Problem 64 - Algebra A Combined Function

Question

Mrs. Smith balances the company books in 8 hours. It takes her assistant 12 hours to do the same job. If they work together, find how long it takes them to balance the books.

Step-by-Step Solution

Answer

They take 4.8 hours working together.

Step 1: Determine Rates of Work

First, determine the rate at which each person works alone. Mrs. Smith can complete the job in 8 hours, so her rate is \( \frac{1}{8} \) of the job per hour. The assistant can complete the job in 12 hours, so their rate is \( \frac{1}{12} \) of the job per hour.

Step 2: Combine Rates

Add the rates of work together to find the combined rate when they work simultaneously. Mrs. Smith's rate is \( \frac{1}{8} \) per hour and her assistant's rate is \( \frac{1}{12} \) per hour. Their combined rate is \( \frac{1}{8} + \frac{1}{12} \). To add these, find a common denominator. The least common denominator of 8 and 12 is 24, so the combined rate is: \[ \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \] of the job per hour.

Step 3: Solve for Time

Once you have the combined rate, set it equal to \( \frac{1}{t} \), where \( t \) is the time it takes them working together to complete 1 job. Thus, \( \frac{5}{24} = \frac{1}{t} \). Solve for \( t \) by taking the reciprocal: \( t = \frac{24}{5} \).

Step 4: Calculate Time in Hours

Convert \( \frac{24}{5} \) to a decimal or a mixed number to find how many hours it takes them. Dividing 24 by 5 gives 4.8. Therefore, it takes them 4.8 hours working together.

Key Concepts

Rates of Work

Understanding the concept of "Rates of Work" is crucial in solving work-related problems. Each worker has their own speed or rate at which they complete a task, expressed as a fraction of the task per unit of time, usually per hour. For instance, if Mrs. Smith can complete a task in 8 hours, her work rate becomes \( \frac{1}{8} \) of the task per hour. Similarly, if her assistant takes 12 hours, their rate is \( \frac{1}{12} \). These individual work rates tell us part of the work done in an hour. By grasping these fundamentals, you can determine how long it takes for any number of workers to complete a job when working together. By finding each worker's rate, you can then move on to calculate the combined work rate.

Here's a helpful tip: always express work rates in fractions, as they represent "parts of the task" rather than percentages.

Combined Work Rate

Once we know each worker's rate, the next step is to calculate the "Combined Work Rate." This is the rate at which they work together to accomplish the job. We do this by simply adding their individual work rates. For instance, Mrs. Smith’s rate is \( \frac{1}{8} \), and the assistant's is \( \frac{1}{12} \). To add these, it’s important to find a common denominator. Here, the smallest common denominator for 8 and 12 is 24. So, we convert these rates:

• Mrs. Smith: \( \frac{3}{24} \)
• Assistant: \( \frac{2}{24} \)

Adding these gives a combined work rate of \( \frac{5}{24} \) of the task per hour. Knowing the combined rate lets you easily determine how long it will take them together to complete the task by using time calculation.

Time Calculation

With the combined work rate calculated, the next step is "Time Calculation." This tells you how long it takes for both workers to finish the task together. You can think of it like this: if their combined work rate is \( \frac{5}{24} \), they work through \( \frac{5}{24} \) of the task each hour. The amount of time \( t \) needed to complete the full job can be represented by the equation:\[ \frac{5}{24} = \frac{1}{t} \]This equation shows that the full task, or 1 job, is divided by the time. Solve for \( t \) by taking the reciprocal, thus \( t = \frac{24}{5} \) hours. To make it more comprehensible, convert \( \frac{24}{5} \) into a decimal, resulting in 4.8 hours. This means it takes both Mrs. Smith and her assistant a total of 4.8 hours to finish the task when working together. By mastering simple arithmetic, fraction addition, and reciprocal operations, such mathematical "Time Calculation" becomes straightforward.