x% оf y is y% of:
x
Understanding the Percentage Problem
The question asks us to find a value such that "x% of y" is equal to "y% of" that value. This is a classic percentage problem that can be solved by setting up an equation. Let's break down the phrases into mathematical terms.
Translating Percentages to Equations
Percentages are essentially fractions out of 100. So, "x%" can be written as \( \frac{x}{100} \) and "y%" can be written as \( \frac{y}{100} \). The word "of" in mathematics usually means multiplication.
- "x% of y" translates to \( \frac{x}{100} \times y \) or \( \frac{xy}{100} \).
- "y% of [something]" translates to \( \frac{y}{100} \times z \), where \( z \) is the unknown value we are trying to find.
The problem states that these two expressions are equal:
\( \text{x% of y} = \text{y% of z} \)
\( \frac{xy}{100} = \frac{yz}{100} \)
Solving the Percentage Equation
Now we have the equation \( \frac{xy}{100} = \frac{yz}{100} \). Our goal is to isolate \( z \).
- Multiply both sides of the equation by 100 to eliminate the denominators:
\( 100 \times \frac{xy}{100} = 100 \times \frac{yz}{100} \)
\( xy = yz \)
- Now, to find \( z \), we need to divide both sides of the equation by \( y \). We assume \( y \) is not zero, as percentages are usually non-zero in such problems.
\( \frac{xy}{y} = \frac{yz}{y} \)
\( x = z \)
So, the unknown value \( z \) is equal to \( x \). This means "x% of y" is equal to "y% of x".
Comparing with Options
We found that the missing value is \( x \). Let's look at the provided options:
| Option Number | Option Value | Does it Match \( x \)? |
|---|---|---|
| 1 | \( 100x \) | No |
| 2 | \( x \) | Yes |
| 3 | \( \frac{x}{10} \) | No |
| 4 | \( \frac{y}{10} \) | No |
Option 2 matches our calculated value of \( x \).
Conclusion
By setting up the equation based on the definition of percentages, we found that "x% of y" is equal to "y% of x".
Revision Table: Percentage Concepts
| Concept | Explanation | Mathematical Form |
|---|---|---|
| Percentage | A fraction out of 100 | \( p\% = \frac{p}{100} \) |
| 'of' | Indicates multiplication | \( a \text{ of } b = a \times b \) |
| p% of q | 'p' percent of 'q' | \( \frac{p}{100} \times q \) |
Additional Information: Properties of Percentages
The problem highlights an interesting property of percentages: the symmetry between the base and the rate. That is, x% of y is always equal to y% of x.
Let's see why this is always true:
- x% of y = \( \frac{x}{100} \times y = \frac{xy}{100} \)
- y% of x = \( \frac{y}{100} \times x = \frac{yx}{100} \)
Since \( xy = yx \) (due to the commutative property of multiplication), it follows that \( \frac{xy}{100} = \frac{yx}{100} \). This property is useful for quickly solving or verifying certain percentage problems. For example, 10% of 50 is \( \frac{10}{100} \times 50 = 5 \). And 50% of 10 is \( \frac{50}{100} \times 10 = 5 \). The results are the same.
Important Questions from Percentage
Radha saves 25% of her income. If her expenditure increases by 20% and her income increases by 29%, then her savings increase by;
The income of A is 45% more than the income of B and the income of C is 60% less than the sum of the incomes of A and B. The income of D is 20% more than that of C. If the difference between the incomes of B and D is Rs. 13200, then the income (in Rs.) of C is:
The price of cooking oil increased by 25%. Find by how much percentage a family must reduce its consumption in order to maintain the same budget.
The population of a city increased by 30% in the first year and decreased by 15% in the next year. If the present population is 11,050 then population 2 years ago was:
The income of A is 30% less than the income of B and the income of B is 137.5% more than that of C. If the income of A is Rs. 28500 less than that of B, then the income (in Rs.) of C is:
