Question

A man and a boy together dig \(600 \mathrm{~kg}\) of earth in an hour \((600 \mathrm{~kg} / \mathrm{hr})\) working independently but simultaneously. How long would the man take working alone? (1) The man moves \(200 \mathrm{~kg} / \mathrm{h}\) more than the boy (2) The boy works twice as long as the man A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

The man would take 1.5 hours working alone, determined using statement (1) alone.

Step-by-step solution

- Interpret the Problem

Understand that the combined rate of digging is given, and individual rates need to be determined. The goal is to find out how long the man takes working alone.

- Break Down Given Information

From the problem, we know: 1. Man and boy together dig 600 kg/hour. 2. Man's rate (M) is 200 kg/hour more than the boy's rate (B).

- Set Up Equations from Given Information

From statement (1): Let the boy’s rate be B kg/hr. Then the man's rate is: M = B + 200 Since they work together, we have: M + B = 600.

- Solve for Individual Rates Using Equation

Substitute M into the combined rate: (B + 200) + B = 600 2B + 200 = 6002B = 400 B = 200 Thus, the boy's rate is 200 kg/hr and the man's rate is 400 kg/hr.

- Evaluate Using Statement 2

Statement (2) says the boy works twice as long as the man. This information isn't necessary to solve for individual rates, but that statement alone doesn't help find how long the man takes to dig alone. We already found the rates from statement (1).

- Determine Time for Man Alone

Given the man alone can dig at 400 kg/hour rate, the time taken to dig 600 kg is: Time = Total work / Rate = 600 / 400 = 1.5 hours.

Key concepts

Rate of Work

The rate of work is a key concept in problems involving tasks completed over time. Here, the rate represents how much work is done per unit of time.
In this problem, the rate is given in terms of kilograms of earth dug per hour. The combined rate is the sum of the individual rates of both workers. Understanding each worker's rate separately helps in determining how long each would take to complete the task individually. Essentially, if you know one worker's rate, you can deduce the other's rate using the combined rate.

Combined Rates

When two or more workers or machines are involved, their combined rate is the sum of their individual rates.
In this exercise, we know that the man and boy together have a combined rate of 600 kg/hr. This means that if they work together for one hour, they will collectively dig 600 kg of earth. Using this combined rate, along with additional information, we can separate their rates:
- Man's rate (M) and boy's rate (B) add up to 600 kg/hr.
- Given one rate in relation to the other, we can substitute it back into the combined rate equation to find individual rates.

Equations

Equations are crucial for solving work-rate problems. They help in forming relationships between different variables.
In this problem, we set up two key equations:
- From the combined rate: \(M + B = 600\)
- From the information given: \(M = B + 200\)
By solving these equations together, we can isolate each variable. Substituting one equation into the other allows us to find specific values for individual rates:
- Substitute \(M\) in the combined rate equation to get: \(B + 200 + B = 600\)
- Simplifying, we find \(2B + 200 = 600\) and eventually \(B = 200\)
This gives us the boy's rate, and then, using \(M = B + 200\), we find the man's rate as \(M = 400\) kg/hr.
Through these steps, we can determine how long each would take to complete the task working alone.