/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 16 persons can reap \(\frac{1}{5... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 42

16 persons can reap \(\frac{1}{5}\) th field in 6 days. How many persons (with same efficiency) are required to reap rest of the field in 8 days? (a) 27 (b) 54 (c) 48 (d) 64

Short Answer

Expert verified
Answer: 48 persons.

Step by step solution

01

Calculate the person-days required to reap the entire field

Let x be the number of person-days to reap the entire field. Since 16 persons can reap \(\frac{1}{5}\)th of the field in 6 days, we can represent this as: $$\frac{1}{5}x = 16 \times 6$$
02

Solve for x

Now we will solve for x: $$x = 5 (16 \times 6) = 5(96) = 480$$ So the entire field requires 480 person-days to be reaped.
03

Subtract the person-days required for the first \(\frac{1}{5}\)th of field

As calculated above, \(\frac{1}{5}x = 16 \times 6 = 96\) person-days were required for the first \(\frac{1}{5}\)th of the field. We need to find the person-days required for the remaining \(\frac{4}{5}\)th of the field: $$480 - 96 = 384$$ So, 384 person-days are required for the remaining \(\frac{4}{5}\)th of the field.
04

Calculate the number of persons

Since we have to reap the remaining field in 8 days, we need to find out the number of persons required for these 384 person-days. Let's denote the required number of persons as y: $$y \times 8 = 384$$
05

Solve for y

Now we will solve for y: $$y = \frac{384}{8} = 48$$ So, 48 persons are required to reap the remaining \(\frac{4}{5}\)th of the field in 8 days. The correct answer is (c) 48.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Person-Days Calculation
In work and time problems, the concept of person-days is crucial. Person-days calculate the total amount of work done based on the number of people working and the number of days they work. Think of it as a measure of the total effort needed to complete a task.

To calculate person-days, you multiply the number of persons by the number of days they work. For instance, if it takes 16 people 6 days to complete \(\frac{1}{5}\) of a task, you would calculate the person-days as follows:
  • Number of persons = 16
  • Number of days = 6
  • Person-days for \(\frac{1}{5}\) of the task = 16 \times 6 = 96 person-days
Understanding person-days helps in determining how much work can be completed by any number of people over a set period. It essentially quantifies labor in terms of time and effort.
Fraction of Work
In these problems, the work is often divided into fractions to facilitate calculation. Fractions of work denote portions of the task completed, usually shown as a part of the whole task. For example, if a task is divided into 5 parts, each representing \(\frac{1}{5}\) of the total job, completing one part is depicted as having completed \(\frac{1}{5}\) of the work.

Applying this to our problem:
  • The initial work done by 16 persons in 6 days is \(\frac{1}{5}\) of the entire task.
  • This allows us to gauge how much effort is theoretically needed to complete the remaining \(\frac{4}{5}\) of the task.
By breaking down tasks into fractions, it simplifies calculating efforts needed to finish or plan the completion of various sections of the work.
Efficiency in Work
Efficiency refers to the rate at which work is done. In this context, it implies how effectively the labor force completes a given portion of the task in a certain timeframe. If all the workers have the same rate of working, their collective efficiency can be considered uniform.

For our exercise:
  • Given that 16 people can complete \(\frac{1}{5}\) of the job in 6 days encapsulates their efficiency.
  • The efficiency remains constant whether calculated over \(\frac{1}{5}\) or \(\frac{4}{5}\) of the work.
This efficiency helps predict outcomes for similar tasks under varying conditions, such as changes in the number of workers or the total amount of time available.
Algebraic Equations
Algebraic equations serve a powerful role in solving work and time problems. They offer an organized method for expressing unknowns and knowns to find solutions related to manpower and timeframe.

In the given problem:
  • Equations were set up to find the total person-days necessary to finish the complete work.
  • By expressing unknown outputs like total person-days (\(x\)) and persons required (\(y\)), calculations became manageable.
  • The equations \(\frac{1}{5}x = 16 \times 6\) and \(y \times 8 = 384\) were central to determining \(x\) and \(y\).
These equations simplify complex work-related questions into solvable steps through logical operations, enhancing comprehension and problem-solving abilities.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the value of \(\frac{p^{2}+q^{2}}{r^{2}+s^{2}}\), if \(p: q:: r: s\). (a) \(\frac{1}{4}\) (b) \(\frac{1}{9}\) (c) \(\frac{p s}{r q}\) (d) \(\left(\frac{p-q}{r-s}\right)^{2}\)

Time period \((T)\) of pendulum is directly proportional to the square root of length of string by which bob is attached to a fixed point and inversely proportional to the square root of gravitational constant ' \(g\) '. Time period of a bob is 3 seconds when the gravitational constant \(g\) is \(4 \mathrm{~m} / \mathrm{sec}^{2}\) and length of string is 9 metre, what is the time period of a bob having a string of length 64 metre and gravitational constant \(16 \mathrm{~m} / \mathrm{sec}^{2} ?\) (a) 4 seconds (b) 12 seconds (c) 16 seconds (d) 10 seconds

The speeds of rickshaw, car and scooter are in the ratio of \(3: 5: 6\). What is the ratio of time taken by each one of them for the same distance? (a) \(6: 5: 3\) (b) \(10: 6: 5\) (c) \(12: 7: 6\) (d) data insufficient

The angles of a triangle are in the ratio of \(2: 3: 4\). Find the measurement of greatest angle : (a) \(30^{\circ}\) (b) \(60^{\circ}\) (c) \(100^{\circ}\) (d) \(80^{\circ}\)

Four numbers are in proportion. The sum of the squares of the four numbers is 50 and the sum of the means is 5 . The ratio of first two terms is \(1: 3\) What is the average of the four numbers? (a) 2 (b) 3 (c) 5 (d) 6
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.