Chapter 1 Review: Mathematical Operations, Equations, and Applications

Basic Arithmetic Operations

  • Problem 1: Signed Integer Subtraction

    • Expression: −15−8-15 - 8

    • Step-by-step Solution:

    • Subtracting a positive integer is equivalent to adding its negative counterpart: −15−8=−15+(−8)-15 - 8 = -15 + (-8)

    • Summing two negative numbers yields a negative result: −15+(−8)=−23-15 + (-8) = -23

    • Final Answer: −23-23

  • Problem 2: Signed Decimal Addition

    • Expression: −45.9+−8.5-45.9 + -8.5

    • Step-by-step Solution:

    • Both terms are negative, so combine their absolute values and retain the negative sign: −(45.9+8.5)-(45.9 + 8.5)

    • Summing the decimal values: 45.9+8.5=54.445.9 + 8.5 = 54.4

    • Applying the negative sign: −54.4-54.4

    • Final Answer: −54.4-54.4

  • Problem 3: Fraction Multiplication

    • Expression: 316×415\frac{3}{16} \times \frac{4}{15}

    • Step-by-step Solution:

    • Multiply numerators together and denominators together: 3×416×15=12240\frac{3 \times 4}{16 \times 15} = \frac{12}{240}

    • Simplify the fraction by dividing numerator and denominator by their greatest common divisor, 1212: 12÷12240÷12=120\frac{12 \div 12}{240 \div 12} = \frac{1}{20}

    • Decimal equivalent: 0.050.05

    • Final Answer: 120\frac{1}{20} (or 0.050.05)

  • Problem 4: Polynomial Expression Multiplication

    • Expression: (3x2y4)(4x3y2)(3x^2 y^4)(4x^3 y^2)

    • Step-by-step Solution:

    • Group and multiply numerical coefficients: 3×4=123 \times 4 = 12

    • Apply the product rule for exponents (am⋅an=am+na^m \cdot a^n = a^{m+n}) to variable xx: x2⋅x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5

    • Apply the product rule for exponents to variable yy: y4⋅y2=y4+2=y6y^4 \cdot y^2 = y^{4+2} = y^6

    • Combine all factors: 12x5y612x^5 y^6

    • Final Answer: 12x5y612x^5 y^6

  • Problem 5: Order of Operations with Nested Grouping

    • Expression: [6+(−2)(4)]−8−2[6 + (-2)(4)] - 8 - 2

    • Step-by-step Solution:

    • Evaluate multiplication inside the brackets first: (−2)(4)=−8(-2)(4) = -8

    • Simplify inside the brackets: 6+(−8)=−26 + (-8) = -2

    • Perform remaining subtractions from left to right: −2−8−2=−10−2=−12-2 - 8 - 2 = -10 - 2 = -12

    • Final Answer: −12-12

  • Problem 6: Absolute Value Operations

    • Expression: ∣−5+2∣+∣7−(−3)∣|-5 + 2| + |7 - (-3)|

    • Step-by-step Solution:

    • Evaluate the first absolute value term: ∣−5+2∣=∣−3∣=3|-5 + 2| = |-3| = 3

    • Evaluate the second absolute value term: ∣7−(−3)∣=∣7+3∣=∣10∣=10|7 - (-3)| = |7 + 3| = |10| = 10

    • Add the results together: 3+10=133 + 10 = 13

    • Final Answer: 1313

Decimal Conversions and Expression Simplification

  • Problem 7: Changing Fraction to Decimal Form

    • Expression: 32\frac{3}{2}

    • Step-by-step Solution:

    • Divide numerator by denominator: 3÷2=1.53 \div 2 = 1.5

    • Final Answer: 1.51.5

  • Problem 8: Changing Percentage to Decimal Form

    • Expression: 454%454\%

    • Step-by-step Solution:

    • Convert percentage to decimal by dividing by 100100: 454100=4.54\frac{454}{100} = 4.54

    • Final Answer: 4.544.54

  • Problem 9: Simplifying Algebraic Expressions (Distributive Property)

    • Expression: 3a−4b+2(a+b)3a - 4b + 2(a + b)

    • Step-by-step Solution:

    • Apply distributive property to 2(a+b)2(a + b): 2a+2b2a + 2b

    • Rewrite expression: 3a−4b+2a+2b3a - 4b + 2a + 2b

    • Group like terms: (3a+2a)+(−4b+2b)(3a + 2a) + (-4b + 2b)

    • Combine like terms: 5a−2b5a - 2b

    • Final Answer: 5a−2b5a - 2b

  • Problem 10: Simplifying Linear Expressions

    • Expression: 4x−25(x+2)−104x - 25(x + 2) - 10

    • Step-by-step Solution:

    • Distribute −25-25 through (x+2)(x + 2): −25x−50-25x - 50

    • Rewrite expression: 4x−25x−50−104x - 25x - 50 - 10

    • Combine like terms: (4x−25x)+(−50−10)=−21x−60(4x - 25x) + (-50 - 10) = -21x - 60

    • Final Answer: −21x−60-21x - 60

Algebraic Expression Evaluation

  • Given Variable Values:

    • w=13w = 13

    • x=−2x = -2

    • y=−7y = -7

    • z=6z = 6

  • Problem 11: Perimeter Formula Evaluation

    • Formula: P=2w+2zP = 2w + 2z

    • Step-by-step Solution:

    • Substitute w=13w = 13 and z=6z = 6 into the formula: P=2(13)+2(6)P = 2(13) + 2(6)

    • Perform multiplications: P=26+12P = 26 + 12

    • Sum the products: P=38P = 38

    • Final Answer: 3838

  • Problem 12: Complex Grouped Expression Evaluation

    • Expression: [(8x+2)+2x]+y[(8x + 2) + 2x] + y

    • Step-by-step Solution:

    • Substitute x=−2x = -2 and y=−7y = -7:

    • Evaluate innermost term 8x+28x + 2: 8(−2)+2=−16+2=−148(-2) + 2 = -16 + 2 = -14

    • Add 2x2x inside brackets: −14+2(−2)=−14+(−4)=−18-14 + 2(-2) = -14 + (-4) = -18

    • Add outer yy value: −18+(−7)=−25-18 + (-7) = -25

    • Final Answer: −25-25

Solving Equations

  • Problem 13: One-Step Fractional Linear Equation

    • Equation: 21=516x21 = \frac{5}{16}x

    • Step-by-step Solution:

    • Multiply both sides by reciprocal 165\frac{16}{5}: x=21×165x = 21 \times \frac{16}{5}

    • Calculate numerator: 21×16=33621 \times 16 = 336

    • Divide by denominator: x=3365=67.2x = \frac{336}{5} = 67.2

    • Final Answer: 67.267.2

  • Problem 14: Single-Variable Multiplication Equation

    • Equation: −4x=57-4x = \frac{5}{7}

    • Step-by-step Solution:

    • Divide both sides by −4-4: x=5/7−4=−528x = \frac{5/7}{-4} = -\frac{5}{28}

    • Convert to decimal: x≈−0.17857x \approx -0.17857

    • Round to nearest tenth: −0.2-0.2

    • Final Answer: −0.2-0.2

  • Problem 15: Two-Step Linear Equation

    • Equation: 2x−8=72x - 8 = 7

    • Step-by-step Solution:

    • Add 88 to both sides: 2x=7+8=152x = 7 + 8 = 15

    • Divide both sides by 22: x=152=7.5x = \frac{15}{2} = 7.5

    • Final Answer: 7.57.5

  • Problem 16: Multi-Step Linear Equation

    • Equation: 42+29x=31742 + 29x = 317

    • Step-by-step Solution:

    • Subtract 4242 from both sides: 29x=317−42=27529x = 317 - 42 = 275

    • Divide both sides by 2929: x=27529≈9.4827x = \frac{275}{29} \approx 9.4827

    • Round to nearest tenth: 9.59.5

    • Final Answer: 9.59.5

Algebraic Transformations and Literal Equations

  • Problem 17: Eliminating Fractions Using Exponents

    • Expression: BH\frac{B}{H}

    • Step-by-step Solution:

    • Apply negative exponent rule 1an=a−n\frac{1}{a^n} = a^{-n} to shift denominator term to numerator: BH−1B H^{-1}

    • Final Answer: BH−1B H^{-1}

  • Problem 18: Solving Volume Formula for Height

    • Formula: V=BH3V = \frac{B H}{3}

    • Step-by-step Solution:

    • Clear fraction by multiplying both sides by 33: 3V=BH3V = B H

    • Isolate HH by dividing both sides by BB: H=3VBH = \frac{3V}{B}

    • Final Answer: H=3VBH = \frac{3V}{B}

Word Problems and Real-World Applications

  • Problem 19: Percentage Savings Calculation

    • Problem Statement: Ben saved $12.46 on a watch that is usually $67.93. What percent did he save?

    • Given Values:

    • Savings Amount = $12.46

    • Regular Price = $67.93

    • Formula:     Percent Saved=(SavingsOriginal Price)×100%\text{Percent Saved} = \left(\frac{\text{Savings}}{\text{Original Price}}\right) \times 100\%

    • Calculation:     Percent Saved=(12.4667.93)×100%≈0.183422×100%=18.3422%\text{Percent Saved} = \left(\frac{12.46}{67.93}\right) \times 100\% \approx 0.183422 \times 100\% = 18.3422\%

    • Final Answer (nearest tenth of a percent): 18.3%18.3\%

  • Problem 20: Commission Earnings Calculation

    • Problem Statement: Brenda receives 18% of everything she sells. If she sold $19,000 in merchandise, how much money did she make?

    • Given Values:

    • Commission Rate = 18%=0.1818\% = 0.18

    • Total Sales = $19,000

    • Formula:     Earnings=Commission Rate×Total Sales\text{Earnings} = \text{Commission Rate} \times \text{Total Sales}

    • Calculation:     Earnings=0.18×19,000=$3,420\text{Earnings} = 0.18 \times 19,000 = \$3,420

    • Final Answer: $3,420

  • Problem 21: Home Loan Principal Calculation

    • Problem Statement: Mr. Kai pays a $425 closing fee, which is 0.25% of his home loan. How much was the loan?

    • Given Values:

    • Closing Fee = $425

    • Fee Percentage = 0.25%=0.00250.25\% = 0.0025

    • Formula:     Loan Amount=Closing FeeFee Percentage\text{Loan Amount} = \frac{\text{Closing Fee}}{\text{Fee Percentage}}

    • Calculation:     Loan Amount=4250.0025=$170,000\text{Loan Amount} = \frac{425}{0.0025} = \$170,000

    • Final Answer: $170,000

  • Problem 22: Percentage Decrease in Attendance

    • Problem Statement: The youth group attendance last month was 44. Because of conflicting schedules, only 35 were able to attend this month. What was the percent decrease to the nearest percent?

    • Given Values:

    • Original Attendance = 4444

    • New Attendance = 3535

    • Step-by-step Solution:

    • Calculate decrease amount: 44−35=944 - 35 = 9

    • Calculate percent decrease ratio: Percent Decrease=(944)×100%\text{Percent Decrease} = \left(\frac{9}{44}\right) \times 100\%

    • Calculate percentage: 0.204545×100%=20.4545%0.204545 \times 100\% = 20.4545\%

    • Round to nearest percent: 20%20\%

    • Final Answer: 20%20\%

  • Problem 23: Percentage Increase in Play Attendance

    • Problem Statement: Find the percent change in attendance at a school's annual play if 150 people attended the first year and 234 attended the second year.

    • Given Values:

    • Year 1 Attendance = 150150

    • Year 2 Attendance = 234234

    • Step-by-step Solution:

    • Calculate increase amount: 234−150=84234 - 150 = 84

    • Calculate percent change ratio: Percent Change=(84150)×100%\text{Percent Change} = \left(\frac{84}{150}\right) \times 100\%

    • Calculate percentage: 0.56×100%=56%0.56 \times 100\% = 56\%

    • Final Answer: 56%56\%

  • Problem 24: Distance Travelled Calculation

    • Problem Statement: Find the distance Rayna travels by car in 4.1 hours if her average speed is 55 mi./hr.

    • Given Values:

    • Time (tt) = 4.1 hours4.1\,\text{hours}

    • Speed (vv) = 55 mi./hr.55\,\text{mi./hr.}

    • Formula:     d=v×td = v \times t

    • Calculation:     d=55 mi./hr.×4.1 hours=225.5 milesd = 55\,\text{mi./hr.} \times 4.1\,\text{hours} = 225.5\,\text{miles}

    • Final Answer: 225.5 miles225.5\,\text{miles}