The relationship between the list price, $\displaystyle L=\$600$, price rung up at the register, $\displaystyle S=\$400$, the listed discount $\displaystyle D=10\%$ and the unknown employee discount, $\displaystyle E$ is

$\displaystyle (1-E/100) =\frac{S}{L(1-D/100)}$

Substituting values

$\displaystyle (1-E/100) =\frac{400}{600\cdot(1-10/100)}$

Solving for $\displaystyle E$, $\displaystyle E=25.9\%$

If the initial price is $\displaystyle L=\$400$ and two discounts $\displaystyle D=40\%$ and $\displaystyle E=10\%$ are applied, the final sale price can be computed using

$\displaystyle L\cdot (1-D/100)\cdot (1-E/100)) = S$

Inserting values this is:

$\displaystyle \$400\cdot(1-40/100)\cdot(1-10/100))\% =\$216=S$

The savings off list price is

$\displaystyle L-S=\$400- \$216=\$184$.

The only item of clothing among Paris's items is the dress. It costs $89, so Paris will get a 10% discount - that is, she will pay 90% of the price, or

$\displaystyle \$89 \cdot 0.90 = \$80.10$

She will pay full price for the other items, so she will pay a total of

$\displaystyle \$80.10 + \$32.00 + \$42.00 + \$128.00 = \$282.10$.

Let $\displaystyle Y$ be the price of the hat, before discount and tax. 

Taking $\displaystyle D$ percent off the price of the hat is equivalent to charging $\displaystyle 100-D$ percent of the price of the hat, or multiplying the price of the hat by $\displaystyle \frac{100-D}{100}$.

 Analogously, adding $\displaystyle T$ percent to the amount paid as tax is the same as multiplying the amount paid by $\displaystyle 100+T$ percent, or $\displaystyle \frac{100+T}{100}$.

Therefore, the amount paid is

$\displaystyle X = Y \cdot \frac{100-D}{100}\cdot \frac{100+T}{100}$

To find $\displaystyle Y$ in terms of the other variables:

$\displaystyle Y \cdot \frac{100-D}{100}\cdot \frac{100+T}{100} = X$

$\displaystyle Y \cdot \frac{100-D}{100}\cdot \frac{100+T}{100} \cdot \frac{100}{100-D}\cdot \frac{100}{100+T}= X \cdot \frac{100}{100-D}\cdot \frac{100}{100+T}$

$\displaystyle Y = X \cdot \frac{100}{100-D}\cdot \frac{100}{100+T}$

To calculate a $\displaystyle 20\%$ discount, simply multiply the total by .2 and then subtract it from the original price.

$\displaystyle \$12.82\cdot.2=\$2.56$

$\displaystyle \$12.82-\$2.56=\$10.26$

The discount will apply to the jeans and the shirts, which, together, cost

$\displaystyle \$45 + \$45 + \$32 + \$32 = \$154$

Clothing purchases between $\displaystyle \$100$ and $\displaystyle \$199.99$ are discounted 20%, so Bryan will pay $\displaystyle 80\%$ of the price of the items, or 

$\displaystyle \$154 \cdot 0.80 = \$123.20$

Add to this the $18 cologne, which is not discounted:

$\displaystyle \$123.20 + \$18.00 = \$141.80$.

Harvey will receive a discount on the watch and the cologne; their prices total, before discount,

$\displaystyle \$19 + \$69 = \$ 88$.

Since this purchase totals under $100, he will receive 10% off - that is, he will pay 90% of the original price, or

$\displaystyle \$ 88 \cdot 0.90 = \$ 79.20$.

Add to this the price of the shirts, for which he will pay full price:

$\displaystyle \$ 79.20 + \$49.00 + \$49. 00+ \$49. 00 = \$226.20$