First, note that the complete cost of the shirt is \(\displaystyle \$13\).  Therefore, the store makes \(\displaystyle \$20 - \$13 = \$7\) in profit on the sale of the shirt.  At this point, the problem merely becomes one of percentage change.  They make \(\displaystyle \frac{\$7}{\$13}\) percent profit— approximately \(\displaystyle 0.54\) or \(\displaystyle 54\%\).

The first thing to do is figure out the sale price.  There are two ways to do this:

First way: \(\displaystyle 45 - 0.1 * 45 = 45 - 4.5 = 40.5\).

Second way: \(\displaystyle (0.9) * 45 = 40.5\)

(The second way is like saying, "The new price is 90% of the original."  This is really the easiest way to do problems like this and should be learned.)

Now, to find the profit, merely subtract the original amount, namely \(\displaystyle \$30\): \(\displaystyle 40.5 - 30 = \$10.5\).

Be careful. Based on the question, we apply the percent profit to the her cost to produce a cookie NOT to the $2.00 in revenue.  So our equation would be:

\(\displaystyle .23x+x=2.00\)

where \(\displaystyle x\) represents how much it costs Jessica to make a cookie.

We can simplify this to:

\(\displaystyle 1.23x=2.00\)

\(\displaystyle x=1.63\)

So it would cost Jessica $1.63 to make a cookie.

Be careful with this calculation.  If an item is making 10% profit, that means that it is making that profit on the original price.  

Therefore, if \(\displaystyle p\) is the original price, we can write:

\(\displaystyle 1.1 * p = 15.4\).  

Solving for \(\displaystyle p\), we get \(\displaystyle p = 14\).  

This means that the profit per item is $1.40.  Now, divide 10,000 by 1.4 and get 7,142.857.  Since you cannot sell partial items, you will need to sell 7,143 items in order to get a $10,000 profit.

If an item is selling for $17.10, its original price \(\displaystyle p\) can be found by the following equation: \(\displaystyle p * 1.14 = 17.1.\) 

Solving for \(\displaystyle p\), we get $15.00 as the original price.

If the item is sold at a 20% loss, this means that the retailer is getting only 80% of the original price \(\displaystyle p\).  This can be represented as

\(\displaystyle 0.8 * p = 22\).  

Solving for \(\displaystyle p\), we get

\(\displaystyle p = 27.50\)

There are two steps to this problem.  First, we must calculate how much profit is made per item.  The general form of the question we are asking is, "How much is 15% of $30?"  This can be rewritten:

\(\displaystyle p = 0.15 * 30 = 4.5\)

Now, if you make $4.5 per item, you need to divide the total profit desired ($25,000) by $4.5:

\(\displaystyle < items> = \frac{25,000}{4.5} = 5555.56\)

Be careful!  You need to sell 5,556 items, NOT 5,555.  At 5,555, you will have only made $24,997.5 in profit. 

This problem implies that some original price \(\displaystyle p\) was marked up by \(\displaystyle 17\%\).  

This can be represented in an equation as \(\displaystyle s = 1.17 * p\), where \(\displaystyle s\) is the final sale price. 

Therefore, we can rewrite the equation:

\(\displaystyle 37 = 1.17 * p\)

Divide both sides by 1.17:

\(\displaystyle p = 31.6239\)

Rounding, we can say that the cost of the item is $31.62.  Subtract this from $37 to get a profit of $5.38.

\(\displaystyle \\ \textup{Call \textit{c} the unit cost, then \textit{c} is also the unit profit, since 2\textit{c} is the unit price.}\\\textup{If 1,000 units are sold, 1,000\textit{c} profit is made, which we know equals \$5,000.}\\\textup{Solving the equation } 1000c=5000 \textup{ gives the desired solution } c=\$5.\)

\(\displaystyle 15\%\) of the total proceeds can be calculated as such:

\(\displaystyle \frac{15\%}{100\%}\cdot\$3250.00=\$487.50\)