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MAT 108 Social Science Mathematics Exam 1 Review

CUNY John Jay College of Criminal Justice - Exam 1 Review Notes

Overview of the Exam Structure

  • Course: MAT 108 Social Science Mathematics
  • Review Document for Fall 2023
  • Important Instructions:
    • Follow directions clearly.
    • Indicate methods used as you will be scored on correctness, accuracy, and completeness of both results and explanations.
    • Show work both with and without using a calculator.
    • Formulas will be provided during the exam.
    • Express answers as simplified fractions unless otherwise noted.

Section 6-1: Applications of Linear Equations

  1. Application Problem:

    • Problem Statement: The difference of a number and 8 is the same as 46 less the number.
    • Solution Method: Set up the equation as follows:
      • Let the number be represented as x.
      • Equation: x - 8 = 46 - x.
      • Solve for x.

  2. Inventory Problem:

    • Problem Statement: A storehouse stores 450 different inventory items. \frac{3}{5} of these items are perishable.
    • Solution Method:
      • Find \frac{3}{5} of 450:
        \frac{3}{5} \times 450 = 270 perishable items.

  3. Garage Capacity Problem:

    • Problem Statement: Center City East Garage has a capacity of 253 cars more than Center City West Garage. Combined capacity is 1231 cars.
    • Solution Method:
      • Let capacity of Center City West Garage be y, then Center City East is y + 253.
      • Set up equation: y + (y + 253) = 1231.
    • Solve for y, then find each garage's capacity.

  4. Marble Distribution Problem:

    • Problem Statement: 30 marbles are to be divided into three bags; second bag has three times as many as the first, third has twice as many as the first.
    • Define Variables:
      • First bag: x marbles
      • Second bag: 3x marbles
      • Third bag: 2x marbles
    • Equation: x + 3x + 2x = 30.
    • Solve for x and find the number in the second bag: 3x.

  5. Car Rental Problem:

    • Problem Statement: Luxury car rental for $29.95/day and $0.19/mile, with a budget of $200 for 2 days.
    • Cost Breakdown: Cost for 2 days = 2 \times 29.95 + Mileage Costs.
    • Set Up Inequality: 2(29.95) + 0.19m \leq 200, where m is miles.
    • Solve for maximum whole miles m.

Section 6-2: Ratio and Proportion

  1. Unit Price Calculation:

    • Problem Statement: A 1.5-lb package of seedless grapes costs $3.95.
    • Unit Price: \frac{3.95}{1.5} = 2.63 per pound (rounded).

  2. Patient Schedule Problem:

    • Problem Statement: Dr. Wong sees 12 patients in 3 hours.
    • Rate Calculation: Patients per hour = \frac{12}{3} = 4.
    • Time for 72 Patients: \frac{72}{4} = 18 hours.

  3. Committee Ratio Problem:

    • Problem Statement: 13 people in a committee, 8 are women.
    • Ratio Calculation: Men = 13 - 8 = 5.
    • Ratio of Women to Men: \frac{8}{5}.

  4. Equation Solving:

    • Equation: 3x - 1 = 7x + 2.
    • Solution Steps: Rearrange and combine like terms to solve for x.

  5. Self-Tanning Application Problem:

    • Problem Statement: 4-oz bottle provides 5 applications, Sarah has a 13-oz bottle.
    • Application Rate: \frac{5}{4} applications per ounce.
    • Total Applications Expected: \frac{13\times5}{4} (Round down to whole number).

Section 6-3: The Rectangular Coordinate System and Linear Equations in Two Variables

  1. Slope Calculation:

    • Problem Statement: Find the slope between points (6, -9) and (-1, 1).
    • Slope Formula: m = \frac{y2 - y1}{x2 - x1} = \frac{1 - (-9)}{-1 - 6} = \frac{10}{-7} = -\frac{10}{7}.

  2. Graphing by Intercepts:

    • Equation: -4x - 24y - 24 = 0.
    • Find intercepts: Set y=0 to find x-intercept; set x=0 to find y-intercept.

  3. Graphing with Slope and Intercept:

    • Equation: 3x + 2y = 6.
    • Find y-intercept and slope, identify if slope is positive, negative, zero, or undefined.

  4. Find Linear Equation:

    • Committee Balloon Inflation Problem: 30 balloons already inflated, inflating 60/hour.
    • Equation for Total Balloons: y = 60x + 30, where x is the number of hours.
    • Calculate Time for 270 Balloons: Solve 60x + 30 = 270 for x.