Percentage Decrease When Dividing by 6 Instead of Multiplying by 6

The question at hand is straightforward but offers a deeper mathematical insight into the difference between multiplication and division operations. Specifically, we seek to understand the percentage decrease when a number is divided by 6 instead of being multiplied by 6. This article will explain the calculations step by step using various examples and approaches to ensure clarity.

Mathematical Representation

Let's assume the initial number is x. We will represent the two operations involved:

Multiplying by 6: ( x times 6 6x ) Dividing by 6: ( x div 6 frac{x}{6} )

The difference between these two results can be expressed as:

( 6x - frac{x}{6} frac{36x - x}{6} frac{35x}{6} )

Calculating the Percentage Decrease

To find the percentage decrease, we use the formula:

( text{Percentage Decrease} frac{text{Original Value} - text{New Value}}{text{Original Value}} times 100 )

Substituting our values, we get:

( frac{6x - frac{x}{6}}{6x} times 100 frac{frac{35x}{6}}{6x} times 100 frac{35x}{6} times frac{1}{6x} times 100 )

Which simplifies to:

( frac{35}{36} times 100 frac{3500}{36} approx 97.22% )

Verification Through Different Approaches

Let's verify the above calculation through different methods to ensure accuracy and understanding:

Example 1: X 6

Let ( x 6 ):

Multiplying by 6: ( 6 times 6 36 ) Dividing by 6: ( frac{6}{6} 1 ) Decrease: ( 36 - 1 35 ) Percentage Decrease: ( frac{35}{36} times 100 approx 97.22% )

Example 2: X 600

Let ( x 600 ):

Multiplying by 6: ( 600 times 6 3600 ) Dividing by 6: ( frac{600}{6} 100 ) Decrease: ( 3600 - 100 3500 ) Percentage Decrease: ( frac{3500}{3600} times 100 approx 97.22% )

General Case: Y 6x, Z x/6

Let's consider ( y 6x ) and ( z frac{x}{6} ):

Difference: ( 6x - frac{x}{6} frac{35x}{6} ) Percentage Decrease: ( frac{frac{35x}{6}}{6x} times 100 frac{35x}{6} times frac{1}{6x} times 100 frac{350}{36} times 100 approx 97.22% )

Therefore, the percentage decrease is consistently calculated as 97.22%, confirming the initial result.

Conclusion

The mathematical theory and multiple examples have shown that when a number is divided by 6 instead of being multiplied by 6, the percentage decrease is consistently 97.22%. This provides insight into the significant difference between these two operations, allowing for precise predictions and calculations in various fields of mathematics and real-life applications.

Understanding such operations is crucial for students, mathematicians, and professionals who require accurate calculations. Whether in finance, engineering, or educational settings, this knowledge can be invaluable in solving complex problems.