All Algebra II Resources
Example Questions
Example Question #1 : Other Mathematical Relationships
Sarah notices her map has a scale of . She measures
between Beaver Falls and Chipmonk Cove. How far apart are the cities?
is the same as
So to find out the distance between the cities
Example Question #2 : Direct Proportionality
If an object is hung on a spring, the elongation of the spring varies directly as the mass of the object. A 20 kg object increases the length of a spring by exactly 7.2 cm. To the nearest tenth of a centimeter, by how much does a 32 kg object increase the length of the same spring?
Let be the mass of the weight and the elongation of the spring. Then for some constant of variation
,
We can find by setting
from the first situation:
so
In the second situation, we set and solve for
:
which rounds to 11.5 centimeters.
Example Question #1 : Other Mathematical Relationships
Sunshine paint is made by mixing three parts yellow paint and one part red paint. How many gallons of yellow paint should be mixed with two quarts of red paint?
(1 gallon = 4 quarts)
First set up the proportion:
x =
Then convert this to gallons:
Example Question #1 : Other Mathematical Relationships
Sally currently has 192 books. Three months ago, she had 160 books. By what percentage did her book collection increase over the past three months?
To find the percentage increase, divide the number of new books by the original amount of books:
She has 32 additional new books; she originally had 160.
Example Question #1 : Proportionalities
Find for the proportion
.
To find x we need to find the direct proportion. In order to do this we need to cross multiply and divide.
From here we mulitply 100 and 1 together. This gets us 100 and now we divide 100 by 4 which results in
Example Question #1 : Basic Single Variable Algebra
On a map of the United States, Mark notices a scale of
. If the distance between New York City and Los Angeles in real life is
, how far would the two cities be on Mark's map?
If the real distance between the two cities is
, and
=
, then we can set up the proportional equation:
Example Question #1 : Basic Single Variable Algebra
If and
, find
and
.
We cannot solve the first equation until we know at least one of the variables, so let's solve the second equation first to solve for . We therefore get:
With our , we can now find x using the first equation:
We therefore get the correct answer of and
.
Example Question #8 : Direct Proportionality
If an object is hung on a spring, the elongation of the spring varies directly with the mass of the object. A 33 kilogram object increases the length of a spring by exactly 6.6 centimeters. To the nearest tenth of a kilogram, how much mass must an object posess to increase the length of that same spring by exactly 10 centimeters?
Let be the mass of the weight and the elongation of the spring, respectively. Then for some constant of variation
,
.
We can find by setting
:
Therefore .
Set and solve for
:
kilograms
Example Question #9 : Direct Proportionality
If is directly proportional to
and when
at
, what is the value of the constant of proportionality?
The general formula for direct proportionality is
where is the proportionality constant. To find the value of this
, we plug in
and
Solve for by dividing both sides by 12
So .
Example Question #10 : Direct Proportionality
The amount of money you earn is directly proportional to the nunber of hours you worked. On the first day, you earned $32 by working 4 hours. On the second day, how many hours do you need to work to earn $48.
The general formula for direct proportionality is
where is how much money you earned,
is the proportionality constant, and
is the number of hours worked.
Before we can figure out how many hours you need to work to earn $48, we need to find the value of . It is given that you earned $32 by working 4 hours. Plug these values into the formula
Solve for by dividing both sides by 4.
So . We can use this to find out the hours you need to work to earn $48. With
, we have
Plug in $48.
Divide both sides by 8
So you will need to work 6 hours to earn $48.
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