Foundations of Mathematical Reasoning
Student Assignment 3.4.B
Assignment 3.4 Part B
Questions
Your Resource Algebraic Terminology contains the following information:
Formulas are a type of an algebraic equation. You have probably seen algebraic
equations in previous math classes. For example:
y = x+3
Each side of this equation is called an algebra expression.
expression
expression
y
= x+3
So “ x + 3” is an expression and “ y ” is an expression. The equal sign indicates that the
two expressions are equal, thus forming an equation.
So an equation must have an equal sign with expressions on each side.
Expression = Expression
[Note that this is exactly the same as x + 3 = y ]
An equation defines a sequence of calculations, often using algebra to shorten the
information. In the earlier example, this sequence is:
1. Start with x
2. Add three to x
3. The result is y
Notice how much shorter y = x + 3 is than the three listed steps.
The word formula is usually used to express important and non-changing relationships,
especially in contexts such as science, business, medicine, sports, or statistics. For
example, in Lesson 3.2 you used the formula for the area of a rectangle, A = L iW . This
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Foundations of Mathematical Reasoning
Student Assignment 3.4.B
is a formula because the relationship between area and the length and width of a
rectangle is always the same.
An example of an equation would be if you had a job in which you make $12 per hour.
This relationship could be written algebraically as P =12h where P is your pay in
dollars, and h is the number of hours you work. But if you get a raise, the relationship
would change. You also might call the equation a model because it models a situation
using mathematics.
Question 1: Refer to the information above to answer the following question.
1) The equation y =
x
- 5 represents which sequence of calculations?
4
Answer Choices:
Start with x , subtract 5, divide the result by 4.
Start with x , divide by 4, subtract 5 from the result.
Start with x , divide by 4, subtract from 5.
Questions 2 through 5: Lenders such as banks, credit unions, and mortgage companies make
loans. The person receiving the loan usually pays the loan off in small payments over a long
period of time. The lender earns money by charging interest, which is based on a percentage of
the amount that is borrowed. There are different types of interest. Short-term loans are often
calculated using the formula for simple interest. The total amount repaid is based on the value
of the original loan, called the principal, and the interest. The formula for the total dollars
needed to repay the loan, with interest, is found using the formula
A = P + P×r ×t
or
A = P(1+ r ×t)
where




A is the amount (total principal plus interest) required to repay the loan
P is the amount borrowed, the principal
r is the annual interest rate, quoted as a percent, but used as a decimal
t is the time, in years (so six months would be 1/2 year)
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Foundations of Mathematical Reasoning
Student Assignment 3.4.B
2) Suppose you get a loan of $5,000 at an annual interest rate of 4.25%. Use the given
information to write the formula for the total amount to be repaid in t years.
3) Make a table of values that shows the payoff amount (A) for 4 months, 6 months, 1 year, 3
years, and 6 years.
t (years)
A ($)
0
[A]
[G]
4 months
[B]
[H]
6 months
[C]
[I]
1 year
[D]
[J]
3 years
[E]
[K]
6 years
[F]
[L]
4) Estimate the time to repay the loan if you want the total payoff to be less than $7,000. Round
to the nearest tenth of a year.
5) How can these two formulas both represent simple interest? Write a brief explanation.
A = P + P×r ×t
or
A = P(1+ r ×t)
Questions 6 through 10: The tuition at a daycare center is based on family income. A reduced
tuition has a subsidy. There are three levels of tuition:



Full subsidy—the family does not pay any tuition
Partial subsidy—the family pays part of the tuition
No subsidy—the family pays the full tuition
The data for the daycare center for each age level is given below. For example, the
center receives full subsidies for 17 children in the 3-year-old class. Answer the
questions below. Round to the nearest whole percent.
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Foundations of Mathematical Reasoning
Student Assignment 3.4.B
Full Subsidy
Partial Subsidy
No Subsidy
Total
3 year-olds
17
13
8
[D]
4 year-olds
22
14
15
[E]
5 year-olds
15
16
11
[F]
Total
[A]
[B]
[C]
[G]
6) Complete the last column and last row in the table above.
7) What percentage of 3-year-olds received a full or partial subsidy?
8) What percentage of those who receive no subsidy are 5 years old?
9) What percentage of the students are 3 years old?
10) The daycare center’s income for one term comes from federal funding for the subsidy and
the tuition paid by families based on the formula below. Find the funding for the center.
I = 1,530F + 1,750P + 1,875N
where
I = total income
F = number of children receiving a full subsidy
P = number of children receiving a partial subsidy
N = number of children receiving no subsidy
Based on the data given, the daycare center’s total income for one term is $_________.
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Foundations of Mathematical Reasoning
Student Assignment 3.4.B
The following problem should be printed out, completed, and placed in
your binder.
11) Search the Internet for mathematical formulas. Record two formulas (for example,
compound interest) that you have not yet used in this course. Tell what each variable
represents and give an occupation that might use this formula.
Hints
Hint #1: Refer to Resource Algebraic Terminology, if needed.
Hint #2: Refer to Resource Algebraic Terminology, if needed.
Hint #3: Refer to Resource Algebraic Terminology, if needed.
Hint #4: Remember, you want the total payoff for the loan, which includes the total principal plus
interest, to be less than $7,000. Refer to Resource Algebraic Terminology, if needed.
Hint # 5: Refer to Resource Properties, if needed.
Hint #6: Refer to Resource Understanding Visual Displays of Information, if needed.
Hint # 7: Refer to Resource Fractions, Decimals, and Percentages, if needed.
Hint #8: Refer to Resource Fractions, Decimals, and Percentages, if needed.
Hint #9: Refer to Resource Fractions, Decimals, and Percentages, if needed.
Hint # 10: Refer to Resource Algebraic Terminology, if needed.
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