Today we will explore the Essential Question, “What is the effect
of changing a cost parameter in a given situation?"
Prices and incomes are continually rising and falling. Often
people are concerned with the effects of these changes on their
finances.
In a relation or function, an unknown quantity is called a variable
and a known quantity is called a constant or a parameter.
If the cost of x items is $5 per item, then the cost of all of the x
items in dollars is 5x. In the expression 5x, the 5 is the constant
or parameter and the x is the variable. In today's lesson we will
solve problems that involve changing a cost parameter and
determining its effect.
The strategy is:
(1) Analyze before the change occurred.
(2) Analyze after the change occurred.
(3) Answer the question.
Example 1: Vanessa spent $171 on gasoline last month when the
average cost of gasoline was $1.90 per gallon. If she uses the same
number of gallons of gasoline next month when the average cost of a
gallon is $2.55, how much more money will she spend next month on
gasoline?
(1) Before
(2) After
Total cost: 90($2.55) = $229.50
Total cost: $171
Cost per gallon:
$1.90
Gallons used: 171  1.90  90
(3) Answer the question:
Cost per Gallon: $2.55
Gallons used:
$229.50 - $171 = $58.50
The gasoline will cost Vanessa $58.50 more due to the price increase.
Example 2: A telephone company charges a flat fee of $10.00 per month
plus $0.25 per minute of usage. The bill for a customer one month was
$22.25. Then the charge per minute of usage was changed to $0.20 per
minute but the flat monthly fee did not change. If the same number of
minutes is used the next month, what will the customer's savings be, in
dollars, for the total bill?
(1) Before
(2) After
Total bill:
$22.25
Cost of minutes only:
$22.25  $10  $12.25
Cost per minute: $0.25
Total bill: $10 + $9.80 = $19.80
Minutes used: 12.25  0.25  49
Cost of minutes only:
49(0.20) = $9.80
Cost per minute: $0.20
Minutes used:
(3) Answer the question: $22.25 - $19.80 = $2.45
The customer’s savings will be $2.45 due to the price decrease.
Guided Practice Problems:
1. The cost to rent a bus for a class trip is $1080. Each student who goes
on the trip has to pay an equal share of the cost. When the bus was
ordered, 48 students had signed up to go on the trip but only 40 students
actually went. Only those who rode on the bus had to pay. How much less
would it have cost per student if all 48 who had signed up had ridden the
bus?
(1) Before
(2) After
Total bill:
Total bill: $1080
$1080
Number of students:
48
Cost per student:
$1080  48  $22.50
Number of students: 40
Cost per student:
$1080  40  $27
(3) Answer the question: $27 - $22.50 = $4.50
The savings per student would be $4.50.
2. The girls volleyball team needs to earn $2,000 to buy uniforms.
Last year they raised $1,600 by selling popcorn in tins at $8 per tin.
By how much would they have to increase the price of a tin of
popcorn in order to achieve their goal if they can sell the same
number of tins of popcorn this year?
(1) Before
Total raised:
(2) After
$1,600
Total to be raised:
$2,000
$2000  200  $10
Price of one tin: $8
Price of one tin:
Number tins sold:
1600  8  200
Number of tins sold: 200
(3) Answer the question: $10 - $8 = $2
The price of one tin of popcorn would have to increase by $2.00.
3. Last year 4 local bands played at a spring break concert for a local
school. Tickets were sold at $5 each and the money was used to pay each
band $100 with the remaining $2725 going to the school's recreation fund.
Next year 6 bands will play and be paid $100 each. If the tickets are still $5
each, how many tickets will have to be sold next year in order to raise the
same amount of money as last year for the school's recreation fund?
(1) Before
(2) After
Number of bands:
4
Number of bands: 6
Amount paid each band: $100
Amount paid each band: $100
Total to bands: 4($100) = $400
Total to bands: 6($100) = $600
Amount to school rec fund: $2725 Amount to school rec fund: $2725
Total amount made:
$400 + $2725 = $3125
Total needed:
$600 + $2725 = $3325
(3) Answer the question: 3325  5  665 tickets
Next year 665 tickets will have to be sold.
4. A telephone company charges a flat fee of $20.00 per month plus $0.25
per minute of usage. The bill for a customer one month was $70.00. Then
the charge per minute of usage was changed to $0.20 per minute but the flat
monthly fee did not change. If the same number of minutes is used the next
month, what will the customer's savings be, in dollars, for the total bill?
(1) Before
Total bill:
$70.00
Cost of minutes only:
$70  $20  $50
Cost per minute: $0.25
Minutes used: 50  0.25  200
(2) After
Total bill:
$20 + $40 = $60
Cost of minutes only:
200(0.20) = $40
Cost per minute: $0.20
Minutes used:
200
(3) Answer the question: $70 - $60 = $10
The customers savings will be $10 due to the price decrease.
Independent Practice Problems:
1. A pizza delivery person is paid $5 per hour worked plus $3 for each pizza
delivered. Last week he worked 35 hours and earned $265. This week he
wants to earn $295. If he works the same number of hours this week, how
many more pizzas must he deliver this week than he delivered last week?
(2) After
(1) Before
Pay per hour: $5
Pay per hour: $5
Hours worked: 35
Hours worked: 35
Pay for hours worked: $5(35)=$175
Pay for hours worked: $5(35)=$175
Pay for pizzas: $265 - $175 = $90
Pay for pizzas: $295 - $175 = $120
Number of pizzas delivered: 90  3  30
Number of pizzas delivered:
120  3  40
(3) Answer the question: 40 – 30 = 10 more pizzas
2. A newspaper carrier earns $0.12 for each paper he delivers daily. He
now delivers papers to 100 customers each day. If he increases the daily
number of customers to120, how much more money would he earn each
day? (1) Before
(2) After
Pay per paper: $0.12
Pay per paper: $0.12
Papers delivered: 120
Papers delivered: 100
Amount earned per day: $0.12(100)=$12 Amount earned per day:
$0.12(120)=$14.40
(3) Answer the question: $14.40 - $12 = $2.40 more he will earn each day.