Mathematics Sample Problems

Area of a Square Containing a Circle

• Question: What is the area, in cm², of the smallest square that will contain a circle of radius 4 cm?
• Formula: The diameter of the circle = 2 × radius = 2 × 4 cm = 8 cm.
• Calculation of Area: The side length of the square must equal the diameter of the circle.
• Area of the Square: Area = side² = $8^2 = 64$ cm².

Minimum Colors Needed to Paint a Cube

• Question: What is the least number of colors needed to paint a cube so that no two adjacent faces have the same color?
• Answer Choices:
  • A) 4
  • B) 8
  • C) 16
  • D) 32
  • E) 64
• Solution: To ensure no two adjacent faces share the same color, a minimum of 2 colors is required (checkerboard pattern).

Fraction of Time Calculation

• Question: What fraction of 4 hours is 15 seconds?
• Conversion: 4 hours = 4 × 60 minutes × 60 seconds = 14400 seconds.
• Fraction: The fraction of time = $\frac{15}{14400}$ which simplifies to:
  • A) $\frac{1}{16}$
  • B) $\frac{1}{64}$
  • C) $\frac{1}{760}$
  • D) $\frac{1}{960}$
  • E) $\frac{1}{9600}$

Numerical Value of a Fraction

• Question: Find the numerical value of the fraction: 1/9.
• Answer Choices:
  • A) 2
  • B) 3
  • C) 4
  • D) 5
  • E) 6
• Solution: Numerical value = $1/9$ is approximately 0.11, which does not directly correlate to the options.

Estate Division Analysis

• Situation: Homer Simpson left an estate of $2.4 million. Distribution:
  • 10% to Springfield University
  • 15% to the First Church of Springfield
  • $600,000 to his wife, Marge
  • Remainder to three children
• Calculations:
  1. 10% of $2.4 million = $0.10 × 2400000 = 240000$
  2. 15% of $2.4 million = $0.15 × 2400000 = 360000$
  3. Total given away = 240000 + 360000 + 600000 = 1200000
  4. Remainder for children = 2400000 - 1200000 = 1200000
  5. Amount for each child = $\frac{1200000}{3} = 400000$
• Answer Choices for Each Child:
  • A) $240,000
  • B) $300,000
  • C) $360,000
  • D) $400,000
  • E) $440,000

Glue Tube Requirement for Boat Construction

• Question: It takes 7 tubes of glue to construct 15 model boats. Calculate for 10 model boats.
• Rates:
  • $\frac{7\text{ tubes}}{15\text{ boats}}$ = $\frac{x\text{ tubes}}{10\text{ boats}}$
• Cross-Multiplication: 15x = 70 → x = 70/15 = 4.67 (approx. 5 tubes).
• Answer Choices:
  • A) $\frac{1}{6}$
  • B) $\frac{3}{10}$
  • C) $\frac{5}{6}$
  • D) $\frac{7}{10}$
  • E) $\frac{2}{3}$

Day of the Week Calculation

• Question: If January 28, 2001 is a Sunday, what day of the week is February 28, 2001?
• Details: January has 31 days, therefore the days from January 28 to January 31 = 3 days.
• February Days: Therefore February 1, 2001 is a Wednesday. Continuing:
  • 28 days in February: 28 + 3 = 31 days later.
• Conclusion: February 28, 2001 is a Wednesday.
• Answer Choices:
  • A) Wednesday
  • B) Tuesday
  • C) Monday
  • D) Friday

Concentration of Sugar Solution

• Question: What is the concentration of sugar solution made by mixing 1 kg of 12% sugar solution with 2 kg of water?
• Sugar Mass Calculation: 12% of 1 kg = 0.12 kg sugar.
• Total Solution Mass: Total = 1 kg + 2 kg = 3 kg.
• Concentration Formula: Concentration = $\frac{\text{sugar mass}}{\text{total mass}} = \frac{0.12}{3}$ = 0.04 or 4%.
• Answer Choices:
  • A) 3%
  • B) 4%
  • C) 6%
  • D) 8%
  • E) 9%