If X is 12.25% more than Y. then Y is approximately_____ less than X.

Question

Prepp Team · Accepted Answer

Understanding the Percentage Relationship Between X and Y

The question asks us to find out by what percentage Y is less than X, given that X is 12.25% more than Y. This is a common type of percentage change problem where we need to switch the base of the percentage calculation.

Step-by-Step Calculation of Percentage Less Than

Let's define our variables:

Let Y be the initial value.
Let X be the value that is 12.25% more than Y.

According to the problem statement, X is 12.25% more than Y. This can be written mathematically as:

$\text{X} = \text{Y} + 12.25\% \text{ of Y}$

$\text{X} = \text{Y} + \left(\frac{12.25}{100}\right) \times \text{Y}$

$\text{X} = \text{Y} + 0.1225 \text{Y}$

$\text{X} = \text{Y} (1 + 0.1225)$

$\text{X} = 1.1225 \text{Y}$

Now, we need to find out by what percentage Y is less than X. The difference between X and Y is $\text{X} - \text{Y}$. We want to express this difference as a percentage of X. The formula for percentage less than X is:

$\text{Percentage less than X} = \frac{\text{X} - \text{Y}}{\text{X}} \times 100$

We know that $\text{X} = 1.1225 \text{Y}$. Let's substitute this into the formula:

$\text{Percentage less than X} = \frac{1.1225 \text{Y} - \text{Y}}{1.1225 \text{Y}} \times 100$

$\text{Percentage less than X} = \frac{(1.1225 - 1) \text{Y}}{1.1225 \text{Y}} \times 100$

$\text{Percentage less than X} = \frac{0.1225 \text{Y}}{1.1225 \text{Y}} \times 100$

The term Y cancels out from the numerator and the denominator:

$\text{Percentage less than X} = \frac{0.1225}{1.1225} \times 100$

Now, we perform the calculation:

$\frac{0.1225}{1.1225} \times 100 = \frac{12.25}{112.25} \times 100$

Let's calculate the value:

$\frac{12.25}{112.25} \approx 0.109131$

$\text{Percentage less than X} \approx 0.109131 \times 100$

$\text{Percentage less than X} \approx 10.9131\%$

Rounding to one decimal place, this is approximately 10.9%.

Comparing with the Options

Let's compare our calculated value (approximately 10.91%) with the given options:

3%
10.9%
12%
15.6%

Our calculated value is closest to 10.9%.

Conclusion on Percentage Difference

If X is 12.25% more than Y, then Y is approximately 10.9% less than X. This shows that the percentage increase from Y to X is different from the percentage decrease from X to Y when measured relative to different bases (Y and X respectively).

Revision Table: Key Percentage Concepts

Concept	Formula (A vs B)	Explanation
A is P% more than B	$A = B + \frac{P}{100}B = B(1 + \frac{P}{100})$	A is calculated by adding P% of B to B.
A is P% less than B	$A = B - \frac{P}{100}B = B(1 - \frac{P}{100})$	A is calculated by subtracting P% of B from B.
Percentage A is of B	$\frac{A}{B} \times 100\%$	Compares A directly to B as a base.
Percentage Increase (from B to A)	$\frac{A - B}{B} \times 100\%$ (if A > B)	Change relative to the initial value (B).
Percentage Decrease (from A to B)	$\frac{A - B}{A} \times 100\%$ (if A > B)	Change relative to the new value (A).

Additional Information: Why Percentages Differ

It's important to understand why the percentage increase from Y to X (12.25%) is not the same as the percentage decrease from X to Y (approximately 10.9%). The reason is the base value used for the percentage calculation.

When we say X is 12.25% more than Y, the percentage is calculated with respect to Y as the base.
When we calculate how much less Y is than X, the percentage is calculated with respect to X as the base.

Since X is a larger value than Y, the same absolute difference (X - Y) will represent a smaller percentage when calculated with respect to the larger base (X) compared to the smaller base (Y).