How to Identify a Defective Machine Using a Weighting Machine Once

Have you ever faced the challenge of identifying a single defective machine that produces 120 gram balls among a group of 10 machines, each producing a different number of 100 gram balls, using a weighting machine only once? This article will guide you through the solution step by step, ensuring that you understand the process and can implement it effectively.

Understanding the Problem

You have 10 machines, and all of them produce 100 gram (GM) balls, except one that produces 120 GM balls. The task is to use a weighting machine only once to identify the defective machine. This problem requires a clever strategy to determine the defective machine based on the total weight of the balls taken from each machine.

Strategy to Identify the Defective Machine

The solution involves a well-planned strategy to collect a certain number of balls from each machine, combine them, and then weigh the total. Here’s how you can do it:

Step 1: Label the Machines

Label the machines from 1 to 10. Each machine produces a different number of balls corresponding to its label. That is, machine 1 will produce 1 ball, machine 2 will produce 2 balls, and so on up to machine 10, which will produce 10 balls.

Step 2: Collect Balls from Each Machine

Take a different number of balls from each machine as follows:

Machine 1: 1 ball Machine 2: 2 balls Machine 3: 3 balls Machine 4: 4 balls Machine 5: 5 balls Machine 6: 6 balls Machine 7: 7 balls Machine 8: 8 balls Machine 9: 9 balls Machine 10: 10 balls

Step 3: Calculate the Total Expected Weight

This step involves calculating the expected weight of the balls if all machines were producing the correct weight (100 GM). The total number of balls is:

[ text{Total balls} 1 2 3 ... 10 frac{10 times 11}{2} 55 text{ balls} ]

The expected weight, assuming all balls weigh 100 GM, would be:

[ text{Expected weight} 55 text{ balls} times 100 text{ GM} 5500 text{ GM} ]

Step 4: Weigh the Balls

Weigh the 55 balls you collected from all the machines. The actual weight will be different if there is a defective machine.

Step 5: Determine the Defective Machine

The difference between the expected weight and the actual weight will be used to find the defective machine. If the actual weight is (W), the difference can be calculated as:

[ text{Difference} 5500 - W ]

Since the defective machine produces 20 GM more per ball, divide the difference by 20 to find the defective machine:

[ text{Machine Number} frac{text{Difference}}{20} ]

For example, if the actual weight is 5520 GM:

Calculate the difference: (5500 - 5520 -20 text{ GM} ) Find the machine number: ( frac{-20}{20} -1 )

This indicates that machine 1 is the defective machine producing 120 GM balls.

Alternative Method

Another way to find the defective machine is to collect all ten balls and then remove one ball at a time while weighing. When the weight changes significantly, the defective ball has been removed.

Conclusion

By using this method, you can efficiently identify the defective machine in a single weighing. The problem requires careful planning and calculation, but the solution is both elegant and effective. This technique can be applied to various similar scenarios in quality control and industrial settings, ensuring that only the correct products are distributed to consumers.

Keywords

defective machine, weighing machine, quality control