Understanding the Percentage Relationship Between X and Y
This article explores the mathematical relationship between two variables, X and Y, and how to express one in terms of the other using percentage calculations. We will delve into algebraic manipulations to find out what percentage of X is Y, given that X is some fraction of Y.
Case 1: Expressing Y as a Percentage of X
Given that ( X 80% text{ of } Y ), we can write:
( X frac{80}{100}Y frac{4}{5}Y )
To express Y in terms of X, we rearrange:
( Y frac{5}{4}X )
Now, to find what percentage of X is Y, we use the formula:
( frac{Y}{X} times 100 frac{frac{5}{4}X}{X} times 100 frac{5}{4} times 100 125% )
Therefore, Y is 125% of X.
Case 2: Expressing Y in Terms of X Given a Specific Percentage
Given that X is 70% of Y, we start with:
( X frac{70}{100}Y )
Let Y be some percentage of X, represented by ( z% ) of X. Hence,
( X frac{z}{100}Y )
Substituting the given relationship:
( frac{7}{10}Y frac{z}{100}Y )
( frac{7}{10} frac{z}{100} )
( 700 10z )
( z frac{700}{10} 70 times 10 142.85 )
Hence, Y is approximately 142.85% of X.
General Case: Expressing Y as a Percentage of X
Given an equation like ( X frac{a}{b}Y ), to find what percentage of X is Y, we use the formula:
( text{Percentage} frac{Y}{X} times 100 )
For example, if ( X 0.7Y ), we can write:
( text{Percentage} frac{Y}{0.7Y} times 100 frac{1}{0.7} times 100 approx 142.86% )
This shows that Y is approximately 142.86% of X.
Algebraic Manipulation Simplified
Considering the manipulation given:
( X frac{80}{100}Y frac{4}{5}Y )
Expressing Y in terms of X:
( Y frac{5}{4}X )
To convert to a percentage:
( frac{5}{4} times 100 125% )
Thus, Y is 125% of X.
Conclusion
In conclusion, understanding the relationship between X and Y through percentage calculations involves algebraic manipulations and the application of the percentage formula. These techniques are useful in various mathematical and real-world scenarios, such as finance, physics, and engineering.