Problem 54 - Algebra and Trigonometry

Question

Sharing a Job Stan and Hilda can mow the lawn in 40 min if they work together. If Hilda works twice as fast as Stan, how long does it take Stan to mow the lawn alone?

Step-by-Step Solution

Answer

Stan takes 120 minutes to mow the lawn alone.

Step 1: Define the variables

Let Stan's mowing rate be \( s \) lawns per minute. Hilda works twice as fast, so her mowing rate is \( 2s \) lawns per minute.

Step 2: Set up the equation for combined work

Together, Stan and Hilda can mow the lawn in 40 minutes, which means in one minute they mow \( \frac{1}{40} \) of the lawn. Thus, the equation for their combined rates is:\[ s + 2s = \frac{1}{40} \].

Step 3: Solve for Stan's rate

Simplify the equation from Step 2: \( 3s = \frac{1}{40} \). Solve for \( s \) by dividing both sides by 3: \( s = \frac{1}{120} \).

Step 4: Calculate the time Stan takes to mow alone

Stan's rate corresponds to him mowing \( \frac{1}{120} \) of the lawn in one minute. To find the total time it takes him to mow one entire lawn, calculate the reciprocal of his rate: \( 120 \) minutes.

Key Concepts

Algebra

Algebra involves using letters and symbols to represent numbers and quantities in equations and formulas. It helps us model and solve practical problems. In this context, we're dealing with rates of work, which are easy to manage with algebraic expressions.

First, we define variables to represent unknown quantities. For Stan's mowing rate, we use the variable \( s \) to denote the portion of the lawn he can mow per minute. Since Hilda works twice as fast, her rate becomes \( 2s \). This setup allows us to express their combined work rate algebraically, simplifying the problem.

By using algebra, we break down complex problems into manageable steps. The variables and their relationships build a clear path to find the desired solution. From this foundation, we can utilize equations to decipher how long Stan takes to mow the lawn by himself.

Equation Solving

Solving equations is a crucial skill in algebra, allowing us to find unknown values. Once we have established the rates of Stan and Hilda, the next step is to set up and solve an equation representing their combined work.

We know that together, they mow the lawn in 40 minutes. This means that in one minute, they complete \( \frac{1}{40} \) of the lawn. The equation we form is \( s + 2s = \frac{1}{40} \), representing their combined rates. Here, Stan contributes \( s \), and Hilda contributes \( 2s \) to the total rate.

Solving the equation involves simplifying it by adding like terms, resulting in \( 3s = \frac{1}{40} \). We then isolate \( s \) by dividing both sides by 3, leading to \( s = \frac{1}{120} \). This solution step is essential, as it directly tells us how much of the lawn Stan mows per minute.

Collaborative Work

Collaborative work problems demonstrate how individuals can complete tasks faster by working together. The given problem showcases Stan and Hilda's teamwork, achieving a goal more efficiently than working alone.

In collaborative scenarios, each person's work rate contributes to the completion of a task. By understanding that Hilda's rate is \( 2s \) while Stan's is \( s \), we can analyze how their combined efforts equal the reciprocal of time taken when working jointly. Their shared rate is \( 3s \), summing up the portion of the task fulfilled per minute.

This shows how recognizing and calculating individual work rates can help solve larger problems. Learning this concept is essential when faced with similar collaborative tasks, making it applicable to real-world situations where teamwork leads to efficient outcomes.