Whoa, you think we can understand all that?

Whatever or however it is, the furniture store uses hocus-pocus with their Accountants, or with a resident Math guy, to hoodwink us on the apparent store loss in trying to sell the unattractive dining room set at a huge discount. No wonder furnitures are like jewels.

Anyway, let us see through the abracadabra.

"A furniture store bought a dining room suite which must be sold for $5250 to cover the wholesale price."

Meaning, wholesale price, W, = $5250.

"Overhead is 50%of wholesale price and normal profit is 25%of wholesale price."

-----Overhead, OH, = 0.5W

-----Normal Profit, P, = 0.25W

"The suite is marked so that the store can allow a 20% discount while still receiving the recquired retail price."

Okay.

Say the Sale Price is N.

So, if marked down 20%, then marked down price is 0.8N.

Retail price = W +OH +P

Hence, 0.8N = W +OH +P

0.8N = W +0.5W +0.25W

0.8N = 1.75W

N = (1.75 /0.8)w = 2.1875W ------Sale Price, or "List Price".

"When the suite remain unsold the store owner decided to mark the suite down for an inventory clearance sale."

So, new sale price, or Clearance Sale = x*N

where x = 1 -r

and r = rate of markdown

Hence, Clearance Sale = (1 -r)N ------------------(i)

"To arrive at the rate of mark down the owner decided that the store's profit should have to be no less than 20% of the normal profit..."

Aha, a big part of the hocus pocus.

Okay.

Normal Profit, P, = 0.25W

20% of P = 0.2(0.25W) = 0.05W -------------new profit, minimum profit.

"...and that part of the mark down would be covered through a reduction in the commission paidto the sales person.This was done by reducing the normalcommission(which accounts for 40%of the overhead)by 33.5%."

The biggest part of the confuse-the-enemy.

OH = 0.5W

Normal commission for Sales persons = 40% of OH = 0.4(0.5W) = 0.2W

Reduce that by 33.5% means the new commission is 66.5% of 0.2W. That is 0.665(0.2W) = 0.133W

Hence, (0.5W -0.2W) +0.133W = 0.433W --------------------new OH.

So, new retail price = W +(new OH) +(minimum profit)

New retail price = W +0.433w +0.05W = 1.483W ---------for maximum reduction because of minimum profit.

"What is the maximum rate of mark down that can be advertised instead of the usual 20% ?"

Max Clearance Sale Price = new retail price = 1.483W

So,

(1 -r)N = 1.483W

(1 -r)(2.1875W) = 1.483W

1 -r = 1.483 / 2.1875

1 -r = 0.677943

r = 1 -0.677943

r = 0.322

Therefore, instead of the usual 20% discount, the store can go as low as 32.2% discount. So the store can advertise a maximum of 32.2% marked down. ----------------answer.

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[In real world, do they ever advertize mark-downs in fractions of a percent?]