Answer:
The clearance price is $17.28.
The clearance price will not give the store any profit.
Step-by-step explanation:
We have been given that a store pays a manufacturer $20 for a hat. To make a profit, the store prices the hat 60% higher than the amount they paid.
The price of the hat after increasing the price by 60% will be:
![\text{Price of the hat after an increase of 60\%}=20+(\frac{60}{100}*20)](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20an%20increase%20of%2060%5C%25%7D%3D20%2B%28%5Cfrac%7B60%7D%7B100%7D%2A20%29)
![\text{Price of the hat after an increase of 60\%}=20+(0.60*20)](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20an%20increase%20of%2060%5C%25%7D%3D20%2B%280.60%2A20%29)
![\text{Price of the hat after an increase of 60\%}=20+12](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20an%20increase%20of%2060%5C%25%7D%3D20%2B12)
![\text{Price of the hat after an increase of 60\%}=32](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20an%20increase%20of%2060%5C%25%7D%3D32)
So, the price of hat after the increase of 60% will be $32.
As the sale reduces the price of hat by 10%, so the price of hat after sale will be:
![\text{Price of the hat after sale of 10\%}=32-(\frac{10}{100}*32)](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20sale%20of%2010%5C%25%7D%3D32-%28%5Cfrac%7B10%7D%7B100%7D%2A32%29)
![\text{Price of the hat after sale of 10\%}=32-(0.10*32)](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20sale%20of%2010%5C%25%7D%3D32-%280.10%2A32%29)
![\text{Price of the hat after sale of 10\%}=32-3.2](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20sale%20of%2010%5C%25%7D%3D32-3.2)
![\text{Price of the hat after sale of 10\%}=28.8](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20sale%20of%2010%5C%25%7D%3D28.8)
So the price of hat after 10% sale will be $28.8.
As the clearance sale reduces the price by 40%, so price of hat after clearance sale will be 28.8 minus 40% of 28.8.
![\text{Price of the hat after clearance sale}=28.8-(\frac{40}{100}*28.8)](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20clearance%20sale%7D%3D28.8-%28%5Cfrac%7B40%7D%7B100%7D%2A28.8%29)
![\text{Price of the hat after clearance sale}=28.8-(0.40*28.8)](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20clearance%20sale%7D%3D28.8-%280.40%2A28.8%29)
![\text{Price of the hat after clearance sale}=28.8-11.52](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20clearance%20sale%7D%3D28.8-11.52)
![\text{Price of the hat after clearance sale}=17.28](https://tex.z-dn.net/?f=%5Ctext%7BPrice%20of%20the%20hat%20after%20clearance%20sale%7D%3D17.28)
Therefore, the clearance price is $17.28.
As store has purchased the hat from manufacturer for $20 and sold it $17.28, therefore, the store will not have a profit.