Problem 23 - Elementary Algebra

Question

A delivery boy, working alone, can deliver all his goods in 6 hours. Another delivery boy, working alone, can deliver the same goods in 5 hours. How long will it take the boys to deliver all the goods working together?

Step-by-Step Solution

Answer

Answer: The delivery boys working together will take approximately 2.73 hours to deliver all the goods.

Step 1: Find the individual work rate of each delivery boy

To find the individual work rate, we can use the formula work_rate = work_done / time_taken. For each delivery boy, the work_done will be equal to 1, as they are completing the full task individually. So, for the first boy, his work rate is (1/6) of the task per hour, and for the second boy, his work rate is (1/5) of the task per hour.

Step 2: Calculate the combined work rate

To calculate the combined work rate, we simply add the individual work rates of both delivery boys. So, the combined work rate is (1/6) + (1/5). To add the fractions, we need the least common multiple (LCM) of 5 and 6, which is 30. Therefore, the combined work rate is (5+6)/30, which equals 11/30.

Step 3: Find the time needed when working together

Since we know the combined work rate (11/30), we can now find the time it will take for the boys to deliver all the goods working together. We can use the formula time_taken = work_done / work_rate. In this case, the work_done is still 1 task and the work_rate is 11/30. So, the time_taken will be (1) / (11/30) which equals (30/11) hours.

Step 4: Final answer

The boys will be able to deliver all the goods together in (30/11) hours or approximately 2.73 hours.

Key Concepts

Least Common Multiple

Finding the least common multiple (LCM) is an essential skill, especially when you deal with fractions and collaborative tasks like work rate problems. The LCM of two numbers is the smallest number that is a multiple of both. For example, if we take the numbers 5 and 6, their LCM can be found by listing their multiples:

• Multiples of 5: 5, 10, 15, 20, 25, 30…
• Multiples of 6: 6, 12, 18, 24, 30…

The smallest common multiple here is 30. Finding the LCM is crucial in work rate problems because it helps us to add fractions with different denominators, allowing us to combine individual work rates effectively.

Fraction Addition

Adding fractions might sound tricky, but it's quite straightforward once you know the steps. When you have fractions like \(\frac{1}{6}\) and \(\frac{1}{5}\), you need to find their sum. The first step is to determine the least common multiple, which in our case is 30. After identifying the LCM, you convert each fraction to have the same denominator:

• Convert \(\frac{1}{6}\) to \(\frac{5}{30}\) because \(6 \times 5 = 30\).
• Convert \(\frac{1}{5}\) to \(\frac{6}{30}\) because \(5 \times 6 = 30\).

Now that the denominators are the same, you can easily add the fractions: \(\frac{5}{30} + \frac{6}{30} = \frac{11}{30}\). This technique allows you to combine work rates, which are crucial in collaborative work situations.

Collaborative Work

Collaborative work problems often involve calculating how long it takes different individuals or machines, each working at their own pace, to complete a task together. These problems highlight the concept that combined efforts can accomplish tasks more efficiently.In this exercise, each delivery boy has a unique work rate: one can deliver alone in 6 hours, and the other in 5 hours. To find how long they work together, you first convert their work abilities into rates:

• First boy: \(\frac{1}{6}\) of the job per hour
• Second boy: \(\frac{1}{5}\) of the job per hour

Adding these rates gives the combined work rate. This idea illustrates the power of collaboration, showing how joint efforts can lead to faster completion of tasks.

Time Calculation

Time calculation is fundamental in understanding how long a task will take when combined forces are involved. Once you have the combined work rate, you can easily calculate the time needed to finish a job together. The total task is measured as '1' complete job. When the boys work together, their combined work rate is \(\frac{11}{30}\) of the job per hour. To find out how long they'll take together, you use the formula \(\text{time} = \frac{\text{work done}}{\text{work rate}}\).In this case, \(\text{work done} = 1\) and \(\text{work rate} = \frac{11}{30}\). So, \(\text{time} = \frac{1}{\frac{11}{30}} = \frac{30}{11}\). This tells you the boys need approximately 2.73 hours to deliver all the goods together, showing how effectively you can calculate collaboration advantages.