Dividing a Sum of Money Among Laborers in a Given Ratio

Introduction

Suppose Mr. X has a certain amount of money with him that he needs to distribute among five laborers in the ratio 1/2 : 1/3 : 1/4 : 1/5 : 1/7. The question arises: how much money should Mr. X have to ensure that each laborer receives an exact number of rupees?

Understanding the Problem

Mathematically, we need to find the minimum amount of money Mr. X should have to distribute among the laborers in such a way that each laborer receives an exact number of rupees in the specified ratio. This problem involves the use of ratios and the least common multiple (LCM).

Step-by-Step Solution

Step 1: Finding the LCM of the Denominators

First, we need to find the least common multiple (LCM) of the denominators 2, 3, 4, 5, and 7.

Prime Factorization:

2 21
3 31
4 22
5 51
7 71

The LCM is calculated as follows:

LCM 22 * 31 * 51 * 71 4 * 3 * 5 * 7 420

Step 2: Converting the Fractions to a Common Denominator

Next, we convert each part of the ratio so that they share a common denominator of 420:

1/2 210/420
1/3 140/420
1/4 105/420
1/5 84/420
1/7 60/420

Step 3: Summing the Parts

We then sum all these parts:

210/420 140/420 105/420 84/420 60/420 599/420 3999/420

Step 4: Determining the Minimum Amount

The minimum amount Mr. X should have to distribute the money according to the given ratio, ensuring each laborer receives an exact amount, is 3999/420 rupees. To make it a whole number, the total amount must be a multiple of 420, and the sum of the parts is 3999 rupees.

Conclusion

Therefore, Mr. X should have a minimum of 3999 rupees to distribute the amount among the five laborers in the ratio 1/2 : 1/3 : 1/4 : 1/5 : 1/7 and ensure that each laborer receives an exact number of rupees.