Problem 1 - A History of Mathematics: An Introduction

Question

Of three workmen, \(A\) can finish a given job once in three weeks, \(B\) can finish it three times in eight weeks, while \(C\) can finish it five times in twelve weeks. How long will it take for the three workmen to complete the job together? (This exercise and the next two are from Newton's Universal Arithmetic.)

Step-by-Step Solution

Answer

Answer: The three workers will complete the job together in 8/9 weeks.

Step 1: Find the rate of each worker separately

A can complete the job once in 3 weeks, so his rate is: (1 job) / (3 weeks) = 1/3 job/week B can complete the job three times in 8 weeks, so his rate is: (3 jobs) / (8 weeks) = 3/8 job/week C can complete the job five times in 12 weeks, so his rate is: (5 jobs) / (12 weeks) = 5/12 job/week

Step 2: Find the combined rate of all three workers

Now, we need to find the combined rate of all three workers when they work together. To do this, we simply add their individual rates: Combined rate = A's rate + B's rate + C's rate Combined rate = (1/3 + 3/8 + 5/12) jobs/week

Step 3: Simplify the combined rate expression

To simplify the combined rate expression, we need to get a common denominator for all the fractions. The least common multiple of 3, 8, and 12 is 24. We can now rewrite the fractions with the same denominator: Combined rate = (8/24 + 9/24 + 10/24) jobs/week

Step 4: Add the fractions

Now, add the fractions together: Combined rate = (8/24 + 9/24 + 10/24) jobs/week Combined rate = (8 + 9 + 10) / 24 jobs/week Combined rate = 27/24 jobs/week

Step 5: Find the time it takes for all three workers to complete the job together

Since Rate = Work / Time, we can rearrange the equation to solve for time: Time = Work / Rate In this case, the work is to complete 1 job, and the combined rate of all three workers is 27/24 jobs/week. Therefore: Time = (1 job) / (27/24 jobs/week) Time = (1 * 24) / 27 weeks Time = 24/27 weeks

Step 6: Simplify the result

Finally, we simplify the fraction: Time = 24/27 weeks Time = 8/9 weeks The three workmen will be able to complete the job together in 8/9 weeks.

Key Concepts

Arithmetic

Arithmetic is a foundational mathematical skill that deals with the basic operations of numbers, such as addition, subtraction, multiplication, and division. In this exercise, we deal primarily with the addition of fractions to solve a work rate problem, where three workers contribute to completing a job. When working with their rates, expressed as fractions, arithmetic helps us combine these rates to find a total work rate. This demonstrates a real-world application of arithmetic, allowing students to see its relevance beyond theoretical exercises.

Key operations in finding the combined rate include:

• Identifying each workman's rate as a fraction.
• Adding these fractions to find a total or combined work rate.

These operations require careful handling of fractions and a clear understanding of how to manage different denominators.

Fraction Simplification

Fraction simplification is an essential skill, particularly when dealing with problems involving work rates. In this exercise, simplification is vital to ensure we interpret rates correctly and end with a clear, concise answer.

• Each worker's rate is initially calculated as a fraction.
• These fractions must be converted to equivalent fractions with a common denominator to be added together.
• Simplification occurs at the stage where we combine the rates.
• Finally, the result is expressed in its simplest form, emphasizing a concise representation of the answer.

The simplification process helps make complicated calculations manageable and results easier to understand. Remember, whether you're adding, subtracting, or even dividing fractions, always look for opportunities to simplify your fractions to keep your work neat and precise.

Least Common Multiple

The least common multiple (LCM) is a key concept when dealing with fractions, especially in the context of operations that involve adding or subtracting them. The LCM is the smallest number that is a multiple of two or more numbers. In this exercise, determining the LCM allows us to find a common denominator for the rates expressed as fractions.

Here’s how the least common multiple is used in the solution:

• Identify the denominators of the workers' rates: 3, 8, and 12.
• Calculate the LCM for these numbers, which in this case is 24.
• Adjust each fraction so that it has 24 as a common denominator, enabling straightforward addition.

This concept is crucial because it turns potentially challenging operations with different denominators into simpler calculations. Once common denominators are established using the LCM, fractions can be added easily and the total can be found, leading to accurate results in any work rate problem.