## Friday, October 04, 2013

### your easy algebra problem for the day (with GRE bonus)

The easy problem:

By August 31, 2014, Kevin will be 45, and will have lived nine years in Korea. In what future year can Kevin claim, on August 31, to have lived one-fourth of his life in Korea?

The GRE bonus problem (from the MGRE website):

A retailer previously bought an item from a wholesaler for \$20 and sold it to consumers for a retail price of \$35. After a wholesale cost reduction, the retailer reduces the retail price by 10%, yet the retailer’s profit on the item still increases by 20%. By what percent did the wholesale cost decrease?

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Charles said...

So, a little under four years!

With the help of Wolfram-Alpha (God I love that thing), I figured that in a little over four years (12 January 2018 to be precise), if I am still in Korea... I will have spent half my life here. Or, at least, I will have called Korea home for half my life, as I have spent time off the peninsula since moving here.

So I've got that going for me, which is nice.

Kevin Kim said...

The first problem is easy enough to set up. Make a fraction: [#years in Korea = 9] / [Kevin's age = 45]. Add "x" to the top and bottom to indicate the number of years that Kevin needs to live in Korea (while aging at the same rate), and set that fraction equal to 1/4. To wit:

(9 + x)/(45 + x) = 1/4

Cross-multiply:

45 + x = 4(9 + x)

Distribute:

45 + x = 36 + 4x

Combine like terms:

3x = 9

So:

x = 3 more years.

By that point, Kevin will be 48 on August 31, and 2014 + 3 = 2017.

QED.

The GRE problem is a bit trickier. We need to remember that the wholesale cost of an item is the "raw" cost, i.e., the cost of the item right as it comes out of the factory, and before it gets shipped to a store, where the price will be marked up and the item will be sold for a profit. Let's start defining terms, then:

Phase 1—
Orig. wholesale price (x) = \$20.
Orig. retail price (x + 15) = \$35.
Retailer's first profit = \$15.

Phase 2—
Wholesale cost reduction = ?%
Reduced wholesale cost = X
Retailer's 10% reduced price (y) = .9(35) = \$31.50.
Retailer's new profit = 1.2(15) = \$18.

The formula for calculating a raw profit is:

retail - wholesale = [raw-dollar profit]

The formula for calculating a percentage profit is:

(retail/wholesale - 1)•100 = %profit.

We have to make sure we answer the question, which is asking by what percent the wholesale cost decreased. I think the easiest way to get that answer is to ask, first, what the reduced wholesale cost was.

So:

[new retail] - [new wholesale] = [new profit]

Easy enough. Plug in the numbers and variables:

31.50 - X = 18.

So:

X = 31.50 - 18 = 13.50

The new wholesale price is \$13.50. The previous wholesale price was \$20. It's now just a matter of calculating the percentage change to get the answer we need.

Percentage change:

((new - old)/(old))•100 = ∆X%

(13.50 - 20)/20 = -0.325

That's a change of 32.5%.

QED.