Question

Anand can do a piece of work in 45 days, but Bahuguna can do the same work in 5 days less, than Anand, when working alone. Anand and Bahuguna both started the work together but Bahuguna left after some days and Anand finished the remaining work in 56 days with half of his efficiency but he did the work with Bahuguna with his complete efficiency. For how many days they had worked together? (a) 6 (b) 8 (c) 9 (d) 12

Short answer

After Bahuguna left, Anand finished the remaining task with half of his efficiency in 56 days. Answer: Anand and Bahuguna worked together for 8 days.

Step-by-step solution

Calculate Work rate of Anand and Bahuguna

Let's assume the total work is comprised of 360 units. Then, Anand's work rate can be found by dividing his share of the total work by the time he takes to complete it. Similarly, Bahuguna's work rate can be found using the same method. Anand's work rate: \[ \frac{360}{45} = 8 \text{ units/day} \] Bahuguna's work rate: \[ \frac{360}{40} = 9 \text{ units/day} \]

Set up the equation for the days they worked together

Let's denote the number of days they worked together as 'x'. So, Bahuguna worked with Anand for x days, and Anand worked alone for 56 days after that with half of his efficiency. With half of Anand's efficiency, his work rate would be: \[ \frac{8}{2} = 4 \text{ units/day} \] The equation for the total units of work will be: \[ 8x + 9x + 4 (56) = 360 \]

Solve the equation for x

Simplifying the equation, we get: \[ 17x + 224 = 360 \] \[ 17x = 136 \] \[ x = 8 \] So, Anand and Bahuguna worked together for 8 days. The correct answer is (b) 8.

Key concepts

Work Rate Calculation

Understanding work rate calculation is fundamental to solving problems involving time and work. The work rate, sometimes referred to as efficiency, is generally the amount of work done per unit of time. In the context of our exercise, we consider work as a quantifiable amount of tasks, such as units that can be completed.

For Anand, the work rate was calculated by dividing the total work, arbitrarily set as 360 units, by the 45 days he takes to complete it alone, which results in a work rate of 8 units per day. Bahuguna works faster, completing the same amount in 40 days, hence his work rate is 9 units per day. By determining their work rates, we can set up linear equations to solve complex work and time problems, like the one presented.

Linear Equations

Solving linear equations is a crucial skill in math that allows us to find unknown values, such as the number of days two people work together. These equations are linear because the variables in them are to the first power and graph as a straight line.

In our problem, we construct a linear equation to represent the total work done by Anand and Bahuguna together and by Anand working alone at half his efficiency. The general form of a linear equation is usually given as ax + b = c. In context, 'x' represented the number of days they worked together, and by substituting the work rates and given days, we get 17x + 224 = 360. Solving for 'x' here answered the problem by isolating the variable and finding its value.

Efficiency in Work

The concept of efficiency in work applies to how effectively a person or machine performs a job. In terms of our example, efficiency is represented by the work rate. Anand’s efficiency is halved in the latter part of the problem, where he completes the work at 4 units per day instead of the usual 8 units.

This reduction in efficiency is a twist in the problem that affects the calculation. When his efficiency is halved, it takes Anand longer to complete the same amount of work, which influences our equation setup. Understanding how efficiency alters work rate can help solve more complex time and work problems, as it is essential to adapt efficiency changes when setting up equations.

Cooperative Work Problem Solving

In many real-world situations, tasks are completed faster and more efficiently when people work together - this is the essence of cooperative work problem solving. Our problem presents a situation where two individuals, Anand and Bahuguna, work together, combining their work rates to complete a task more quickly.

To solve such problems, it's crucial to calculate the individual work rates and then combine them for the period they work together. The sum of their individual work rates, multiplied by the number of days they work collaboratively, gives us the total work done together. In the given problem, the cooperative work period is what we're solving for, and it requires careful reading and application of the information about their individual and combined work rates to set up the correct equation.