TAX TYPES IN DETAIL - FLAT TAX,
PROGRESSIVE TAX AND REGRESSIVE TAX
Study Notes
2.2 Learning Objectives
Determine a flat tax amount
Governments collect taxes to pay for the services they provide. In the United
States, federal income taxes help fund the military, the environmental
protection agency, and thousands of other programs. Property taxes help
fund schools. Gasoline taxes help pay for road improvements. While very few
people enjoy paying taxes, they are necessary to pay for the services we all
depend upon.
Taxes can be computed in a variety of ways, but are typically computed as a
percentage of a sale, of one’s income, or of one’s assets.
Example 1
The sales tax rate in a city is 9.3%. How much sales tax will you pay on a
$140 purchase?
Solution
The sales tax will be 9.3% of $140. To compute this, we multiply $140 by the
percent written as a decimal:
. $140(0.093) = $13.02
When taxes are not given as a fixed percentage rate, sometimes it is
necessary to calculate the effective rate.
Effective rate
The effective tax rate is the equivalent percent rate of the tax paid out of the
dollar amount the tax is based on.
Example 2
Joan paid $3,200 in property taxes on her house valued at $215,000 last
year. What is the effective tax rate?
Solution
We can compute the equivalent percentage: 3200 , or about 1.49%
effective rate. 215000 = 0.01488
Taxes are often referred to as progressive, regressive, or flat.
Tax categories
A flat tax, or proportional tax, charges a constant percentage rate.
A progressive tax increases the percent rate as the base amount increases.
A regressive tax decreases the percent rate as the base amount increases.
Example 3
The United States federal income tax on earned wages is an example of a
progressive tax. People with a higher wage income pay a higher percent tax
on their income.
Solution
For a single person in 2011, adjusted gross income (income after deductions)
under $8,500 was taxed at 10%. Income over $8,500 but under $34,500 was
taxed at 15%.
A person earning $10,000 would pay 10% on the portion of their income
under $8,500, and 15% on the income over $8,500, so they’d pay:
2.1.1 146717 8500(0.10) = 850 10% of 8500 1500(0.15) = 225 15%
of the remaining $1500 of income Total tax: = $1075
The effective tax rate paid is 1075 = 10.75%
A person earning $30,000 would also pay 10% on the portion of their income
under $8,500, and 15% on the income over $8,500, so they’d pay:
8500(0.10) = 85010% of 8500 21500(0.15) = 3225 15% of the remaining
$21500 of income
Total tax: = $4075
The effective tax rate paid is . 30000 = 13.58% Notice
that the effective rate has increased with income, showing this is a
progressive tax.
Example 4
A gasoline tax is a flat tax when considered in terms of consumption, a tax
of, say, $0.30 per gallon is proportional to the
amount of gasoline purchased. Someone buying 10 gallons of gas at $4 a
gallon would pay $3 in tax, which is$3 . $40 = 7.5% Someone
buying 30 gallons of gas at $4 a gallon would pay $9 in tax, which is $9
, the same effective rate. $120 = 7.5% Solution
However, in terms of income, a gasoline tax is often considered a regressive
tax. It is likely that someone earning $30,000 a year and someone earning
$60,000 a year will drive about the same amount. If both pay $60 in gasoline
taxes over a year, the person earning $30,000 has paid 0.2% of their
income, while the person earning $60,000 has paid 0.1% of their income in
gas taxes.
Try it Now 1
A sales tax is a fixed percentage tax on a person’s purchases. Is this a flat,
progressive, or regressive tax?
Answer
While sales tax is a flat percentage rate, it is often considered a regressive
tax for the same reasons as the gasoline tax.
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2.1.2 146717 2.1.1: Income Taxation
2.2.1 Learning Objectives
Compare flat tax, modified flat tax, and progressive tax
Many people have proposed various revisions to the income tax collection in
the United States. Some, for example, have claimed that a flat tax would be
fairer. Others call for revisions to how different types of income are taxed,
since currently investment income is taxed at a different rate than wage
income.
The following two projects will allow you to explore some of these ideas and
draw your own conclusions.
Project 1: Flat tax, Modified Flat Tax, and Progressive Tax.
Imagine the country is made up of 100 households. The federal government
needs to collect $800,000 in income taxes to be able to function. The
population consists of 6 groups:
Group A: 20 households that earn $12,000 each
Group B: 20 households that earn $29,000 each
Group C: 20 households that earn $50,000 each
Group D: 20 households that earn $79,000 each
Group E: 15 households that earn $129,000 each
Group F: 5 households that earn $295,000 each
This scenario is roughly proportional to the actual United States population
and tax needs. We are going to determine new income tax rates.
The first proposal we’ll consider is a flat tax – one where every income group
is taxed at the same percentage tax rate.
Determine the total income for the population (all 100 people together)
Determine what flat tax rate would be necessary to collect enough
money.
The second proposal we’ll consider is a modified flat-tax plan, where
everyone only pays taxes on any income over $20,000. So, everyone in
group A will pay no taxes. Everyone in group B will pay taxes only on $9,000.
Determine the total taxable income for the whole population
Determine what flat tax rate would be necessary to collect enough
money in this modified system
Complete this table for both the plans
The third proposal we’ll consider is a progressive tax, where lower income
groups are taxed at a lower percent rate, and higher income groups are
taxed at a higher percent rate. For simplicity, we’re going to assume that a
household is taxed at the same rate on
2.1.1.1 146718 all their income.
Set progressive tax rates for each income group to bring in enough
money. There is no one right answer here – just make sure you bring in
enough money!
Discretionary income is the income people have left over after paying
for necessities like rent, food, transportation, etc. The cost of basic expenses
does increase with income, since housing and car costs are higher, however
usually not proportionally. For each income group, estimate their essential
expenses, and calculate their discretionary income. Then compute the
effective tax rate for each plan relative to discretionary income rather than
income.
8) Which plan seems the most fair to you? Which plan seems the least fair to
you? Why?
Project 2: Calculating Taxes.
Visit www.irs.gov, and download the most recent version of forms 1040, and
schedules A, B, C, and D.
Scenario 1: Calculate the taxes for someone who earned $60,000 in standard
wage income (W-2 income), has no dependents, and takes the standard
deduction.
Scenario 2: Calculate the taxes for someone who earned $20,000 in standard
wage income, $40,000 in qualified dividends, has no dependents, and takes
the standard deduction. (Qualified dividends are earnings on certain
investments such as stocks.)
Scenario 3: Calculate the taxes for someone who earned $60,000 in small
business income, has no dependents, and takes the standard deduction.
Based on these three scenarios, what are your impressions of how the
income tax system treats these different forms of income (wage, dividends,
and business income)?
Scenario 4: To get a more realistic sense for calculating taxes, you’ll need to
consider itemized deductions. Calculate the income taxes for someone with
the income and expenses listed below.
2.1.1.2 146718 Married with 2 children, filing jointly
Wage income: $50,000 combined
Paid sales tax in Washington State
Property taxes paid: $3200
Home mortgage interest paid: $4800
Charitable gifts: $1200
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2.1.1.3 146718 2.1.2: Exercises
Section 2.2.2 Exercises
Laars earns an annual salary of $60,000. Determine his gross earnings
per pay period under each of the following payment frequencies:
Monthly
Semi-monthly
Biweekly
Weekly
A city has a sales tax of 10.25%. What would be the sales tax on a
$2575.57 purchase? Round to the nearest cent.
The tax on gas in California went up to $0.511 cents per gallon in July
2021. If an average gallon of gas costs $4.31 per gallon, what is the effective
tax rate of the gas tax? Round the percent to two decimal places.
Samantha and Anahi carpool to work to split the cost of gas. They each
pay $120 per month to cover the cost of the commute. If Samantha makes
$3200 a month and Anahi makes $4100, what is the effective tax rate on
their income? Would this be considered a progressive, regressive, or flat tax
rate? Alex bought a new gaming system with several games that retailed at
$435 before tax. They noticed the tax charged was $42.41. What was the
sales tax rate of that city? Use the 2021 Federal Income Tax Brackets at
the end of this page for the following questions.
Determine the federal income tax on $23,325 for a single individual.
Determine the federal income tax on $152,904 for married individuals
filing jointly.
Determine the federal income tax on $102,350 for a single individual.
Adrian made $40,525 in 2021 and is filing as single. Melany made
$40,526 in 2021 and is also filing as single. Melany thinks that since both she
and Adrian make approximately the same amount of money, they should be
in the same tax bracket. Using the tax bracket chart, determine if Melany’s
claim is true. Find the difference in the amount of taxes they paid.
Brenda made $41,685 in 2021 and is filing single. Luz made $51,892
filing as Head of Household. Brenda claims to Luz that since she is making
less money, that she will pay less in taxes. Using the tax chart below,
determine if Brenda’s claim is true. Find the difference in the amount of
taxes they paid and state who paid more. What is each of their effective tax
rates?
A lawmaker proposed that everyone will pay the same amount of
taxes. Fill in the following table by writing out each rate. Determine if this is
an example of a progressive, regressive, or flat tax rate.
Rate For Single Individuals Amount of Tax Paid $6,500
$2,500 $9,951 $2,500 $40,526 $2,500
$86,376 $2,500 $164,926 $2,500
$209,426 $2,500 $523,601 $2,500
2021 Federal Income Tax Brackets and Rates for Single Filers, Married
Couples Filing Jointly, and Heads of Households
Rate For Single Individuals For Married Individuals Filing For Heads of
Households Joint Returns 10% of any
amount up to $9,950 of any amount up to $19,900 of any amount up to
$14,200 12% of the portion above $9,950 up to of the
portion above $19,900 up of the portion above $14,200 up to $40,525
plus $995 to $81,050 plus $1990 $54,200 plus $1420 22%
of the portion above $40,525 up of the portion above $81,050 up to
of the portion above $54,200 up to to $86,375 plus $4664 $172,750
plus $9328 $86,350 plus $6220
2.1.2.1 146719
Rate For Single Individuals For Married Individuals Filing For Heads of
Households Joint Returns
24% of the portion above
$86,375 up to of the portion above $172,750 to of the portion above
$86,350 up to $164,925 plus $14751 $329,850 plus $29502
$164,900 plus $13296 32% of the
portion above $164,925 up of the portion above $329,850 up of the
portion above $164,900 up to $209,425 plus $33603 to
$418,850 plus $ 67206 to $209,400 plus $32148
35% of the portion above
$209,425 up of the portion above $418,850 up of the portion above
$209,400 up to $523,600 plus $47843 to $628,300 plus
$95686 to $523,600 plus $46388
37% of the portion above $523,600 or of the portion above
$628,300 or of the portion above $523,600 or more plus
$157804 more plus $168994 more plus 156358
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2.1.2.2 146719 2.2: Modeling with Linear Functions
When modeling scenarios with a linear function and solving problems
involving quantities changing linearly, we typically follow the same problem
solving strategies that we would use for any type of function:
Problem Solving Strategy
Identify changing quantities, and then carefully and clearly define
descriptive variables to represent those quantities. When appropriate, sketch
a picture or define a coordinate system. Carefully read the problem to
identify important information. Look for information giving values for the
variables, or values for parts of the functional model, like slope and initial
value. Carefully read the problem to identify what we are trying to find,
identify, solve, or interpret.
Identify a solution pathway from the provided information to what we
are trying to find. Often this will involve checking and tracking units, building
a table or even finding a formula for the function being used to model the
problem. When needed, find a formula for the function.
Solve or evaluate using the formula you found for the desired
quantities.
Reflect on whether your answer is reasonable for the given situation
and whether it makes sense mathematically.
Clearly convey your result using appropriate units, and answer in full
sentences when appropriate.
Example
Emily saved up $3500 for her summer visit to Seattle. She anticipates
spending $400 each week on rent, food, and fun. Find and interpret the
horizontal intercept and determine a reasonable domain and range for this
function.
Solution
In the problem, there are two changing quantities: time and money. The
amount of money she has remaining while on vacation depends on how long
she stays. We can define our variables, including units.
Output: , money remaining, in dollars
Input: , time, in weeks t Reading the problem, we identify two important
values. The first, $3500, is the initial value for . The other value appears to
be a rate of change – the units of dollars per week match the units of our
output variable divided by our input variable. She is spending money each
week, so you should recognize that the amount of money remaining is
decreasing each week and the slope is negative.
To answer the first question, looking for the horizontal intercept, it would be
helpful to have an equation modeling this scenario. Using the intercept and
slope provided in the problem, we can write the equation:
M(t) = 3500 − 400t
To find the horizontal intercept, we set the output to zero, and solve for the
input:
3500 − 400t 3500 400 = 8.75
The horizontal intercept is 8.75 weeks. Since this represents the input value
where the output will be zero, interpreting this, we could say: Emily will have
no money left after 8.75 weeks.
When modeling any real life scenario with functions, there is typically a
limited domain over which that model will be valid – almost no trend
continues indefinitely. In this case, it certainly doesn’t make sense to talk
about input values less than zero. It is also likely that this model is not valid
after the horizontal intercept (unless Emily’s going to start using a credit
card and go into debt).
The domain represents the set of input values and so the reasonable domain
for this function is
0 ≤ t ≤ 8.75
. However, in a real world scenario, the rental might be weekly or nightly.
She may not be able to stay a partial week and so all options should be
considered. Emily could stay in Seattle for 0 to 8 full weeks (and a couple of
days), but would have to go
2.2.1 146720 into debt to stay 9 full weeks, so restricted to whole
weeks, a reasonable domain without going in to debt would be , or if she
went into debt to finish out the last week.0 ≤ t ≤ 8 0 ≤ t ≤ 9
The range represents the set of output values and she starts with $3500 and
ends with $0 after 8.75 weeks so the corresponding
range is . If we limit the rental to whole weeks, however, the range would
change. If she left after 8 weeks 0 ≤ M(t) ≤ 3500 dollars left after 8
because she didn’t have enough to stay for a full 9 weeks, she would have
weeks, giving a range of M(8) = 3500 −400(8) = $300
. If she wanted to stay the full 9 weeks she would be $100 in debt
giving a range of 300 ≤ M(t) ≤ 3500 . −100 ≤ M(t) ≤ 3500
Most importantly remember that domain and range are tied together, and
what ever you decide is most appropriate for the domain (the independent
variable) will dictate the requirements for the range (the dependent
variable).
Exercise
A database manager is loading a large table from backups. Getting
impatient, she notices 1.2 million rows had been loaded. Ten minutes later,
2.5 million rows had been loaded. How much longer will she have to wait for
all 80 million rows to load?
Answer
Letting be the number of minutes since she got impatient, and N be the
number rows loaded, in millions, we have two t points: (0, 1.2) and (10, 2.5).
The slope is 2.5 −1.2 1.3 million rows per
minute. m = 10 − 0 = 10 = 0.13 We know the
intercept, so we can write the equation: N = 0.13t
+1.2 To determine how long she will have to wait, we need to solve for
when . N =80
= 0.13t +1.2 = 80
0.13t = 78.8
78.8 t = 0.13 = 606
. She’ll have to wait another 606 minutes, about 10 hours.
Example
Jamal is choosing between two moving companies. The first, U-Haul, charges
an up-front fee of $20, then 59 cents a mile. The second, Budget, charges an
up-front fee of $16, then 63 cents a mile(Rates retrieved Aug 2, 2010 from
www.budgettruck.com and http://www.uhaul.com/). When will U-Haul be the
better choice for Jamal?
Solution
The two important quantities in this problem are the cost, and the number of
miles that are driven. Since we have two companies to consider, we will
define two functions:
Input:, miles driven
Outputs:
cost, in dollars, for renting from U-Haul Y (m) cost, in dollars, for
renting from Budget B(m) Reading the problem carefully, it appears that we
were given an initial cost and a rate of change for each company. Since our
outputs are measured in dollars but the costs per mile given in the problem
are in cents, we will need to convert these quantities to match our desired
units: $0.59 a mile for U-Haul, and $0.63 a mile for Budget.
2.2.2 146720 Y (m) Y (m) Looking to what we’re trying to find, we
want to know when U-Haul will be the better choice. Since all we have to
make that
decision from is the costs, we are looking for when U-Haul will cost less, or
when . The solution pathway will Y (m) < B(m) function is smaller. lead
us to find the equations for the two functions, find the intersection, then look
to see where the Using the rates of change and initial charges, we can write
the equations:
(m) = 20 +0.59m
B(m) = 16 +0.63m
These graphs are sketched to the right, with Y(m) drawn dashed.
To find the intersection, we set the equations equal and solve:
Y (m) B(m) 20 +0.59m 16 +0.63m 0.04m
This tells us that the cost from the two companies will be the same if 100
miles are driven. Either by looking at the graph, or noting that is growing
at a slower rate, we can conclude that U-Haul will be the cheaper price when
more than 100 miles
are driven.
Example
A town’s population has been growing linearly. In 2004 the population was
6,200. By 2009 the population had grown to 8,100. If this trend continues,
Predict the population in 2013
When will the population reach 15000?
Solution
The two changing quantities are the population and time. While we could use
the actual year value as the input quantity, doing so tends to lead to very
ugly equations, since the vertical intercept would correspond to the year 0,
more than 2000 years ago!
To make things a little nicer, and to make our lives easier too, we will define
our input as years since 2004:
Input: , years since 2004
Output: , the town’s population (t) The problem gives us two input-output
pairs. Converting them to match our defined variables, the year 2004 would
correspond
to , giving the point (0, 6200). Notice that through our clever choice of
variable definition, we have “given” ourselves the vertical intercept of the
function. The year 2009 would correspond to , giving the point (5, 8100).
2.2.3
146720 To predict the population in 2013 ( ), we would need an
equation for the population. Likewise, to find when the population would
reach 15000, we would need to solve for the input that would provide an
output of 15000. Either way, we need an equation. To find it, we start by
calculating the rate of change:
8100 −6200 m =
5 − 0 = = 380 people per year Since we already know the vertical intercept
of the line, we can immediately write the equation:
P (t) = 6200 +380t To predict the population in 2013, we evaluate our
function at t = 9
P(9) = 6200 +380(9) = 9620
If the trend continues, our model predicts a population of 9,620 in 2013.
To find when the population will reach 15,000, we can set and solve for . P
(t) = 15000
6200 +380t
23.158 Our model predicts the population will reach 15,000 in a little
more than 23 years after 2004, or somewhere around the year 2027.
Example
Anna and Emanuel start at the same intersection. Anna walks east at 4 miles
per hour while Emanuel walks south at 3 miles per hour. They are
communicating with a two-way radio with a range of 2 miles. How long after
they start walking will they fall out of radio contact?
Solution
In essence, we can partially answer this question by saying they will fall out
of radio contact when they are 2 miles apart, which leads us to ask a new
question: how long will it take them to be 2 miles apart?
In this problem, our changing quantities are time and the two peoples’
positions, but ultimately we need to know how long will it take for them to be
2 miles apart. We can see that time will be our input variable, so we’ll define
Input: , time in hours. t Since it is not obvious how to define our output
variables, we’ll start by drawing a picture.
Because of the complexity of this question, it may be helpful to introduce
some intermediary variables. These are quantities that we aren’t directly
interested in, but seem important to the problem. For this problem, Anna’s
and Emanuel’s distances from the starting point seem important. To notate
these, we are going to define a coordinate system, putting the “starting
point” at the intersection where they both started, then we’re going to
introduce a variable, , to represent Anna’s position, and define it A to
be a measurement from the starting point, in the eastward direction.
Likewise, we’ll introduce a variable, , to represent E Emanuel’s position,
measured from the starting point in the southward direction. Note that in
defining the coordinate system we specified both the origin, or starting point,
of the measurement, as well as the direction of measure. While we’re at it,
we’ll define a third variable, , to be the measurement of the distance
between Anna and Emanuel. Showing the variables on the picture is often
helpful: 2.2.4 146720
Looking at the variables on the picture, we remember we need to know how
long it takes for , the distance between them, to equal 2 miles.
Seeing this picture we remember that in order to find the distance between
the two, we can use the Pythagorean Theorem, a property of right triangles.
From here, we can now look back at the problem for relevant information.
Anna is walking 4 miles per hour, and Emanuel is walking 3 miles per hour,
which are rates of change. Using those, we can write formulas for the
distance each has walked. They both start at the same intersection and so
when , the distance travelled by each person should also be 0, so given the
rate for each, and the initial value for each, we get: t = 0 A(t) = 4t
E(t) = 3t Using the Pythagorean theorem we get:
D(t)2 = A(t)2 + E(t)2
substitute in the function formulas
D(t)2 = (4t)2 +(3t)2 = 16 2 +9 2 = 25 2
solve for
D(t)
using the square root
D(t) = ±√
2
= ±5|t|
Since in this scenario we are only considering positive values of t and our
distance
D(t)
will always be positive, we can simplify this answer to D(t) = 5t
Interestingly, the distance between them is also a linear function. Using it,
we can now answer the question of when the distance between them will
reach 2 miles:
D(t)
= 0.4
They will fall out of radio contact in 0.4 hours, or 24 minutes.
Example
There is currently a straight road leading from the town of Westborough to a
town 30 miles east and 10 miles north. Partway down this road, it junctions
with a second road, perpendicular to the first, leading to the town of
Eastborough. If the town of Eastborough is located 20 miles directly east of
the town of Westborough, how far is the road junction from Westborough?
Solution
It might help here to draw a picture of the situation. It would then be helpful
to introduce a coordinate system. While we could place the origin anywhere,
placing it at Westborough seems convenient. This puts the other town at
coordinates (30, 10), and Eastborough at (20, 0).
2.2.5 146720 Using this point along with the origin, we can find the
slope of the line from
Westborough to the other town: 10 − 0 . This gives the
equation of m = 30 − 0 =
the road from Westborough to the other town to be
. W(x) = 3 x From
this, we can determine the perpendicular road to Eastborough will have slope
. Since the town of Eastborough is at the point (20, 0), we can = −3
find the equation:
E(x) = −3x + b
plug in the point (20, 0)
0 = −3(20) + b
b = 60
E(x) = −3x +60
We can now find the coordinates of the junction of the roads by finding the
intersection of these lines. Setting them equal,
= −3x +60
10 = 60
= 18 Substituting this back into (x)
y = W(18) = 3 (18) = 6
The roads intersect at the point (18, 6). Using the distance formula, we can
now find the distance from Westborough to the junction:
dist = √ (18 − 0)2 +(6 − 0)2 ≈ 18.934 miles
Important Topics of this Section
The problem solving process
Identify changing quantities, and then carefully and clearly define
descriptive variables to represent those quantities. When appropriate, sketch
a picture or define a coordinate system. Carefully read the problem to
identify important information. Look for information giving values for the
variables, or values for parts of the functional model, like slope and initial
value. Carefully read the problem to identify what we are trying to find,
identify, solve, or interpret.
Identify a solution pathway from the provided information to what we
are trying to find. Often this will involve checking and tracking units, building
a table or even finding a formula for the function being used to model the
problem. When needed, find a formula for the function.
2.2.6 146720 Solve or evaluate using the formula you found for the
desired quantities.
Reflect on whether your answer is reasonable for the given situation
and whether it makes sense mathematically.
Clearly convey your result using appropriate units, and answer in full
sentences when appropriate.
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SA 4.0 license and was authored, remixed, and/or curated by David Lippman
& Melonie Rasmussen (The OpenTextBookStore) via source content that was
edited to the style and standards of the LibreTexts platform; a detailed edit
history is available upon request.
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2.2.7 146720 2.2E: Modeling with Linear Functions (Exercises)
section 2.3 exercise
In 2004, a school population was 1001. By 2008 the population had
grown to 1697. Assume the population is changing linearly.
How much did the population grow between the year 2004 and 2008?
How long did it take the population to grow from 1001 students to
1697 students?
What is the average population growth per year?
What was the population in the year 2000?
Find an equation for the population, P, of the school t years after 2000.
Using your equation, predict the population of the school in 2011.
In 2003, a town’s population was 1431. By 2007 the population had
grown to 2134. Assume the population is changing linearly.
How much did the population grow between the year 2003 and 2007?
How long did it take the population to grow from 1431 people to 2134?
What is the average population growth per year?
What was the population in the year 2000?
e. Find an equation for the population, , of the town years after 2000. f.
Using your equation, predict the population of the town in 2014.
A phone company has a monthly cellular plan where a customer pays a
flat monthly fee and then a certain amount of money per minute used on the
phone. If a customer uses 410 minutes, the monthly cost will be $71.50. If
the customer uses 720 minutes, the monthly cost will be $118.
a. Find a linear equation for the monthly cost of the cell plan as a function of
, the number of monthly minutes used.
b. Interpret the slope and vertical intercept of the equation.
c. Use your equation to find the total monthly cost if 687 minutes are used.
A phone company has a monthly cellular data plan where a customer
pays a flat monthly fee and then a certain amount of money per megabyte
(MB) of data used on the phone. If a customer uses 20 MB, the monthly cost
will be $11.20. If the customer uses 130 MB, the monthly cost will be $17.80.
a. Find a linear equation for the monthly cost of the data plan as a function
of , the number of MB used.
b. Interpret the slope and vertical intercept of the equation.
c. Use your equation to find the total monthly cost if 250 MB are used.
In 1991, the moose population in a park was measured to be 4360. By
1999, the population was measured again to be 5880. If the population
continues to change linearly,
a. Find a formula for the moose population, . P b. What does your model
predict the moose population to be in 2003?
In 2003, the owl population in a park was measured to be 340. By
2007, the population was measured again to be 285. If the population
continues to change linearly,
a. Find a formula for the owl population, .
b. What does your model predict the owl population to be in 2012?
The Federal Helium Reserve held about 16 billion cubic feet of helium
in 2010, and is being depleted by about 2.1 billion cubic feet each year.
a. Give a linear equation for the remaining federal helium reserves, , in terms
of , the number of years since 2010. b. In 2015, what will the helium reserves
be? c. If the rate of depletion doesn’t change, when will the Federal Helium
Reserve be depleted?
Suppose the world’s current oil reserves are 1820 billion barrels. If, on
average, the total reserves is decreasing by 25 billion barrels of oil each
year:
2.2E.1 146721 a. Give a linear equation for the remaining oil
reserves, , in terms of , the number of years since now.
b. Seven years from now, what will the oil reserves be?
c. If the rate of depletion isn’t change, when will the world’s oil reserves be
depleted?
9. You are choosing between two different prepaid cell phone plans. The first
plan charges a rate of 26 cents per minute. The
second plan charges a monthly fee of $19.95 11 cents per minute. How
many minutes would you have to use in a month plus
in order for the second plan to be preferable?
You are choosing between two different window washing companies.
The first charges $5 per window. The second charges a base fee of $40 plus
$3 per window. How many windows would you need to have for the second
company to be preferable? When hired at a new job selling jewelry, you
are given two pay options:
Option A: Base salary of $17,000 a year, with a commission of 12% of your
sales
Option B: Base salary of $20,000 a year, with a commission of 5% of your
sales
How much jewelry would you need to sell for option A to produce a larger
income?
12. When hired at a new job selling electronics, you are given two pay
options:
Option A: Base salary of $14,000 a year, with a commission of 10% of your
sales
Option B: Base salary of $19,000 a year, with a commission of 4% of your
sales
How much electronics would you need to sell for option A to produce a larger
income?
13. Find the area of a triangle bounded by the axis, the line 6
, and the line perpendicular to that passes through the origin.
y f(x) = 9 − 7 x f(x)
14. Find the area of a triangle bounded
by the axis, the line , and the line
perpendicular to that passes
through the origin. x f(x) = 12 − 3 x f(x)
15. Find the area of a
parallelogram bounded by the axis, the line , the line
, and the line parallel to passing through (2, 7) y
f(x) = 1 +2x
f(x)
16. Find the area of a parallelogram bounded by the axis,
the line , the line , and the line parallel to passing
through (6, 1) g(x) = 2 f(x) = 3x f(x)
17. If and , then the
line cuts off a triangle from the first quadrant. Express the area of
that f(x) = b + mx
triangle in terms of m and b. [UW]
18. Find the value of m so the linesand and the
-axis form a triangle with an area of 10. [UW] f(x) = mx +5
g(x) = x y The median home values in
Mississippi and Hawaii (adjusted for inflation) are shown below. If we assume
that the house values are changing linearly,
Year
1950
2000
Mississippi
25200
71400
Hawaii
74400
272700
a. In which state have home values increased at a higher rate?
b. If these trends were to continue, what would be the median home value in
Mississippi in 2010?
c. If we assume the linear trend existed before 1950 and continues after
2000, the two states’ median house values will be (or were) equal in what
year? (The answer might be absurd)
The median home value ins Indiana and Alabama (adjusted for
inflation) are shown below. If we assume that the house values are changing
linearly,
Year
1950
2000
Indiana
37700
94300
Alabama
27100
85100
2.2E.2 146721 a. In which state have home values increased at
a higher rate?
b. If these trends were to continue, what would be the median home value in
Indiana in 2010?
c. If we assume the linear trend existed before 1950 and continues after
2000, the two states’ median house values will be (or were) equal in what
year? (The answer might be absurd)
Pam is taking a train from the town of Rome to the town of Florence.
Rome is located 30 miles due West of the town of Paris. Florence is 25 miles
East, and 45 miles North of Rome. On her trip, how close does Pam get to
Paris? [UW]
You’re flying from Joint Base Lewis-McChord (JBLM) to an undisclosed
location 226 km south and 230 km east. Mt. Rainier is located approximately
56 km east and 40 km south of JBLM. If you are flying at a constant speed of
800 km/hr, how long after you depart JBLM will you be the closest to Mt.
Rainier?
Answer
1a. 696 people
b. 4 years
c. 174 people per year
d. 305 people
e. P (t) = 305 +174t f. 2219 people.
3a. C(x) = 0.15x +10 b. The flat monthly fee is $10 and there is an additional
$0.15 fee for each additional minute used c. $113.05
5a. P (t) = 190t +4170 b. 6640 moose
7a. R(t) = 16 − 2.1t b. 5.5 billion cubic feet
c. During the year 2017
9. More than 133 minutes
More than $42857.14 worth of jewelry
20.012 square units
6 square units
b2 A = − 2m 19a. Hawaii
$80640
During the year 1933
21. 26.225 miles
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2.2E.3 146721 2.3: Simple Interest
2.3 Learning Objectives
Find the simple interest for a loan
Discussing interest starts with the principal, or amount your account starts
with. This could be a starting investment, or the starting amount of a loan.
Interest, in its most simple form, is calculated as a percent of the principal.
For example, if you borrowed $100 from a friend and agree to repay it with
5% interest, then the amount of interest you would pay would just be 5% of
100: The total amount you would repay would be $105, the original principal
plus the interest. $100(0.05) = $5
Simple One-time Interest
I = 0 r (2.3.1) A= 0+I= 0 + 0 r = 0 (1 + r) (2.3.2) where
is the interest is the end amount: principal plus interest
is the principal (starting amount) 0 is the
interest rate (in decimal form. Example: ) Example 1
A friend asks to borrow $300 and agrees to repay it in 30 days with 3%
interest. How much interest will you earn? Solution
0 = $300 the principal r = 0.03 3% rate = $300(0.03) = $9. You will
earn $9 interest.
One-time simple interest is only common for extremely short-term loans. For
longer term loans, it is common for interest to be paid on a daily, monthly,
quarterly, or annual basis. In that case, interest would be earned regularly.
For example, bonds are essentially a loan made to the bond issuer (a
company or government) by you, the bond holder. In return for the loan, the
issuer agrees to pay interest, often annually. Bonds have a maturity date, at
which time the issuer pays back the original bond value.
Example 2
Suppose your city is building a new park, and issues bonds to raise the
money to build it. You obtain a $1,000 bond that pays 5% interest annually
that matures in 5 years. How much interest will you earn?
Solution
Each year, you would earn 5% interest: in interest. So over the course of
five years, you would earn a total $1000(0.05) = $50 of $250 in interest.
When the bond matures, you would receive back the $1,000 you originally
paid, leaving you with a total of $1,250.
We can generalize this idea of simple interest over time.
Simple Interest over Time
I = P0 rt
A = 0 + I = 0 + 0 rt = 0 (1 + rt) where
is the interest I
2.3.1 146464 is the end amount: principal plus interest
is the principal (starting amount) 0 is the interest rate in decimal form
is time
The units of measurement (years, months, etc.) for the time should match
the time period for the interest rate.
Definition: Annual Percentage Rate
Interest rates are usually given as an annual percentage rate (APR) – the
total interest that will be paid in the year. If the interest is paid in smaller
time increments, the APR will be divided up.
For example, a APR paid monthly would be divided into twelve 6% 0.5%
A annual rate paid quarterly would be divided into four payments. 4%
1%
payments.
Example 3 Treasury Notes (T-notes) are bonds issued by the federal
government to cover its expenses. Suppose you obtain a $1,000 T-note with
a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much
interest will you
earn?
Solution
Since interest is being paid semi-annually (twice a year), the 4% interest will
be divided into two 2% payments.
0 = $1000 the principal 2% rate t = 8 4 years = 8 half-years
= $1000(0.02)(8) = $160. You will earn $160 interest total over the
four years.
Try it Now 1
A loan company charges $30 interest for a one month loan of $500. Find the
annual interest rate they are charging.
Answer
of interest = $30 principal = $500 unknown (annual) r = month (1/12
of a year) t = 1
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2.3.2 146464 2.3.1: Exercises
Section 2.3.1 Exercises
You borrow $350 from your aunt and promise to pay her back in one
month with 3% interest. How much extra will she get when you pay her
back? You were charged $487.50 in interest at the end of one year for
money you borrowed from a bank that charged 6.5% interest. How much did
you borrow? You had to pay $618.12 in interest on a loan for $10,750
over the first year. What was the interest rate used? Round the percent to
two decimal places. A friend lends you $200 for a week, which you agree
to repay with 5% one-time interest. How much will you have to repay?
Your brother deposits $1200 in a savings account that pays 4% simple
interest. How much will he have in the account in 4 years if he doesn't touch
it. Suppose you obtain a $3,000 T-note with a 3% annual rate, paid
quarterly, with maturity in 5 years. How much interest will you earn?
Suppose you take a cash advance of $750 on your credit card. The
service charge is $5.50. You pay off the cash advance after 45 days. Your
credit card statement shows the interest on the cash advance was $29.59.
For the interest alone, what is the APR? If you buy a 3-year CD that pays
2.4% simple interest for $75,000, what will it be worth at the end of its term?
If you invest $1500 at 4.25% simple interest and the investment is
later worth $1553.13, what was the term?
A T-bill is a type of bond that is sold at a discount over the face value.
For example, suppose you buy a 13-week T-bill with a face value of $10,000
for $9,800. This means that in 13 weeks, the government will give you the
face value, earning you $200. What annual interest rate have you earned?
Suppose you are looking to buy a $5000 face value 26-week T-bill. If
you want to earn at least 1% annual interest, what is the most you should
pay for the T-bill?
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