Distributing Cement Bags Among Four Warehouses

Understanding the Problem

The problem involves distributing a certain amount of cement in 25 kg bags among four warehouses. The distribution is based on specific fractions of the total amount.

To solve this, we need to determine the total number of cement bags and the exact amount each warehouse gets. Let's break it down step by step.

Setting Up the Problem

Let the total number of 25 kg cement bags be ( X ). The distribution among the warehouses is as follows:

First warehouse: ( frac{X}{3} ) Second warehouse: ( frac{X}{4} ) Third warehouse: ( frac{2}{3} ) of the rest after the first two warehouses Fourth warehouse: 310 tonnes

Solving the Problem

1. **Total Bags Calculation:** [ text{Rest after first two warehouses} X - frac{X}{3} - frac{X}{4} X left( 1 - frac{1}{3} - frac{1}{4} right) X left( frac{12 - 4 - 3}{12} right) frac{5X}{12} ] [ text{Third warehouse gets: } frac{2}{3} times frac{5X}{12} frac{1}{36} frac{5X}{18} ] [ text{Fourth warehouse gets: } 310 text{ tonnes} ] [ text{Therefore: } frac{5X}{18} 310 frac{5X}{36} ] [ 310 frac{5X}{36} ] [ X frac{310 times 36}{5} 2232 text{ tonnes} ]

2. **Converting Tonnes to Bags:** [ 2232 text{ tonnes} times 1000 text{ kg/tonne} div 25 text{ kg/bag} 2232 times 40 89280 text{ bags} ]

3. **Distribution to Each Warehouse:** [ text{First warehouse: } frac{X}{3} times 1000 / 25 2232 / 3 times 40 29760 text{ bags} ] [ text{Second warehouse: } frac{X}{4} times 1000 / 25 2232 / 4 times 40 22320 text{ bags} ] [ text{Third warehouse: } frac{5X}{18} times 1000 / 25 5/18 times 2232 times 40 24800 text{ bags} ] [ text{Fourth warehouse: } 310 times 1000 / 25 12400 text{ bags} ]

Conclusion

The distribution of cement bags among the four warehouses is summarized as follows:

First warehouse: 29,760 bags (744 tonnes) Second warehouse: 22,320 bags (558 tonnes) Third warehouse: 24,800 bags (620 tonnes) Fourth warehouse: 12,400 bags (310 tonnes)

By following the steps above, we have accurately determined the total number of cement bags distributed and the exact amount each warehouse receives.