Secondary Ill
Unit 3.7 Notes: Modeling and Systems
Recall that a rationa l expression is the ratio of two polynomials. Rational expressions can be
used to solve problems that involve compa ring two quantities of the same unit of measure.
Exa mple 1: Delilah is making her own salad dressing out of red vinegar and olive oil. It's a new
recipe so she has to determine the correct proportions. She mixes 10 teaspoons of vinegar and
16 teaspoons of olive oil. Afte r she stirs the mixture, she realizes it's not the consistency she
wants, so she adds more olive oil.
a.
vi ~g_,.v- ::.
I D.
Ho -r to
oil
I D. :::
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b. What is the rati: ~~e~ : ineg: to oli~l if s: aT
c.
f'I']
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What is the ratio of red vinegar to olive oil if she add s 6 teaspoons more of olive oil?
~spoons more of olive oil?
Write an expression to represent the ratio of red vinegar to olive oil. Let
x re pre se nt the
number of additional teaspoons of olive oi l added to the recipe.
lO
d. The recommended ratio of vinegar to olive is 1:7. Determine the amount of olive oil
that she must add to the mixture.
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lU b+X )
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Secondary Ill
Example2:
{
A door-to-door salesperson for TV Bonanza Cable Company offers cable television for
on ly $55.95 per month. However, there is a one-time installation cost of $180.
a. Determine the total cost ofthe cable for the first~ months/
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y;:
TOTti I
C()f
r
180
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b. What is th e average cost per month over th e first two months?
-c.
Write an equation to represent the total cost of cab le for x months.
'I
S5 . q )
-=.
X
-+ I 5 c
d. Write an equation to rep resent the average monthly cost of cab le for x months.
st; . Cf~x
+t86
,)(
e. A competitor offers a simil ar product for $65 per month and no installation charges.
.x- (OY'YI p-<tl-1-0v'..S
Who is offering the better deal? Explain.
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~s--. qs
-c;s. ~ x
·,-s bt+--~ev -to~~
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-=- q. oS.x
'1'. 0 s q. 6 c;
'~o
_g._
1 Lt-
x=
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1er.~cr
A workJ#,roblem i.s a type of problem that involves the rates of several workers and the time
it takes to complete a job. For example, the rate at which two painters work and the total
time it takes them t o paint a house while working together is an example of a work problem.
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ti ~
·,nd 1v ·\ o vCi I 11 WU
+
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Ot-'<:1/
Secondary Ill
Example 3: Mauree n is a community vo lunteer. She vo lunteers by wat eri ng the large vegetab le
garden in her neighborhoo d. Sometim es, Maureen's fri end Sa ndra also volunteers.
c:·
It ta kes Maureen 90 minutes to wat er the garden. When Maureen and Sandra are wo rking
~ together, th ey can complete the job in 4_9 minutes.
a.
Complete th e tab le.
/ ti(alone.
-
I
Time Spent
i
f
Watering
Rate of
Wateri ng
I
i
Minutes
Gardens
Minute
\
\
\
\
I
Ent ire J ob, or
1 Gard en
b.
Write and solve an equ ation to determine the tota l t ime it would take Sandra to water the
garden if she were working alone .
--40
40
-
·t
O,o
4.x
X
t?J x
-t
LJ j
+
~a,) - _LCft<
)< Ct - ( ))qX
~l.oO
-'tlx
-
><:t o
L t.O Cf X
Secondary Ill
A mixture problem is a type of problem that involves the combination of two or more liquids
0/o L. -+ o/tJ L ::
and the concentrations of those liquids.
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Example 4: Manuel is taking a college chemistry cou rse, and some of his time is spent in the chemistry
lab. He is conducting an experiment for which he needs a 6.2isa lt solution. Howeve r, all he can find in
the lab is 120 milliliters (ml) of 10% salt solution.
a.
How many milliliters of sa lt and how many mill iliters of water are in 120mL of 10% salt solution?
J:
• 10 ( I 2.0
• c=:t
b.
o(
1'2.6 )
-
ML elF
1o'b ..-n L.
S.::tl+
~ r= wt:t+< v
What would the concentration of the sa lt solution be if Manuel added 80m L of water?
J'2-
J20+6o
c.
:.
J'2
--
'0~
Write and solve an equation to calculate the amount of water Manuel needs to add to the
120mL of 10% salt solution to make a 2% salt solution. Let x represent t he amount of water
Manuel needs to add.
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2~'-' --+,02.)<
-2· f../
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i2 -::::.
-:o?
Example 5: Keisha is working on a chem istry expe riment. Suppose that the 20mL of a 20% sulfuric acid
is mixed with lOmL of the 5% su lfuric acid solution. What is the concentration of the resulting solution?
+ . OS ( lO) = X
4 + . , s -= 50 x;
• '2. ( 20)
y. t;
Write and solve an equation to
'3tJ)
---0,---3--1~
X'= 15 It tJ (-=o~ r.~V\~
-:.:: 3 QX
~0
---;-
a.
(
..
t\
calcu~te the amount of 5% sulfuric acid solution Keisha added if
the resulting solution is a 12% sulfuric acid solution. Let x represent the amount of 5% sulfuric
acid that she is adding.
~
2- ( """)_ u)
+ .. os (x
Lf + • oS_;.X
_ 2 . '--( - " ccfX
--J~
(c -
.'67
.. o7x
. ~·
) =
"2
-
c
i 2(2o-t x)
.A . -t
;e!.''i
. /lX
- , oS x
-----------------2 7-. 9JS l YY\L ()F)
c;
0/~
Secondary Ill
A distance problem is a type of problem t hat involves
di tance, rate and t ime.
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-r-
L -
Exam ple 6:
A river barge travels 140 miles from a loading dock to a warehouse to deliver supplies.
Then the barge returns to the loading dock. The barge travels with the current to the
warehouse and against the current from the warehouse. The barge's total travel time is
-==-20 hours, and it travels in still water at an average speed of 15 miles per hour.
a.
Use the given information to complete the table. Let x represent the av7age speed of the
'current
·"
1.
,
~
(~istance .Traveled
j
:
Against the
currenv
Round
Miles
Tri~
'
\
'
/
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L
I.
With the Current
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Mj'l~
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\
!
Hours
!
1
/
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/
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..
;
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!
!\ .'
140 i'
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,
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15 +X
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r/
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'-....,_~___; .---
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Averag~ Speed
Time Traveled ·
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!:
b.
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f
/
1
I
i
I
/
i
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\1
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v
'\
//
You are given that the barge's total travel time is 20 hours. Write an algebraic expression, in
terms of the number of hours th e barge travels with the current and the number of hours it
trave ls against the current, that is equivalent to 20 hours.
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-+
1'-l~i
(ts->e:xtS+ X)
c.
Ti ~
•
C'\fl~Oi)T
140~':)
Si .v X.¢ :t I ~
20~ ~.AIS"~LO (\~~-®s--xj
0S- '><)lS+x) ( l ~+x)(~~--1()
'2?. ~ -~)(
Write and solve an equation to calculate the average speed ofthe current.
2 100 -+ ~-+
z,oo
2.0~'2. S -X.,.)
~-=4Soo - zox 2
4 606 - zox 2.
Seconda ry Ill
A cost problem is a type of prob lem that involves the cost of ownership of an item over time.
Example 7:
Melinda has decided that it is time to replace her old refrigerator. She purchases a new
Energy Star certified refrigerator. Energy Star certified refrigerators use less electricity than
those that are not certified. In the long run, the Energy Star refrigerator should cost Melinda
less to operate.
Melinda purchases a new Energy Star refrigerator for $2000. The refrigerator costs
$46 per year to operate.
a.
Write an expression to represent Melinda's average annual cost of own ing the new refrigerator
for x years.
2000
b.
When Melinda's average annual cost of owning the refrigerator is less than $400, she plans to
shop for a new television. When can Me linda shop for a new t elevision?
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