The price of a consumer good increased by p%

VJesus12 wrote:The price of a consumer good increased by p% during 2012 and decreased by 12% during 2013. If no other change took place in the price of the good and the price of the good at the end of 2013 was 10% higher than the price at the beginning of 2012, what was the value of p?

A. −2%
B. 2%
C. 22%
D. 25%
E. Cannot be determined

Let $100 be the original price

The price of a consumer good increased by p% during 2012
p% = p/100, so a p% INCREASE is the same a multiplying the original price by 1 + p/100
So, the new price = ($100)(1 + p/100)

The price then decreased by 12% during 2013
A 12% DECREASE is the same a multiplying the price by 0.88
So, the new price = ($100)(1 + p/100)(0.88)

The price of the good at the end of 2013 was 10% higher than the price at the beginning of 2012
If the original price was $100, then the price at the end of 2013 was $110
So, we can write: $110 = ($100)(1 + p/100)(0.88)
Simplify: $110 = (100 + p)(0.88)
Simplify more: $110 = 88 + 0.88p
Subtract 88 from both sides: 22 = 0.88p
So, p = 22/0.88 = 25

Answer: D

Cheers,
Brent

VJesus12 wrote:The price of a consumer good increased by p% during 2012 and decreased by 12% during 2013. If no other change took place in the price of the good and the price of the good at the end of 2013 was 10% higher than the price at the beginning of 2012, what was the value of p?

A. −2%
B. 2%
C. 22%
D. 25%
E. Cannot be determined

[spoiler]OA=D[/spoiler]

Source: e-GMAT

We let the 2012 price = n and thus, the price at the end of 2013 will be:

(1 + p/100)(0.88)(n)

Since the price at the end of 2013 was 10% higher than at the beginning of 2012, we can create the equation:

1.1n = (1 + p/100)(0.88)(n)

1 = (1 + p/100)(0.8)

1 = 0.8 + 0.8p/100

Multiplying by 100, we have:

100 = 80 + 0.8p

20 = 0.8p

25 = p

Alternate Solution:

Let’s let the price at the beginning of 2012 be 100. Since the price at the end of 2013 was 10% higher, the price at the end of 2013 is 1.1 x 100 = 110. We know the price decreased by 12% and became 110; therefore, before decreasing by 12%, the price was 110/0.88 = 125. Now, the price of 100 increases by p percent and becomes 125; therefore the value of p is [(125 - 100)/100] x 100 = 25.

Answer: D