Question

$X=$ Profit amount of Mobile; $Y=$ Discount amount of Mobile. Decide the relation between $X$ and $Y$.

• A. If $X > Y$
• B. If $X < Y$
• C. If $X \geq Y$
• D. If $X \leq Y$ or can't be determined

Hint:

If profit amount equals discount amount, then $X \geq Y$ is chosen, since equality satisfies the condition.

Answer 1

The Correct Option is C

From earlier calculations:

Mobile: $CP_{\text{Mob}} = \text{₹}\,20833$, $SP_{\text{Mob}} = \text{₹}\,31250$, $MP_{\text{Mob}} = \text{₹}\,41666$

Profit amount of Mobile: 
\[X = SP - CP = 31250 - 20833 = \text{₹}\,10417\]

Discount amount of Mobile:  
\[Y = MP - SP = 41666 - 31250 = \text{₹}\,10416\]

Here, $X = \text{₹}\,10417$ and $Y = \text{₹}\,10416$.  So, $X \geq Y$ is the correct relation.
 

Answer 2

Let the cost price of the Mobile be \( C \). It is sold at a \( 50\% \) profit, so the selling price is \( 1.5C \), which makes the profit amount \( X = 1.5C - C = 0.5C \). A \( 25\% \) discount is given on the marked price to reach this same selling price, so \( 0.75 \times MP = 1.5C \), giving \( MP = 2C \). The discount amount is therefore \( Y = 2C - 1.5C = 0.5C \).

Substituting the actual cost price of the Mobile, \( C = ₹20833.33 \), gives \( X = ₹10416.67 \) and \( Y = ₹10416.67 \), confirming that the two amounts are identically equal for these percentage figures, independent of rounding.

Now check each option:

1. If \( X \gt Y \): This would need \( X \) to exceed \( Y \), but they are equal, so this is false.
2. If \( X \lt Y \): This would need \( X \) to be less than \( Y \), but they are equal, so this is false.
3. If \( X \geq Y \): Since \( X = Y \), the condition \( X \geq Y \) is satisfied.
4. If \( X \leq Y \) or can't be determined: The relation here is fully determinable, since both amounts equal \( 0.5C \), so the can't-be-determined part does not apply, and the determinate relation is better expressed as \( X \geq Y \).

The relation that holds is \( X \geq Y \).

So, the correct answer is If \( X \geq Y \).