Understanding and Solving Ratio-Based Problems

Ratios and fractions are fundamental concepts in mathematics often used to distribute quantities in various fields, from finance to physics. This article aims to clarify the process of solving ratio-based problems, particularly focusing on the example of distributing a total of 9500 according to the ratio 1/2, 1/4, and 1/6.

Mathematical Foundations

In the problem at hand, the total amount of 9500 is to be shared in the ratio 1/2, 1/4, and 1/6. Understanding how to convert these ratios into fractions and then apply them to the total can be crucial for accurate distribution.

Step-by-Step Solution

The first step is to determine the component shares of the total based on the given ratio.

1. Calculating the Shares

Given that the ratio is 1/2, 1/4, and 1/6, we can calculate each share of the 9500 as follows:

1/2 of 9500: [frac{1}{2} times 9500 4750.00] 1/4 of 9500: [frac{1}{4} times 9500 2375.00] 1/6 of 9500: [frac{1}{6} times 9500 1583.65]

By adding these fractions, we get:

[frac{1}{2} frac{1}{4} frac{1}{6} frac{6}{12} frac{3}{12} frac{2}{12} frac{11}{12}]

This means that only (frac{1}{12}) of the total remains undistributed.

The smallest share, which is (frac{1}{12}) of 9500, can be calculated as:

[frac{1}{12} times 9500 791.67]

Thus, the smallest share is 791.67.

Correcting the Misunderstandings

The original phrasing of the problem has some intricacies that might confuse inexperienced solvers. For example, the values 1/2, 1/4, and 1/6 are not ratios but fractions. When these fractions are added, it sums up to (frac{11}{12}), leaving (frac{1}{12}) undistributed.

Alternative Solution

For a correct interpretation, the problem should have used the ratio 6:3:2, which sums up to 11, allowing the distribution to be completed without any remainder:

[frac{6}{11} times 9500 5181.82] [frac{3}{11} times 9500 2590.91] [frac{2}{11} times 9500 1727.27]

In this case, 1727.27 is the smallest share.

Conclusion

Understanding and solving ratio-based problems requires a clear grasp of fractions and the distribution of total amounts. Whether you are distributing payments or resources, the key is to ensure that the sum of the ratios equals the whole (100% or 1), ensuring that no amount is left out. Misunderstandings can lead to discrepancies, but with a thorough understanding of the mathematics involved, such issues can be avoided.