Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

It takes a boy 90 minutes to mow the lawn, but his sister can mow it in 60 minutes. How long would it take them to mow the lawn if they worked together, using two lawn mowers?

Step-by-Step Solution

Answer

36 minutes.

Step 1: Determine the rate of the boy

First, we find out how much of the lawn the boy can mow in one minute. Since the boy can mow the entire lawn in 90 minutes, his rate is \(\frac{1}{90}\) of the lawn per minute.

Step 2: Determine the rate of the sister

Next, we calculate how much of the lawn the sister can mow in one minute. She can mow the entire lawn in 60 minutes, so her rate is \(\frac{1}{60}\) of the lawn per minute.

Step 3: Combine the rates

Now, we add the rates of the boy and his sister to find their combined rate. The combined rate is \(\frac{1}{90} + \frac{1}{60}\).

Step 4: Find the Least Common Denominator

To add the fractions, we find the least common denominator of 90 and 60, which is 180. Re-write the fractions as \(\frac{2}{180}\) and \(\frac{3}{180}\), respectively.

Step 5: Add the fractions

Now, add the fractions: \(\frac{2}{180} + \frac{3}{180} = \frac{5}{180}\). This is the combined rate of the boy and sister working together.

Step 6: Solve for total time together

To find out how long it takes them to mow one lawn together, take the reciprocal of their combined rate: \(\frac{180}{5} = 36\). Thus, it takes them 36 minutes to mow the lawn together.

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and rules for manipulating those symbols. In everyday problems like work problems, algebra helps us find unknown values by forming equations. Here, we use it to calculate how long two people will take to complete a task together. We're given rates and need to determine the total time. Algebra involves setting up equations, such as expressing the rate of work done per minute. In work problems:

• We often assign variables to unknown quantities such as time or rate.
• We use equations to relate these variables, solving for the unknown.
• When combining efforts (like two people working together), we add their rates to find the total.

This process simplifies complex problems and makes them more approachable using logic and equations.

Rates

Rates are a way of measuring one quantity per unit of another. For example, in the exercise, we measure how much of the lawn each person can mow per minute. This is a common scenario in "work problems" where we need to determine how tasks are completed over time.When calculating rates,:

• We divide the total amount of work by the time it takes to complete it.
• In this case, since it takes the boy 90 minutes to mow the entire lawn, his rate is \(\frac{1}{90}\).
• The sister's rate, who completes the task faster in 60 minutes, is \(\frac{1}{60}\).

Using these rates, we can easily calculate how long both will take if they combine their efforts. Adding their rates gives us the combined rate, showing how much they can achieve together per unit time.

Fractions

Fractions are numerical representations of portions and are used extensively in math to express non-whole numbers. In work problems, such as the one involving mowing a lawn, fractions help express rates, making calculations more precise.Understanding fractions involves:

• Numerator: represents a part of the whole.
• Denominator: represents the total number of equal parts the whole is divided into.

In the problem:- The boy's rate is expressed as \(\frac{1}{90}\), meaning he can do 1 part of the task in 90 equal parts in a minute.- The sister's is \(\frac{1}{60}\), or 1 part out of 60 in a minute.Fractions allow us to add their rates to compute the time they take to complete the task together, leading to an efficient solution.

Least Common Denominator

The Least Common Denominator (LCD) is critical when adding fractions with different denominators—common in problems involving rates. It helps us find a common ground to easily add the fractions, streamlining our calculations.To find the LCD:

• Identify the denominators of the fractions you want to combine.
• Determine the smallest number that both denominators can divide into without a remainder.

In the exercise, the denominators are 90 and 60.- The least common denominator here is 180.- By rewriting: the boy's rate becomes \(\frac{2}{180}\) and the sister's \(\frac{3}{180}\).With a shared denominator, we can simply add their rates: \(\frac{2}{180} + \frac{3}{180} = \frac{5}{180}\), easily showing how fast they work together. This method simplifies fractional arithmetic in algebra problems.