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Math Lecture Notes

Jul 23, 2024

Math Lecture Notes

Simplifying Numbers

  • 2 cubed: $2 \times 2 \times 2 = 8$
  • 4 squared: $4 \times 4 = 16$
  • 6 to the 0 power: Any number to the power of 0 is 1
  • 9: 9 (already simplified)
  • 10 to the first: 10 times itself once, or simply 10

Arranging in Order (Least to Greatest)

  1. $6^0$ (1)
  2. $2^3$ (8)
  3. 9
  4. $10^1$ (10)
  5. $4^2$ (16)

The answer is choice d.

Charleston Recycling Company

  • Second most common material: Glass (25%)
  • Paper Recycled Monthly:
    • Paper is 40% of 50,000 tons
    • Calculation: $0.40 \times 50,000 = 20,000$ tons

Dorothy's Age Problem

  • Given:
    • Dorothy is half her sister's age currently
    • In 20 years, Dorothy will be $ rac{3}{4}$ of her sister's age
  • Equations:
    • $D = \frac{1}{2}S$
    • $D+20 = \frac{3}{4}(S+20)$
  • Solving:
    • Substitution: $ rac{1}{2}S + 20 = \frac{3}{4}(S + 20)$
    • Distribute and simplify
    • Solve resulting equation: Dorothy is 10 years old

Chan’s Bonus Problem

  • Situation:
    • Paid 30% in taxes, 30% to charity, 25% to debt
    • Remaining 15% equals $600
  • Equation:
    • $0.15T = 600$
  • Solving:
    • $T = \frac{600}{0.15}$
    • Result: $T = 4000$ (Total bonus $4,000)

Circumference of Tire

  • Given:
    • 50 km/hr (convert to meters: 50,000 meters/hour)
    • 500 revolutions per minute
  • Convert to meters per minute:
    • $\frac{50,000 \text{ meters}}{60 \text{ minutes}}$
  • Find meters per revolution:
    • $\frac{50,000 \text{ meters}}{30,000 \text{ revolutions}} = \frac{100}{60} meters$ per revolution
  • Simplified: $1.67$ meters (approximately)

Combination Lock Problem

  • Possible Combinations: Each digit ($0-9$)
  • Total Combinations:
    • $10 \times 10 \times 10 = 1000$ combinations

Multiplying Binomials - FOIL Method

  • Given: $(a + b)(a - b)$
  • First: $a \times a = a^2$
  • Outer: $a \times (-b) = -ab$
  • Inner: $b \times a = ab$
  • Last: $b \times (-b) = -b^2$
  • Combine terms: $a^2 - b^2$

Ratio of Circle Area to Circumference

  • Area of Circle: $ rac{\pi r^2}{2 \pi r}$
  • Cancel out terms: $ rac{r}{2}$

Calculation of Mean and Mode

  • Mode: Number that appears the most
    • Given numbers: 1, 2, 3, 1, 1, 4, 2. Mode: 1
  • Mean: Average of numbers
    • $\frac{1+2+3+1+1+4+2}{7} = 2$

Jamie's Wallet Problem

  • Initial amount: $6.50
  • Expenses: Drinks $4.25, Magazine $2
  • Remaining: $.25 after expenses
  • Added: Friend repaid $2.50
  • Total left: $2.75

Conversion of Meters to Inches

  • Convert to centimeters first: 1 meter = 100 cm
  • Conversion: 19 m = 1900 cm
  • Convert to inches: 2.54 cm = 1 in
  • Calculation: 1900/2.54 = 748 inches (approximately)

Mrs. Patterson's Classroom

  • Proportion solve:
    • Given: $ rac{2}{5}$ of students are absent
    • 16 seats empty (absent)
    • Cross-multiplication: $ rac{2}{5} = rac{16}{x}$
    • Solve: x = 40 (total students)

Price Calculation for Vegetables

  • Total Spent: $24.15
  • Onions: $3.69 per lb * 2 lbs
  • Carrots: $4.29 per lb * 3 lbs
  • Mushrooms (leftover calculation): $3.90 / 1.5 lbs
  • Price per lb Mushrooms: $2.60

BC to AB Ratio (Number Line)

  • BC Distance: |5 - 8| = 3
  • AB Distance: |-6 - 5| = 11
  • BC to AB Ratio: 3:11

Investment Calculation

  • Principal: $7000
  • Rate: 7.5% (as decimal: 0.075)
  • Time: 1 year
  • Interest: $7000 * 0.075 * 1 = $525

Election Votes Percentage

  • Votes for A: 36800
  • Total Votes: 36800 + 32100 + 2100 = 71000
  • Percentage: $ rac{36800}{71000} ≈ 51.8%$

Francine’s Biking Problem

  • Ratio/Proportion:
    • 16 miles in 45 minutes
    • At that speed, 60 miles
    • $ rac{16}{45}$ = $ rac{60}{x}$
    • Cross-multiplication gives: x = 168.75 minutes

Area of Shaded Region (Quarter Circles)

  • Large Circle Area: $ rac{1}{4} \pi r^2$
  • Between Circles: $A_{outer} - A_{inner}$
  • Calculation: $ rac{1}{4} \pi (9 - 4) = rac{5}{4} \pi$
  • Final: $3.93$ (approx.)

Perimeter of Quadrilateral

  • Add sides: $m + 3 + m + 2 + m + 2m = 5m + 5$

Stock Price Problem

  • Profit Calculation: 22 profit on selling 200 shares
  • Per share profit: 11 cents
  • Buy price subtract: $19.11 - $0.11 = $19.00

Marjorie’s Stock Purchase

  • Stocks & Closure Costs
    • Microsoft: 100 at $45.14
    • Apple: 100 at $16.90
    • Garmin: 200 at $29.30
    • Total Cost: $12,064
    • Total Shares: 400
    • Average Cost per Share: $12,064 / 400 = $30.16

Pradeep’s Investment to Profit

  • Cost per share: $3.50
  • Profit Requirement: 10%
  • Increase calculation: $3.50 + $0.35 = $3.85

Morris' Work Earnings

  • Work hours: From 7 AM to 4:30 PM = 9.5 hours
  • Regular pay: 8 hours at $12/hr = $96
  • Overtime pay:
    • 1.5 hours at 1.5 of $12 = $18 * 1.5 = $27
    • Total: $96 + $27 = $123*

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