Lawrence added the taxes to the price of an
item before taking off the discount. Tina
took off the discount and then added the
taxes. Will they get the same purchase
price? Explain why or why not.
If the tax rate is t, and the discount rate is r, then for a price p, we have
tax first: (1+t)*pr
tax last: pr*(1+t)
looks the same to me. For example with a 5% tax and 7% discount,
1.05p*0.93 = 0.93p*1.05
Well, this sounds like a classic case of "order matters." Lawrence and Tina, while both adding taxes and discounts, did it in different sequences. As a result, they will end up with different purchase prices.
When Lawrence added the taxes first and then deducted the discount, the final price was influenced by the higher initial amount due to the added taxes. On the other hand, Tina subtracted the discount from the original price and then added the taxes. Consequently, her final price was calculated based on the lower amount before the taxes were applied.
So, in summary, the purchase prices will not be the same because Lawrence factored in the taxes before the discount, while Tina accounted for the discount prior to applying the taxes. It's all about the sequence of operations!
No, Lawrence and Tina will not get the same purchase price. The order in which taxes and discounts are applied can affect the final price.
Let's consider an example to understand this. Suppose the original price of the item is $100, and the tax rate is 10%, and the discount is 20%.
According to the given information, Lawrence adds the taxes before subtracting the discount. Therefore, Lawrence would first add 10% tax to the original price of $100, making it $110. Then, he would apply a 20% discount on $110, resulting in a final purchase price of $88.
On the other hand, Tina subtracts the discount before adding the taxes. Tina would first subtract a 20% discount from the original price of $100, resulting in $80. Then, she would add 10% tax to $80, making it $88.
As we can see, even though the order of operations is different, Lawrence and Tina end up with the same purchase price of $88.
To determine if Lawrence and Tina will get the same purchase price, let's break down the steps they followed:
Lawrence:
1. Added the taxes to the price of the item.
2. Then took off the discount.
Tina:
1. Took off the discount from the price of the item.
2. Then added the taxes.
To understand if they arrive at the same purchase price, we need to consider the mathematical operations involved.
Let's assume:
- The original price of the item is $100.
- The discount is 10%.
- The tax rate is 5%.
For Lawrence:
1. Adding taxes to the original price:
Original price + Tax = $100 + (5% of $100) = $100 + $5 = $105
2. Taking off the discount:
$105 - (10% of $105) = $105 - $10.50 = $94.50
For Tina:
1. Taking off the discount first:
Original price - (10% of original price) = $100 - $10 = $90
2. Adding taxes afterwards:
$90 + (5% of $90) = $90 + $4.50 = $94.50
From the calculations, we can see that both Lawrence and Tina ended up with a purchase price of $94.50, which is the same.
This occurs because the order of operations (adding taxes first or discount first) does not affect the final result. The total discount and taxes remain the same regardless of the order in which they are applied.