Bunuel wrote:
A computer company prices its models using the formula \(P = 1.25^sa\), where P is the price per unit, a is a constant, and s is the speed rating, from 1 to 20, of the product. The company also grants a discount of 10% on orders of 100 to 249 units and a discount of 25% on orders of 250 or more units. Lotte orders 300 computers, all with the same speed rating. How many computers could she have purchased for the same price if she had chosen models with a speed rating 2 higher than the ones she chose?
A) 160
B) 192
C) 230
D) 240
E) 675
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:The total amount Lotte spent is:
Total cost = (original # of units purchased)(original price per unit)
Total cost = (300)(original price per unit)
Remember that because Lotte is buying 300 units, she is receiving a discount of 25% of the full price. (You don’t need to do anything with this information right now.)
If Lotte opts for a speed rating 2 higher, she will see an increase in unit price from 1.25^s to 1.25^{s+2}. Find the increase in P as follows:
\(1.25^{s+2}=(1.25^s)(1.25^2)a=(1.25^s)(5/4)^2a=(1.25^s)(25/16)a\)
The price of the faster models is 25/16 that of the price of the slower models. The new purchase would be:
Total cost = (new # of units purchased)(original price per unit)(25/16)
To keep the total cost the same, then, the new number of units purchased must be 16/25 the original number:
Total cost = (original # of units purchased)(16/25)(original price per unit)(25/16)
The new number of units purchased, then is (300)(16/25) = 192 computers.
But wait! You’re not quite done yet. At this quantity her discount drops from 25% to 10%. Because her discount is not as great, she can’t actually buy 192 computers; she has to buy fewer. Glance at the answers; only answer (A) is smaller than 192, so it must be the correct answer.
Here’s how to calculate that (but you don’t need to!). First, Lotte is losing the 25% discount, so divide the original price per unit by 75%, or 0.75: the new price per unit = original price per unit / 0.75. This was the true original price without any discount applied.
Next, Lotte is gaining a 10% discount, so multiply the price by 90%, or 0.9. The new price per unit = the original price per unit × 0.9 / 0.75.
This is now the new price per unit with the 10% discount applied. (Note: you cannot simply subtract and say that 25% – 10% = 15%. The discounts are taken on different starting values!)
Simplify:
New price per unit = 0.9 / 0.75 (original price per unit)
New price per unit = 90 / 75 (original price per unit)
New price per unit = 6 / 5 (original price per unit)
If the new price per unit is 6/5 of the original price per unit, then Lotte has to purchase 5/6 as many computers in order to keep the same total cost:
Total cost = (original # of units purchased)(5/6)(original price per unit)(6/5)
The new number of computers is 192 (5/6) = 160
The correct answer is (A).
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