Problem 41 - Algebra for College Students

Question

Solve each problem. It takes Gina 90 minutes to file the monthly invoices. If Hilda files twice as fast as Gina does, how long will it take them working together?

Step-by-Step Solution

Answer

It will take them 30 minutes working together.

Step 1: Determine Individual Work Rates

First, determine the rate at which Gina and Hilda can file invoices. Gina takes 90 minutes to complete one filing task. Therefore, Gina's rate is \ \(\frac{1}{90}\) of the task per minute.

Step 2: Calculate Hilda's Work Rate

Hilda files twice as fast as Gina. If Gina's rate is \(\frac{1}{90}\), Hilda's rate is twice that, which is \(2 \times \frac{1}{90} = \frac{2}{90} = \frac{1}{45}\) of the task per minute.

Step 3: Combine Work Rates

Add the two individual rates to find their combined work rate. Gina's rate is \(\frac{1}{90}\) and Hilda's rate is \(\frac{1}{45}\). So, the combined rate when they work together is: \ \(\frac{1}{90} + \frac{1}{45}\).

Step 4: Simplify Combined Work Rate

To simplify \(\frac{1}{90} + \frac{1}{45}\), convert them to a common denominator. In this case, the common denominator is 90: \ \frac{1}{90} + \frac{2}{90} = \frac{3}{90} = \frac{1}{30}\.

Step 5: Calculate Time Required

The combined rate \(\frac{1}{30}\) means they complete \(\frac{1}{30}\) of the task per minute. Therefore, to complete the entire task, it takes \(\frac{1}{\frac{1}{30}} = 30\) minutes.

Key Concepts

Individual Work Rates

When solving problems where two or more people work together, it's crucial to first understand individual work rates. This means figuring out how much work each person can do in a unit of time, typically a minute or an hour.

To find someone's individual work rate, divide 1 by the total time it takes for that person to complete the task on their own.

For example, in the given exercise, Gina takes 90 minutes to file all invoices. Therefore, her individual work rate is \(\frac{1}{90}\) of the task per minute. Importantly, Hilda works twice as fast as Gina. To determine her work rate, double Gina's rate: \(\frac{2}{90} = \frac{1}{45}\). So, Hilda can file \(\frac{1}{45}\) of the invoices per minute.

Combined Work Rates

Understanding combined work rates involves adding the individual work rates of all people involved. This gives you a rate at which the task is completed when everyone works together.

In our example, Gina can complete \(\frac{1}{90}\) of the task per minute, and Hilda can do \(\frac{1}{45}\) of the task per minute.
Their combined work rate is found by adding these together: \(\frac{1}{90} + \frac{1}{45}\).

Before you add the rates, ensure the fractions have a common denominator. For this, you often need to convert the work rates to a unified base to make the addition possible.

Common Denominator

To correctly combine fractions like individual work rates, you must use a common denominator. This is the same base for both fractions.

Let's take Gina and Hilda’s rates as an example, \(\frac{1}{90}\) and \(\frac{1}{45}\). To add these, we first find a common denominator for 90 and 45. The smallest common denominator here is 90.
Convert \(\frac{1}{45}\) to \(\frac{2}{90}\). Now, adding the work rates: \(\frac{1}{90} + \frac{2}{90} = \frac{3}{90} = \frac{1}{30}\).

The combined rate \(\frac{1}{30}\) means that together, Gina and Hilda complete the entire task in 30 minutes.