3-4.2 Application Problem P. 85 Answers

Tackling Application Problems in 3-4.2: A Comprehensive Guide

This article provides a detailed walkthrough and explanation of application problems found in section 3-4.2, page 85 (assuming a specific textbook or resource is referenced), focusing on problem-solving strategies and in-depth analysis. While I cannot access specific textbook pages, I will offer a general approach adaptable to various application problem types typically found in this section of mathematics or science textbooks. This will include problems related to rates, proportions, percentages, and other common applications.

Understanding the Context of 3-4.2 Application Problems

Section 3-4.2 likely builds upon the foundational concepts introduced in earlier sections (3-1, 3-2, 3-3, and potentially earlier chapters). This means that successfully solving the application problems in 3-4.2 requires a strong grasp of these prior concepts. These problems often involve translating real-world scenarios into mathematical equations and then solving for unknown variables.

Common Problem Types in 3-4.2

Let's explore some common types of application problems often found in this type of section:

1. Rate Problems

Rate problems involve the relationship between two quantities, often involving distance, time, speed, or work. These often utilize the formula: Rate = Distance / Time or variations thereof. Understanding the units and correctly converting them is crucial.

Example: A train travels 300 miles in 5 hours. What is its average speed?

Solution: We can use the formula: Speed = Distance / Time. Speed = 300 miles / 5 hours = 60 miles per hour.

More Complex Rate Problems:

More challenging rate problems might involve two objects moving at different speeds, approaching each other, or moving away from each other. These often require setting up two equations and solving them simultaneously.

Example: Two cars leave the same point at the same time, traveling in opposite directions. One car travels at 60 mph, and the other at 40 mph. How far apart are they after 3 hours?

Solution: Car 1: Distance = Speed x Time = 60 mph x 3 hours = 180 miles. Car 2: Distance = Speed x Time = 40 mph x 3 hours = 120 miles. Total distance apart: 180 miles + 120 miles = 300 miles.

2. Proportion Problems

Proportion problems deal with ratios and how they relate to each other. They involve setting up two equivalent ratios and solving for an unknown value. A common format is: a/b = c/d

Example: If 3 apples cost $1.50, how much do 5 apples cost?

Solution: Set up the proportion: 3 apples / $1.50 = 5 apples / x. Cross-multiply: 3x = 7.50. Solve for x: x = $2.50.

Advanced Proportion Problems:

More advanced problems might involve scaling recipes, maps, or other real-world scenarios that require understanding ratios and scaling factors.

3. Percentage Problems

Percentage problems focus on determining a percentage of a total amount or finding the total amount given a percentage. These problems require understanding how to convert percentages to decimals and vice versa.

Example: A store offers a 20% discount on an item that costs $50. What is the final price?

Solution: Discount amount: 20% of $50 = 0.20 x $50 = $10. Final price: $50 - $10 = $40.

Complex Percentage Problems:

More complex scenarios might involve multiple discounts, percentage increases, or interest calculations, often involving compound interest formulas.

4. Mixture Problems

Mixture problems involve combining two or more substances with different concentrations or amounts. These often involve setting up equations based on the total amount and the concentration of each component.

Example: How many liters of a 10% saline solution must be mixed with 5 liters of a 25% saline solution to obtain a 15% solution?

Solution: This problem requires setting up a system of equations, typically using variables to represent the unknown quantities. Solving these simultaneous equations will give the required amount of the 10% solution.

5. Work Problems

Work problems involve determining the rate at which individuals or machines complete a task and calculating the time it takes for them to complete the task together or individually.

Example: Person A can paint a house in 6 hours, and Person B can paint the same house in 4 hours. How long would it take them to paint the house together?

Solution: This problem uses the concept of work rates. Find the individual rates (1/6 house per hour for A and 1/4 house per hour for B), add the rates together, and find the total time.

Solving Application Problems: A Step-by-Step Approach

1. Read Carefully: Thoroughly read and understand the problem statement. Identify all known and unknown quantities.

2. Identify Key Concepts: Determine the relevant mathematical concepts and formulas needed (rates, proportions, percentages, etc.).

3. Define Variables: Assign variables to the unknown quantities.

4. Set Up Equations: Translate the problem statement into mathematical equations using the defined variables and relevant formulas.

5. Solve the Equations: Solve the equations using appropriate algebraic techniques.

6. Check Your Answer: Review your solution to ensure it is logical and consistent with the problem statement.

Advanced Problem-Solving Techniques

• Drawing Diagrams: Visual representations can help simplify complex problems, especially in geometry or physics.

• Using Tables: Organizing information in a table can make it easier to identify patterns and relationships.

• Working Backwards: Start with the answer and work backward to see if it satisfies the conditions of the problem.

• Estimation: Before solving, make an estimate of the answer to check for reasonableness.

Conclusion:

Successfully tackling application problems in section 3-4.2 requires a solid understanding of the underlying mathematical concepts, careful problem-solving strategies, and the ability to translate real-world scenarios into mathematical equations. By following a systematic approach and utilizing advanced problem-solving techniques, you can effectively solve a wide variety of application problems. Remember that practice is key; the more problems you work through, the more proficient you will become. This detailed explanation, while not referencing specific page 85 problems, provides a robust framework for approaching the types of problems you’re likely to encounter. Remember to always review your work and ensure your solutions are logical and accurate within the context of the problem.