A store manager sets the marked price of a smart gadget 50% above its cost price and earns a profit of $160 after giving a 20% discount on the marked price. What would be the manager's profit if the gadget were sold at a 10% discount on the marked price?

The correct answer is 280.

Step-by-step explanation:

Step 1: Express Marked Price in terms of Cost Price

Let the Cost Price of the smart gadget be $CP$.

Since the marked price is set 50% above the cost price, we have:

$MP=CP+0.50\times CP=1.5\times CP$

Step 2: Determine Cost Price using the initial profit condition

When a discount of 20% is given on the marked price, the selling price ($S{P}_1$) is:

$S{P}_1=MP\times (1-0.20)=0.8\times MP$

Substituting $MP=1.5\times CP$:

$S{P}_1=0.8\times 1.5\times CP=1.2\times CP$

The profit earned at a 20% discount is given as $160:

$\text{Profit}=S{P}_1-CP$

$160=1.2\times CP-CP$

$160=0.2\times CP$

$CP=\frac{160}{0.2}=800$

So, the cost price of the gadget is $800.

Step 3: Calculate Marked Price

$MP=1.5\times 800=1200$

So, the marked price is $1200.

Step 4: Calculate Profit at a 10% discount

If the gadget is sold at a 10% discount on the marked price, the new selling price ($S{P}_2$) will be:

$S{P}_2=MP\times (1-0.10)=0.9\times 1200=1080$

The manager's profit under this condition is:

$\text{New Profit}=S{P}_2-CP=1080-800=280$

Thus, the profit when sold at a 10% discount is $280.