MTH 098 Practice Exam 1: Step-by-Step Algebra Guidance

Q1. Heather’s Exam Grades: Mean, Median, and Required Fifth Exam Score

Background

Topic: Measures of Central Tendency (Mean and Median), Solving for an Unknown

This question tests your understanding of how to calculate the mean and median of a set of numbers, and how to work backwards to find a required value to achieve a target mean.

Key Terms and Formulas:

• Mean (Average): $\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

• Median: The middle value when the data is arranged in order.

Step-by-Step Guidance

1. List Heather’s first four exam grades: 95, 88, 82, 85.

2. To find the mean, add all four grades and divide by 4: $\text{Mean} = \frac{95 + 88 + 82 + 85}{4}$

3. To find the median, first arrange the grades in order from least to greatest, then find the middle value (or average the two middle values if there is an even number).

4. To find the minimum grade needed on the fifth exam for an average of 80, set up the equation: $\frac{\text{Sum of 5 grades}}{5} = 80$

5. Let $x$ be the fifth exam grade. The sum of the first four grades plus $x$ should be set equal to $5 \times 80$.

Try solving on your own before revealing the answer!

Final Answers:

a) Mean: $\frac{95 + 88 + 82 + 85}{4} = 87.5$

b) Median: Arrange as 82, 85, 88, 95. Median is $\frac{85 + 88}{2} = 86.5$

c) $\frac{95 + 88 + 82 + 85 + x}{5} = 80 \implies x = 50$

Heather needs at least 50 on the fifth exam to average 80.

Q2. Evaluate the Following Arithmetic Expressions

Background

Topic: Order of Operations, Basic Arithmetic

This question checks your ability to perform multiplication, division, subtraction, and addition with whole numbers.

Key Terms and Formulas:

• Order of Operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)

Step-by-Step Guidance

1. a) Multiply 38 by 1011: $38 \times 1011$

2. b) Divide 538 by 114: $538 \div 114$

3. c) Subtract 29 from 712: $712 - 29$

4. d) Add 827 and 313: $827 + 313$

Try solving on your own before revealing the answer!

Final Answers:

a) $38 \times 1011 = 38,418$

b) $538 \div 114 \approx 4.72$

c) $712 - 29 = 683$

d) $827 + 313 = 1,140$

Q3. Classifying Numbers: Natural, Whole, Integer, Rational, Irrational, Real

Background

Topic: Number Sets and Classification

This question tests your understanding of the different sets of numbers in mathematics and how to classify given numbers accordingly.

Key Terms:

• Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...)

• Whole Numbers: Natural numbers plus 0 (0, 1, 2, ...)

• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...)

• Rational Numbers: Numbers that can be written as a fraction of integers

• Irrational Numbers: Numbers that cannot be written as a fraction (e.g., $\sqrt{2}$, $\pi$)

• Real Numbers: All rational and irrational numbers

Step-by-Step Guidance

1. List the numbers: -2, -3, -57, 0, 1.63, 7, 3, 614, 77.

2. For each set, recall the definition and check which numbers fit.

3. Natural numbers: Only positive integers starting from 1.

4. Whole numbers: Natural numbers plus 0.

5. Integers: All whole numbers and their negatives.

6. Rational numbers: Any number that can be written as a fraction.

7. Irrational numbers: Numbers that cannot be written as a fraction (none in this list).

8. Real numbers: All numbers listed are real.

Try classifying each number before checking the answer!

Final Answers:

a) Natural: 7, 3, 614, 77

b) Whole: 0, 7, 3, 614, 77

c) Integers: -2, -3, -57, 0, 7, 3, 614, 77

d) Rational: All except possibly irrational numbers (all listed are rational)

e) Irrational: None

f) Real: All numbers listed

Q4. Evaluate the Following Expressions Using Order of Operations

Background

Topic: Order of Operations (PEMDAS/BODMAS)

This question tests your ability to correctly apply the order of operations to evaluate arithmetic expressions.

Key Terms and Formulas:

• Order of Operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)

Step-by-Step Guidance

1. a) $-4 + 3 - 1 + 12 \div 22$

2. First, perform the division: $12 \div 22$

3. Then, add and subtract from left to right: $-4 + 3 - 1 +$ (result from division)

4. b) $-7 - 56 \div 7 \times 22 + 4$

5. First, perform the division: $56 \div 7$

6. Next, multiply the result by 22.

7. Then, proceed with the subtraction and addition from left to right.

Try solving on your own before revealing the answer!

Final Answers:

a) $-4 + 3 - 1 + 0.545 \approx -1.455$

b) $-7 - 8 \times 22 + 4 = -7 - 176 + 4 = -179$

Q5. Simplify the Expression $a - 2b + 6a - 6b - 3$

Background

Topic: Combining Like Terms

This question tests your ability to simplify algebraic expressions by combining like terms.

Key Terms and Formulas:

• Like Terms: Terms that have the same variable raised to the same power.

Step-by-Step Guidance

1. Identify like terms: $a$ and $6a$; $-2b$ and $-6b$; $-3$ is a constant.

2. Combine the $a$ terms: $a + 6a$

3. Combine the $b$ terms: $-2b - 6b$

4. Write the simplified expression as the sum of the combined terms and the constant.

Try simplifying before checking the answer!

Final Answer:

$7a - 8b - 3$

All like terms have been combined.

Q6. Simple Interest Rate Calculation

Background

Topic: Simple Interest Formula

This question tests your ability to use the simple interest formula to find the interest rate.

Key Terms and Formulas:

• Simple Interest Formula: $I = Prt$

• $I$ = interest earned, $P$ = principal (amount lent), $r$ = interest rate (as a decimal), $t$ = time (in years)

Step-by-Step Guidance

1. Identify the known values: $I = 80$, $P = 2000$, $t = 1$ year.

2. Plug the values into the formula: $80 = 2000 \times r \times 1$

3. Solve for $r$ by dividing both sides by 2000.

4. Convert $r$ to a percentage by multiplying by 100.

Try solving for the interest rate before checking the answer!

Final Answer:

$r = \frac{80}{2000} = 0.04$ or 4%

The interest rate paid was 4%.

Q7. Travel Time at Constant Speed

Background

Topic: Proportions and Unit Rates

This question tests your ability to use proportions or unit rates to solve for an unknown time given a constant speed.

Key Terms and Formulas:

• Speed Formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

• To find time: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Step-by-Step Guidance

1. Find your speed: $\text{Speed} = \frac{25 \text{ miles}}{35 \text{ minutes}}$

2. Set up a proportion or use the speed to find the time for 125 miles: $\text{Time} = \frac{125 \text{ miles}}{\text{Speed}}$

3. Make sure your units are consistent (convert minutes to hours if needed).

Try calculating the time before checking the answer!

Final Answer:

Speed: $\frac{25}{35} = 0.714$ miles per minute

Time: $\frac{125}{0.714} \approx 175$ minutes

It would take about 175 minutes to travel 125 miles at the same speed.

Q8. Solve Each Equation for $x$

Background

Topic: Solving Linear Equations

This question tests your ability to solve for $x$ in various types of linear equations, including those with variables on both sides and with parameters.

Key Terms and Formulas:

• Linear Equation: An equation of the form $ax + b = c$

• Isolate $x$: Use inverse operations to get $x$ alone on one side of the equation.

Step-by-Step Guidance

1. a) $56x - 2 = x - 3$ Subtract $x$ from both sides, then add 2 to both sides.

2. b) $2x - 3 - 2x + 4 = -13 + x$ Combine like terms on the left, then isolate $x$.

3. c) $ax + by + c = 0$ Isolate $x$ in terms of $a$, $b$, $y$, and $c$.

4. d) $9x = 3 - 15$ Simplify the right side, then divide both sides by 9.

Try solving each equation before checking the answer!

Final Answers:

a) $x = -1$

b) $x = -17$

c) $x = \frac{-by - c}{a}$

d) $x = -\frac{4}{9}$