Question

When Andrew does his homework, he always takes 10 minutes to set up his desk and get totally ready to begin. Once he starts working, he is able to complete 1 homework problem every 5 minutes. Assuming that Andrew studies for over 10 minutes, which of the following represents the total number of homework problems, \(p\), Andrew is able to complete in \(m\) minutes? (A) \(p=5 m+10\) (B) \(p=5 m-1\) (C) \(p=\frac{1}{5}(m-10)\) (D) \(p=\frac{1}{10}(m-5)\)

Short answer

The correct option is (C) \( p=\frac{1}{5}(m-10) \).

Step-by-step solution

Understand the Problem

We need to express the total number of homework problems Andrew can complete, denoted as \( p \), in terms of the minutes spent studying, \( m \). Andrew takes 10 minutes initially to set up and then solves one problem every 5 minutes. The equation must represent the total problems completed after the initial setup time.

Identify Equation Components

Andrew spends 10 minutes setting up, which means any additional minutes must be used for solving homework. Thus, \( m - 10 \) gives the number of minutes Andrew actually spends solving problems. Since he solves 1 problem every 5 minutes, we need to express how many 5-minute intervals exist in \( m - 10 \).

Determine Problems Solved

To find the number of problems solved, divide the time available for solving homework by the time it takes to solve each problem. Thus, the number of problems solved is \( \frac{m-10}{5} \).

Determine the Correct Option

Comparing our expression \( \frac{m-10}{5} \) with the given choices, we see that option (C) \( p=\frac{1}{5}(m-10) \) matches. This expression accounts for the setup time and calculates the number of problems solved for the remaining time.

Key concepts

Homework Problem Solving

When tackling math homework, it's crucial to develop a systematic approach.
This not only helps you understand the problem better but also finds the solution more efficiently.
In the exercise, Andrew's task was to find the number of homework problems he could solve within a given time. Here are some steps to effectively solve similar math problems:

• Read Carefully: Always begin by thoroughly understanding what is being asked. Pay attention to details such as time constraints and rates of work.


• Break it Down: Separate the problem into smaller, manageable parts. Identify known and unknown variables.


• Create Equations: Formulate equations from the information given in the problem. Establish relationships between the variables.


• Check Your Work: Once you've computed an answer, verify it by substituting back into your equations or simplifying your understanding of the problem.

In Andrew's case, understanding the setup time versus actual work time was critical.
Breaking the problem down helped in formulating the correct equation.

Time Management in Math

Time management is an essential skill, especially when dealing with math problems that involve timing and rates.
It's about efficiently distributing your time to maximize productivity and comprehending the work involved. In Andrew's scenario, knowing how much of the total time was spent on actual problem-solving and how much was spent on preparation was key to finding a solution.
Here are some strategies for better time management in math:

• Estimate Time: Before starting, estimate how much time each part of the problem might take.


• Prioritize Tasks: Focus on understanding and setting up the problem first, as misinterpretation can lead to errors.


• Use Intervals: Understand productivity intervals, much like Andrew's 5-minute problem-solving slots. This helps in breaking the task into doable parts.


• Review and Revise: Allocate time at the end of your session to review your answers and approaches. Mistakes can be caught when you re-evaluate your work.

Balancing setup time versus working time optimizes your problem-solving ability.
Just like Andrew, proper distribution and understanding of time lead to successful outcomes.

Expressions and Equations

Expressions and equations are foundational concepts in algebra and are crucial for representing relationships in math problems.
An expression represents a mathematical phrase with numbers and operators.
An equation, on the other hand, is a statement of equality involving expressions.In Andrew's exercise, setting up an equation was essential to find the total number of problems solved.Understanding how to deal with expressions like \( m - 10 \) and turning them into equations like \( \frac{m-10}{5} \) is important for many math tasks.Some key concepts to keep in mind:

• Identify Variables: Determine what each variable represents. In Andrew's scenario, \( m \) represents total minutes spent, while \( p \) represents problems solved.


• Simplify Expressions: Simplifying expressions can make equations easier to handle.Use arithmetic to reduce complexity.


• Create Relationships: Formulate a clear relationship between the quantities involved. For Andrew, it was linking time spent with problems solved.


• Practice: Regular practice with forming and solving equations is crucial. It helps in recognizing patterns and applying similar strategies to various problems.

This understanding of expressions and equations aids in translating real-world situations into mathematical terms, as seen with Andrew’s problem.
With practice, forming accurate expressions and equations becomes second nature.