Answer the following questions in full.

$1.$ Explain the difference between a trade discount and a cash discount.

$2.$ Give a mathematical example of a trade discount, whereby the original price is $$150\mathrm{.}00$ and it is discounted by $20\%$ if you buy $10$ units. How much will you pay for the $10$ units? What is the average price for $1$ unit?

$3.$ Give a mathematical example of a double trade discount, whereby the original price is $$200\mathrm{.}00$ and it is discounted by $20\%$ if you pay $10$ units, and then it is discounted and additional $10\%$ if you buy $20$ units. $($This requires two calculations$).$

a$.$ How much will you buy $10$ units?

b$.$ What is the average price for $1$ unit under the first scenario?

c$.$ How much will you pay for the $20$ units?

d$.$ What is the average price for $20$ unit?

$5.$ One example of a Cash Discounts is to receive a $2\%$ discount off of the net price of merchandise if one pays within $10$ days $(2/10,$ n$30).$ Based on this scenario, a merchant offers you $2/10,$ n$30$ on a $$30,000$ purchase made on $9/1.$

a$.$ If you pay your bill on $9/7,$ how much will you owe?

b$.$ If you pay your bill on $9/27,$ how much will you owe?

c$.$ Give a brief explanation in the savings and how this is a good opportunity for both the buyer and the seller?

$6.$ You are a purchasing agent at ABC Inc. You recently made a discounted purchase of $$45,000$ on a $$60,000$ item.

d$.$ Calculate the percentage discount you received on this purchase.

e$.$ Also, show the formula used in your calculation.