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

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 $x$ .
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.
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.
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: $3x$ .
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

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).
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.
Committee Ratio Problem:
- Problem Statement: 13 people in a committee, 8 are women.
- Ratio Calculation: Men = 13 - 8 = 5.
- Ratio of Women to Men: $\frac{8}{5}$ .
Equation Solving:
- Equation: $3x - 1 = 7x + 2$ .
- Solution Steps: Rearrange and combine like terms to solve for $x$ .
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

Slope Calculation:
- Problem Statement: Find the slope between points (6, -9) and (-1, 1).
- Slope Formula: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1} = \frac{1 - (-9)}{-1 - 6} = \frac{10}{-7} = -\frac{10}{7}$ .
Graphing by Intercepts:
- Equation: $-4x - 24y - 24 = 0$ .
- Find intercepts: Set $y=0$ to find $x$ -intercept; set $x=0$ to find $y$ -intercept.
Graphing with Slope and Intercept:
- Equation: $3x + 2y = 6$ .
- Find y-intercept and slope, identify if slope is positive, negative, zero, or undefined.
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 $x$ .