Problem 42 Solve each problem. Painting ... [FREE SOLUTION] (2024)

Short Answer

Rate of Work

Rate of work problems involve finding out how long it takes for one or more individuals to complete a task. The key is understanding that rate of work is the amount of work done per unit of time.
In our example, Julie paints a fence in 12 hours. So, her rate is 1 part of the fence per 12 hours, or \( \frac{1}{12} \) per hour.
Similarly, if two people work together, their combined rate is the sum of their individual rates. If Julie and Betsy can paint together in 5 hours, their combined rate is \( \frac{1}{5} \) per hour.
Recognizing these individual and combined rates is the first step in solving work rate problems.

Algebraic Equations

Algebraic equations help us find unknown values by setting up relationships between known quantities and the unknown. In work rate problems, we often set up an equation involving the rates of the workers and solve for the unknown rate.
For Julie and Betsy, we use their rates to form the equation: \( J + B = \frac{1}{5} \).
Here, \( J \) represents Julie's work rate \( \frac{1}{12} \), and \( B \) is Betsy's rate. We substitute Julie's rate into the equation to isolate Betsy's rate: \[ \frac{1}{12} + B = \frac{1}{5} \].
This equation tells us how much they contribute together versus individually. Solving such equations is crucial in finding the solution.

Fraction Operations

Fractions play a major role in solving rate of work problems, especially when combining or comparing different rates. Adding or subtracting fractions requires a common denominator.
In our exercise, we need to subtract Julie's rate from the combined rate: \[ \frac{1}{5} - \frac{1}{12} \].
To do this, we find a least common denominator, which is 60. Rewriting the fractions, we get: \[ \frac{12}{60} - \frac{5}{60} = \frac{7}{60} \].
Understanding how to operate on fractions allows us to manipulate the rates properly and find the exact values we need.

Reciprocal Calculation

The reciprocal of a number is essential in converting a rate back into time. In this context, the reciprocal of Betsy's rate will tell us how long she takes to complete the task alone.
Since Betsy's rate for painting the fence is \( \frac{7}{60} \) fences per hour, we take its reciprocal to find the time: \[ t = \frac{60}{7} \text{ hours} \].
So, if we convert \( \frac{60}{7} \) to a decimal, it is approximately 8.57 hours.
This demonstrates the practical importance of understanding reciprocals when solving work rate problems.