in 10 years
Erica
x+4
(x + 4) + 10
Alicia
x
x + 10
Write an open sentence:
x + 14 = 2x
Solve:
x + 14 = 2x
-x
-x
14 = x
Alicia is 14 years old now and Erica is 18 years old now.
Check: Reread the problem. Does your answer make sense?
X
Practice 1
Anthony is 9 years older than his sister Marie. Next
year, he will be four times as old as his sister. How
old is Anthony now?
Plan: Find Anthony's age now.
Set Up:
Write an open sentence:
Solve:
Check:
Anthony is 11 years old now.
Practice 2
Cara is six years older than her brother. In three years,
she will be twice as old as her brother will be. How old
is Cara now?
Plan:
Set Up:
Write an open sentence:
Solve:
Check:
Cara is 9 years old now.
6
Bebe is twice as old as Marcus. The sum of
their ages is 57. How old is Bebe?
7
Each sister is two years older than the next.
The oldest sister is twice the age of the
youngest sister, with two sisters in between.
How old are the sisters?
8
Deanna's age is eight years greater than half of
Metri's age. If the sum of their ages is 17, how
old is Deanna?
9
Zach's age is three less than twice Matt's age.
In five years, the sum of their ages will be 19.
How old is Zach now?
10
The son is 28 years younger than his father. The
sum of their ages is 84. How old is dad?
Geometry Problems
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Table of
Contents
Formulas to Remember
Perimeter is the distance around a figure.
Area of a rectangle
A=l w
Area of a triangle
A = 1/2 b h
Example 1 - Area
Find the measure of the area of the shaded region in the
figure below.
x+6
x+4
4x
2x
Plan: Given the length and width of large and small rectangles.
Set Up: A = area of shaded region
Open Sentence: A = 4x(x + 6) - 2x(x + 4)
X
x+6
x+4
4x
2x
Solve:
4x(x + 6) - 2x(x + 4) = A
2
2
4x + 24x - 2x - 8x = A
X
2
2x + 16x = A
2
The area of the shaded region is (2x + 16x) units
Check:
2
Example 2 - Perimeter/Area
The measure of the perimeter of a square is
12a + 16b. Find the measure of the area of the square.
Plan:
Set Up:
Find the area of the square.
2
A=s
s = (12a + 16b) ÷ 4
s = 3a + 4b
Open Sentence: A = (3a + 4b)
2
X
Solve: (3a + 4b) (3a + 4b)
2
9a + 12ab +12ab +16b
2
9a + 24ab + 16b
2
X
2
2
2
2
The area of the square is (9a +24ab + 16b ) units .
Check:
Example 3 - Construction
The length of a rectangular lot is 5 yards less than three times the
width. If the length was decreased by 2 yards and the width
increased by 5 yards, the area would be increased by 17 square
yards. Find the original dimensions of the lot.
Plan:
Find the area of the original lot. It may
help to draw a picture.
Set Up:
w = original width
w + 5 = new width
3w - 5 = original length
3w - 7 = new length
Open Sentence: w(3w - 5) + 17 = (w + 5)(3w - 7)
X
Solve: w(3w - 5) + 17 = (w + 5)(3w - 7)
2
2
2
2
2
2
3w - 5w + 17 = 3w + 15w - 7w - 35
3w - 5w + 17 = 3w + 8w - 35
-3w
-3w
-5w + 17 = 8w - 35
-8w
-8w
-13w + 17 = -35
-17 -17
-13w = -52
-13 -13
w=4
so, 3w - 5 = 7
The original dimensions were 4 yards by
7 yards.
Check:
X
Practice 1
Melissa has a rectangular garden that is 10 feet
longer than it is wide. A brick path that is 3 feet wide
surrounds the garden. The total area of the path is
396 square feet. What are the dimensions of the
garden?
Plan:
Set Up:
Write an open sentence:
Solve:
The width is 25 feet and the length is 35 feet.
Check Your Solution:
Practice 2
A new athletic field is being sodded at Lawrence High
School using 2-yard by 2-yard squares of sod. If the
width of the field is 70 yards less than its length and
its area is 6000 square yards, how many squares of
sod will be needed?
Plan:
Set Up:
Write an open sentence:
Solve:
1500 squares of sod will be needed.
Check Your Solution:
11
The length of a garden is one more foot than
twice the width. The area of the garden is 55
feet. What is the width of the garden?
12
A 3 x 4 picture sits in a picture frame. The total
area of the picture and frame is 56 inches.
How wide is the frame?
-----------------6x--------------------------------
13
x + 15
The garage door is a square that measures 2x feet on
each side. How many square feet of house surrounds
the garage?
14
A square has its side doubled in length. How
much does the area of the square increase?
x
15
An area rug's length is 3 feet less than three
times its width. The area of the rug is 90 square
feet. What is the length of the rug?
Percent Problems
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Table of
Contents
Important Concepts to Recall
Part
Whole
Percent
= 100
List Price - Discount = Sale Price
Simple Interest = Principle x Rate x Time
Percent of Change: Amount of Change
Original Amount
Example 1
Mia bought a sweater at a 15% discount. If she paid
$38.25 for the sweater, what was the original price?
Plan: Find the original price of the sweater. If the sweater
is 15% off, then she paid 85% of the original price.
Set Up: x = original price of the sweater.
Open Sentence: 85% of x = 38.25
X
Solve:
85% of x = 38.25
(.85)(x) = 38.25
.85
.85
x = 45
The original price of the sweater was $45.
Check:
15% of 45 = 6.75
and 45 - 6.75 = 38.25
X
Example 2
The Smiths invest part of $8000 in bank accounts that
pay 5% simple annual interest and the rest in bonds
that pay 12% simple annual interest. How much
money is invested in each account if the total annual
income from these investments is $610?
Plan:
They invest part of the money at 5% and part of
the money at 12%.
Remember: part + part = whole
Set Up:
x = amount invested at 5%
8000 - x = amount invested at 12%
Open Sentence: 0.05x + 0.12(8000 - x) = 610
X
Solve:
0.05x + 0.12(8000 - x) = 610
0.05x + 960 - 0.12x = 610
-0.07x + 960 = 610
- 960 -960
-0.07x = -350
-0.07 -0.07
x = 5000
$5,000 was invested at 5% and $3,000 was invested at 12%.
Check:
5000(0.05) + 3000(0.12) = 610
250 + 360 = 610
610 = 610
X
Practice 1
The O'Connors paid $15,000 in closing costs when
purchasing their new home. If this amount
represents 6% of the purchase price, how much did
they pay for their home?
Plan:
Set Up:
Write an open sentence:
Solve:
They paid $250,000 for their house.
Check Your Solution:
Practice 2
A jacket was on sale for $63.75. If the original selling
price was $75, what was the percent of the discount?
Plan:
Set Up:
Write an open sentence:
Solve:
The discount percent was 15%.
Check Your Solution:
Practice 3
Marco invested $10,000, part at an annual interest
rate of 5% and the rest at an annual rate of 10.25%.
How much money did he invest at each rate if his
total income on the investment for one year was
$867.50?
Plan:
Set Up:
Write an open sentence:
Solve:
He invested $3,000 at 5% and $7,000 at 10.25%
Check Your Solution:
16
The sale price of a dress is $112.50 after a 25%
discount is taken. Find the regular price of the
dress.
17
The bill at the restaurant is $35. You want to
leave a 20% tip. How much should you leave?
18
One store has a $200 bicycle on sale for 40%
off. Another store has the same bicycle for $200
with a 30% off plus an additional 10% off. Will
the bicycles cost the same at both stores?
19
The price of the CD increased from $12 to $15.
What is the percent of increase?
20
Mark invested $20,000, part at an annual
interest rate of 6% and the rest at an annual rate
of 2.5%. How much money did he invest at each
rate if his total income on the investment for
one year was $960.50?
Mixture Problems
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Table of
Contents
Mixture Problems
Sometimes a chemist mixes solutions of different strengths to
obtain a desired solution. Or a business mixes two or more
goods in order to sell a blend at a given price.
Mixture problems are problems related to these situations.
Example 1
Steven bought some 44 cent stamps and some 28
cent stamps. He bought 35 stamps in all and paid
$13.00 for them. How many stamps of each kind did
he buy?
Plan: He bought 2 kinds of stamps and paid $13.00. Find
how many of each kind he bought.
Set Up:
Number of
stamps
Value per
stamp
Total Value
44 cent
stamps
28 cent
stamps
s
$0.44
0.44s
35 - s
$0.28
0.28(35 - s)
Mixture
35
$13.00
X
Open Sentence: 0.44s + 0.28(35 - s) = 13
Solve:
0.44s + 9.80 - 0.28s = 13
0.16s + 9.80 = 13
- 9.80 - 9.80
0.16s = 3.20
0.16 0.16
s = 20
35 - s = 15
He bought 20 forty-four cent stamps and 15 twenty-eight cent
stamps.
Check:
X
Example 2
A 40L solution is 15% salt. How much water must be
added to make it an 8% salt solution?
Plan: How much water must be added to change the solution
from 15% salt to 8% salt
Set Up:
Original
Volume
Of
Solution
% Salt
Volume
of
Salt
40
15%
0.15(40)
s
0%
0(s)
40 + s
8%
0.08(40 + s)
Solution
Water
Added
New
Solution
X
Open Sentence: 0.15(40) + 0(s) = 0.08(40 + s)
Solve:
0.15(40) + 0(s) = 0.08(40 + s)
6 = 3.2 + 0.08s
- 3.2 - 3.2
2.8 = 0.08s
0.08 0.08
35 = s
35 liters of water need to be added to make an 8% salt
solution.
Check:
X
Practice 1
How many pounds of dried apricots worth $7.50 per
pound must be added to 5 pounds of dried bananas
worth $5.25 per pound to form a mixture worth $6.00
per pound?
Plan:
Set Up:
Write an open sentence:
Solve:
2.5 pounds of dried apricots are needed.
Check Your Solution:
Practice 2
A 15L solution is 70% antifreeze. How much
antifreeze must be added to produce a solution that
is 80% antifreeze?
Plan:
Set Up:
Write an open sentence:
Solve:
7.5L of antifreeze must be added.
Check Your Solution:
21
Sofie has twice as many dimes as nickels in her
piggy bank. If the dimes and nickels together
total $18.00, how many dimes does she have?
22
Alberto has 48 mL of solution that is 50% acid.
How many mL of a 15% acid solution should he
add to obtain a solution that is 35% acid?
23
There are 15 lbs. of nuts valued at $7 per lb. The
mixture consists of peanuts, valued at $2.20 per
pound and cashews valued at $8.50 per pound.
Approximately how many pounds of peanuts
are in the mixture?
24
In your chemistry class, you have a bottle of 5%
boric acid solution and a bottle of 2% boric acid
solution. You need 50 mL of a 4% boric acid
solution. How much of the 2% acid solution
must be used?
25
The register at the carnival has 200 bills in $1
and $5 denominations. The value of the money
is $660. How many $5 bills are in the register?
Uniform Motion Problems
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Table of
Contents
Uniform Motion Problems
An object that moves at a constant rate, or speed, is
considered to be in uniform motion.
A formula that is used in solving uniform motion problems is:
rate x time = distance
To apply the formula correctly, the units used for the time and
the distance measurements must be the same as those used
for the rate.
Three types of uniform motion problems are shown in the
following examples.
Example 1 - Motion in the Same Direction
Two speedboats leave from the same dock at the
same time traveling to Point Pleasant. The faster
boat arrives in 6 hours. The slower boat arrives in 9
hours. The slower boat travels at an average speed
that is 15 km/h slower than the faster boat. What is
the average speed of the faster boat?
Plan:
X
Find the rate or speed of each boat.
Set Up:
Rate
(km/h)
Time
(h)
Distance
(km)
Faster
boat
x
6
6x
Slower
boat
x - 15
9
9(x - 15)
Open Sentence:
Solve
6x = 9(x - 15)
6x = 9x - 135
-9x - 9x
-3x = -135
-3
-3
x = 45
x - 15 = 30
The faster boat travels at a rate of 45 km/h and the slower boat
travels at 30 km/h.
Check:
X
Example 2 - Motion in the Opposite Direction
Jazmine and Meghan are 630 meters apart. Jazmine walks
toward Meghan at the rate of 2.5 m/sec and Meghan runs
toward Jazmine. What is Meghan's rate if she reaches
Jazmine in 1.4 min?
Plan:
X
Find Meghan's rate.
Remember 1.4 min = ? sec
Set Up:
Rate
(m/s)
Time
(s)
Distance
(m)
Jazmine
2.5
84
2.5(84)
Meghan
x
84
84x
Open Sentence: 2.5(1.4) + 84x = 630
Solve:
210 + 84x = 630
- 210
-210
84x = 420
84
84
x=5
Meghan runs at a rate of 5 m/s.
Check:
X
Example 3 - Round Trip
Javone is a member of the cross country team at his
school. On Monday he ran from school to Central
Park and back again. On the way to the park he ran
at a rate of 7.5 m/hr and on the return he ran at a rate
of 5 m/hr. If he took 1 hour 15 minutes to run the
entire distance, how far is the school from Central
Park?
Plan: Find the distance from the school to Central Park.
Set Up:
Rate
(m/h)
Time
(h)
Distance
(m)
To the
park
7.5
x
7.5x
Back to
school
5
1.25 - x
5(1.25 - x)
X
Open Sentence: 7.5x = 5(1.25 - x)
Solve:
7.5x = 6.25 - 5x
+ 5x
+ 5x
12.5x = 6.25
12.5 12.5
x = 0.5
The time is 0.5 hours, so the distance from the school to the park
is 7.5(0.5) which equals 3.75 miles.
Check:
X
Practice 1
Jack and Jessie are cycling in the same direction on
the same bike path. Jack's average speed for the trip
is 20 miles per and Jesse's is 14 miles per hour.
After how many hours will they be 7.5 miles apart?
Plan:
Set Up:
Write an open sentence:
Solve:
It will take them 1.25 hours to be 7.5 miles apart.
Check Your Solution:
Practice 2
Jeff and Brian live 1.5 mile apart. They agree to meet
at the library directly between their homes. Jeff needs
12 minutes and Brian needs 18 minutes to get to the
library. If they both travel at the same average speed,
how far do Jeff and Brian live from the library?
Plan:
Set Up:
Write an open sentence:
Solve:
Jeff lives 0.6 miles from the library and
Brian lives 0.9 miles from the library.
Check Your Solution:
Practice 3
Scott delivers newspapers every morning. The trip
delivering them takes 30 minutes. The return trip over the
same route takes 20 minutes. If his average rate going is 6
km/h slower than returning home, how far does he travel
each morning?
Plan:
Set Up:
Write an open sentence:
Solve:
Scott's entire trip is 12 km long.
Check Your Solution:
26
An airplane flies 1500 miles due west in 3 hours
and 1000 miles due south in 2 hours. What is
the average speed of the airplane?
27
Fred and Ted leave their home at the same time,
traveling in opposite directions. Fred travels 50
miles per hour and Ted travels 54 miles per
hour. In how many hours will they be 572 miles
apart?
28
Mark travels at the rate of 32 mph for four
hours. His sister, travels at the rate of 40 miles
per hour. How long will it take her to travel the
same distance as her brother?
29
Joey left home on his bicycle for a long
distance ride. Marcie left 2 hours later on her
motorcycle carrying his lunch. Marcie traveled
at 45 miles per hour and she caught up to Joey
in 1.75 hours. How fast was Joey traveling?
30
Bob and Sally are in a race. Bob is running at
the rate of 8 mph and Sally is running at the
rate of 6 mph. How long will it take them to be
11 miles apart?
Work
Problems
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Table of
Contents
Work Problems
When solving problems that involve finding how long it takes to
complete a task, a constant rate of work is assumed.
Work rate is the fraction of the whole job that can be done per
unit of time.
work rate x time = work done
Another way to do work problems is to think:
Time Together + Time Together = 1 (for one job
Time Alone
Time Alone
done)
Example 1
Maya can paint a room in four hours. Takira can paint
a room in six hours. How long would it take them to
paint the room if they worked together?
Plan: Find the number of hours it will take them to paint the
room together.
Set Up: x = # of hours to paint the room together
Since Maya can paint the whole room in 4 hours, her
work rate is 1/4 of the job per hour.
In x hours, she could paint 1/4 times x of the job, or x/4 of the
job.
Therefore, Takira could do x/6 of the job.
X
Open Sentence:
Maya's work + Takira's work = Job (1 room painted)
x/4 + x/6 = 1
Solve: 12(x/4 + x/6) = 1(12)
3x + 2x = 12
5x = 12
5
5
2
x = 12/5 or 2 or 2.4
5
It would take 2.4 hours to paint the room if they worked together.
Check:
X
Example 2
Kyle can mow the lawn in three hours. Dean can mow the
lawn in two hours. How long would it take them to mow
the lawn if they worked together?
Plan: Find the number of hours it will take them to mow the lawn
together.
Set Up: x = # of hours to mow the lawn together
Kyle can do 1/3 of the work in 1 hour.
Dean can do 1/2 of the work in 1 hour.
X
Open Sentence:
x/3 + x/2 = 1
Solve: 6(x/3 + x/2) = 1(6)
2x + 3x = 6
5x = 6
5
5
1
x = 6/5 or 1 or 1.2
5
It would take 1.2 hours to mow the lawn if they worked
together.
Check:
X
Practice 1
Jake can service a car in 4 hours. Jared can service
a car in 5 hours. How long would it take them to
service nine cars if they worked together?
Plan:
Set Up:
Write an open sentence:
Solve:
If they work together, they can service 9 cars in 20 hours.
Check Your Solution:
Practice 2
Daisy can vacuum and dust the house in 2 hours.
Jessica can do the same job in 1.2 hours. How long
will it take them to vacuum and dust the house if they
work together?
Plan:
Set Up:
Write an open sentence:
Solve:
It will take them 0.75 hours working together.
Check Your Solution:
31
Bob can paint a room in 8 hours. Mark can paint
a room in 10 hours. How long will it take them
to paint the room if they work together?
32
Sal and Sam can row across the lake together
in 3 hours. Alone, it takes Sal 5 hours to row
across the lake. How long does it take Sam to
row across the lake alone?
33
A 500 gallon pool takes 6 days to fill with one
hose. If the neighbors put their hose in the pool
to help fill it, it will take only 4 days. If only the
neighbor's hose is used, how long will it take to
fill the pool?
34
It takes you 30 minutes to clean your room. It
takes your brother 45 minutes to clean his
room. How long does it take the two of you to
clean your rooms if you work together?
35
You can knit two squares for an afghan in 3
hours and your friend can knit 4 squares in 5
hours. How long does it take both of you to knit
10 squares when working together?
Proportionality
Problems
Return to
Table of
Contents
Proportion Problems
Concepts to Remember
Part = Percent
Whole
100
Complementary Angles are two angles whose measures add
up to 90 degrees.
Supplementary Angles are two angles whose measures add
up to 180 degrees.
Example 1
Mr. Jones bought 10 pounds of grass seed to seed an
area of 2000 square feet. At this rate, how much seed
would he need to seed 3200 square feet?
Plan:
Given 10 lbs of seed for 2000 sq ft
Find how much for 3200 sq ft
Set Up:
x = pounds of grass seed for 3200 sq ft
Open Sentence:
10
x
=
2000
3200
X
Solve:
10
x
=
2000
3200
2000x = 10(3200)
2000x = 32000
2000
2000
x = 16
Mr. Jones needs 16 lbs of grass seed.
Check:
X
Example 2
The measures of two complementary angles are in
the ratio of 3:7. Find the measure of each angle in
degrees.
Plan:
The sum of the measures of complementary
angles is 90 degrees. Their ratio is 3:7.
Set Up:
x = measure of smaller angle
90 - x = measure of larger angle
Open Sentence: 3
x
=
7
90 - x
X
Solve:
3 = x
7
90 - x
3(90 - x) = 7x
270 - 3x = 7x
+ 3x +3x
270 = 10x
10 10
27 = x
X
The measures of the two complementary angles are 27 degrees and
63 degrees.
Check:
Practice 1
In a survey of a high school with 1272 students, 7 out of
every 12 students said that they do not like the school
lunches. How many students do like the lunches?
Plan:
Set Up:
Write an open sentence:
Solve:
530 students do like the lunches.
Check Your Solution:
Practice 2
The measures of two supplementary angles are in the
ratio 3:2. Find the measure of each angle.
Plan:
Set Up:
Write an open sentence:
Solve:
108 degrees and 72 degrees.
Check Your Solution:
36
A 96 mile trip requires 8 gallons of gasoline. At
that rate, how many gallons would be required
for a 156 mile trip?
37
The scale on the blueprint for a house is 1 inch
to 3 feet. If the living room on the blueprint is
5.5 inches by 7 inches, what is the area of the
actual room?
38
The measures of two adjacent angles are in the
ratio of 2 to 5. If the sum of the angle measures
is 98 degrees, what is the measure of the larger
angle?
39
The measure of supplementary angles are in
the ratio of 3 to 5. Find the measure of the
smaller angle.
40
The ratio of boys to girls is 4:5. If there are 2000
students in the auditorium, how many are girls?