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
In an examination, 25% of the candidates failed in Mathematics and 12% failed in English. If 10% of the candidates failed in both the subjects and 292 candidates passed in both the subjects, which one of the following is the number of total candidates appeared in the examination?
What is the value of 9% of 5500 + 2.4% of 1100 - 40% of 1600?
Population of a village is 7960 in which 4660 are female. If in that village 60% are literate in which 70% female are literate, then what is the number of literate male ?
The numbers of students of three classes of a school are in the ratio 4 : 5 : 6. If numbers of students in these classes increase by 25%, 20% and 25% respectively, then ratio of numbers of students will become:
In an examination, Ram obtained 20 % more than Ashok but 10% less than Rajesh. If the marks obtained by Ashok is 1080. Then the Percentage marks obtained by Rajesh if the full marks is 2000 ;
