Question

Three men and two women can do a piece of work in 4 days, while two men and three women can do the same work in 5 days. Rs 44 is given to a woman for her contribution towards work, per day. What is the amount received by a man per day? (a) Rs, 88 (b) Rs. 144 (c) Rs. 154 (d) can't be determined

Short answer

Answer: A man receives Rs. 154 per day for his contribution to the work.

Step-by-step solution

Calculate work capacities

We're given that 3 men (M) and 2 women (W) finish the work in 4 days, so we write: 3M + 2W = 4 (total work, as they finish it in one day less) Similarly, we're given that 2 men (M) and 3 women (W) finish the work in 5 days, so we write: 2M + 3W = 5 (total work, as they finish it in one day more)

Find the relationship between M and W

To find the relationship between M and W, we need to solve these two equations simultaneously. We can multiply the first equation by 2 and the second equation by 3 to make the coefficients of W the same: 6M + 4W = 8 6M + 9W = 15 Now, subtract the first equation from the second equation: 5W = 7

Calculate the work capacity of women

We can now find W by dividing both sides of the equation by 5: W = \frac{7}{5}

Calculate the work capacity of men

Insert the value of W found above into the first equation to get the work capacity of men: 3M + 2(\frac{7}{5}) = 4 3M = 4 - \frac{14}{5} 3M = \frac{6}{5} M = \frac{2}{5}

Calculate the ratio of men's work capacity to women's work capacity

Now, we can find the ratio of M to W using the values calculated above. M : W = \frac{2}{5} : \frac{7}{5} (Comparing only the numerators, as both denominators are equal) M : W = 2 : 7

Calculate the amount received by a man per day

A woman earns Rs 44 per day. To find the amount received by a man per day, we'll use the ratio found above: 7W = 2M × Amount earned by a man per day (44) 7 × 44 = 2 × Amount earned by a man per day 308 = 2 × Amount earned by a man per day Amount earned by a man per day = \frac{308}{2} Amount earned by a man per day = 154 So, the correct answer is (c) Rs. 154 per day.

Key concepts

Quantitative Aptitude

Quantitative aptitude refers to the ability to handle numbers and solve problems related to mathematics, particularly in the context of competitive exams and assessments. It includes various topics such as arithmetic, algebra, geometry, and data interpretation. In the context of work and wage problems, quantitative aptitude involves applying mathematical concepts to real-world scenarios, like calculating wages based on work done.

For example, in the provided exercise, the problem required an understanding of the basic principles of algebra to set up and solve equations that represent the relationship between work done by individuals and their wages. This type of problem is common in quantitative aptitude tests, where the goal is often to interpret mathematical data and draw conclusions, like determining the daily wages of workers based on the total work performed and the pay distribution among them.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple unknown variables that are solved together to find a common solution. The challenge lies in finding values for the variables that satisfy all equations at once. In practice, there are various methods to solve these equations, such as substitution, elimination, graphing, and using matrices.

In the provided exercise, the elimination method is used to solve two simultaneous equations representing the work done by men and women. By carefully manipulating the equations to align the coefficients, we can eliminate a variable and solve for the other. This method of solving simultaneous equations is pivotal in cases where the relationships between variables need to be clearly understood and is a fundamental skill in tackling quantitative aptitude problems.

Ratio and Proportion

Ratio and proportion are fundamental concepts in mathematics that not only appear in everyday life but also in a variety of quantitative problems. A ratio is a way to compare two quantities of the same unit, while a proportion is an equation that states that two ratios are equivalent. These concepts are instrumental in making comparisons or determining shares in a given context.

For instance, in the problem we're discussing, once the individual work capacity of men and women was determined, the next step was to express these capacities as a ratio. By understanding how to manipulate and interpret ratios, the final proportion allowed for calculating the man’s wage, based on the known daily wage of a woman. Mastering ratio and proportion is thus crucial for solving work and wage problems and other related quantitative aptitude exercises.