Ratio of Defective to Non-Defective Televisions: Understanding the Calculation

Introduction to the Problem

The problem presented involves a shipment of televisions, where 1/50 of them are defective. This problem requires us to find the ratio of defective to non-defective televisions. Understanding such ratios is crucial in quality control and can provide insights into the manufacturing process and the defect rate.

Definition and Core Concepts

A ratio is a comparison of two quantities, often expressed as a fraction. In this case, we are interested in comparing the number of defective televisions to the number of non-defective ones. The phrase '1/50 of the televisions are defective' provides us with the starting point for our calculation.

Calculation of Defective Televisions

Let's denote the total number of televisions as T.

The number of defective televisions, D, can be calculated as:

[ D frac{1}{50} T ]

This formula tells us that for every 50 televisions, 1 is defective.

Calculation of Non-Defective Televisions

The number of non-defective televisions, N, can be found by subtracting the number of defective televisions from the total:

[ N T - D T - frac{1}{50} T frac{49}{50} T ]

Here, the total number of televisions is reduced by the fraction that is defective, resulting in the remaining non-defective televisions.

Deriving the Ratio

To find the ratio of defective to non-defective televisions, we divide the number of defective televisions by the number of non-defective televisions:

[ text{Ratio} frac{D}{N} frac{frac{1}{50} T}{frac{49}{50} T} ]

The T terms cancel each other out, leaving:

[ text{Ratio} frac{1}{49} ]

This means that for every defective television, there are 49 non-defective ones.

Verification with Examples

To verify our solution, we can use a numerical example. Let's assume there are 100 televisions in the shipment:

Defective televisions: (frac{1}{50} times 100 2) Non-defective televisions: (100 - 2 98)

The ratio of defective to non-defective televisions is:

[ text{Ratio} 2 : 98 1 : 49 ]

This example confirms our earlier calculation.

Conclusion

The ratio of defective to non-defective televisions, given that 1/50 of the televisions are defective, is 1:49. This ratio can be useful in various applications, such as quality control, inventory management, and understanding the defect rate in a production process.