Problem 39 - Algebra 1

Question

Use the following information. You are choosing a business partner for a student lawn-care business you are starting. It takes you an average of 35 minutes to mow a lawn, so your rate is 1 lawn in 35 minutes or \(\frac{1}{35}\) of a lawn per minute. Let \(x\) represent the average time (in minutes) it takes a possible partner to mow a lawn. Write an expression for the partner's rate (that is, the part of a lawn the partner can mow in 1 minute). Then write an expression for the combined rate of you and your partner (the part of a lawn that you both can mow in 1 minute if you work together).

Step-by-Step Solution

Answer

The partner's mowing rate is \(\frac{1}{x}\) lawns per minute and the combined mowing rate is \(\frac{1}{35} + \frac{1}{x}\) lawns per minute.

Step 1: Determine Partner's Rate

Firstly, the rate at which the partner can mow a lawn is expressed as the reciprocal of the time it takes the partner to mow the lawn. So if the partner takes \(x\) minutes to mow a lawn, his or her rate is \(\frac{1}{x}\) lawns per minute.

Step 2: Calculate Combined Rate

The combined rate at which both you and your partner can mow the lawn is the sum of the rate at which you mow it and the rate at which your partner mows it. That is, \(\frac{1}{35} + \frac{1}{x}\) lawns per minute.

Key Concepts

Algebraic Expressions

In mathematical problem-solving, algebraic expressions allow us to represent real-life situations using symbols and numbers. One important aspect of algebraic expressions is their ability to convey the rates at which activities occur. For example, if it takes you 35 minutes to mow one lawn, we can express this as an algebraic expression for your rate: \(\frac{1}{35}\) lawns per minute. This expression shows that in one minute, you mow \(\frac{1}{35}\) of a lawn.
To express your partner's rate, we would use \(\frac{1}{x}\), where \(x\) represents the number of minutes your partner takes to mow a lawn. This expression helps us understand and solve rate problems by creating a straightforward representation of mowing activities.

Reciprocal

The concept of reciprocals plays a crucial role in determining rates, especially in time and work problems. A reciprocal is simply the inverse of a number or expression. If you know the time it takes to complete a task, the reciprocal gives you the rate of completing that task.
For example, if your partner takes \(x\) minutes to mow a lawn, the mowing rate is the reciprocal of \(x\), which is \(\frac{1}{x}\) lawns per minute. Understanding reciprocals allows you to switch from a time-based perspective to a rate-based perspective easily and make meaningful progress in problem-solving.

Combined Rates

When two people work together on a task, their efforts combine to complete the work more quickly. Understanding how to calculate combined rates is essential in estimating how much work can be done in a given time.
In the context of lawn mowing, your rate is \(\frac{1}{35}\) lawns per minute, and your partner's rate is \(\frac{1}{x}\) lawns per minute. To find the combined rate of both working together, you simply add the two rates: \(\frac{1}{35} + \frac{1}{x}\) lawns per minute. This combined rate tells you how much of the lawn can be mowed collectively in a single minute.

Time and Work Problems

Time and work problems are a common type of rate problem in algebra. They often require the calculation of how long it takes for multiple individuals or machines to complete a task when working together.
These problems typically involve setting up expressions for individual rates, like \(\frac{1}{35}\) and \(\frac{1}{x}\), and then finding their sum to determine a combined rate.
Once the combined rate is known, you can calculate how long it will take to complete the job by inverting the combined rate or considering the work done in some specific time duration. By breaking down a complex task into simple rate calculations, these problems become much easier to solve.