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The price of a consumer good increased by p%. . .

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The price of a consumer good increased by p%. . .  [#permalink]

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New post Updated on: 07 Aug 2018, 07:09
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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


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Originally posted by EgmatQuantExpert on 11 Nov 2016, 05:46.
Last edited by EgmatQuantExpert on 07 Aug 2018, 07:09, edited 1 time in total.
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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 11 Nov 2016, 07:09
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EgmatQuantExpert 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:

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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 11 Nov 2016, 12:48
EgmatQuantExpert 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


Price's corresponding to year -

2011 = \(100\)
2012 = \(100 + p\)
2013 = \(\frac{88}{100}(100 + p)\)


Further , \(\frac{88}{100}(100 + p)\) = \(110\)

Or, \(\frac{8}{100}(100 + p)\) = \(10\)

Or, 800 + 8p = 1000

Or, 8p = 200

So, p = 25%

Hence, answer will be (D) 25%

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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 11 Nov 2016, 13:01
Lets start with a number for the original value. 100 is the easiest.

So we're looking for a value of 110 at the end of 2013.

Just by looking at the values we can get an idea of what to start testing. If we're increasing 100 by p% then decreasing it by 12% and the original value is still 10% higher we need a value much higher than 12. 25% is the easiest value to start with.

100+25%=125

125-12%=110

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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 12 Nov 2016, 20:32
For all the algebra loving people out there=>
Let price at the beginning of 2012 be $x
so after the end of 2012=> x[1+p/100]
And finally at the end of 2013 => x[1+p/100][1-12/100]

As per question=> Price was simple 10 percent greater
Hence x[1+10/100] must be the final price.
Equating the two we get
=> x[110/100]=x[1+p/100][88/100]
=> 44p+4400=5500
=> 44p=1100
=> p=1100/44=> 100/4=> 25.
So p must be 25
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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 16 Nov 2016, 09:46
[size=150]Let Price = 100

[size=150]Increased by P% = 100(1+P/100)

Treat it like successive percents;

So a 12% decrease would mean 88% of (1+P/100) of 100

The key words are no other change took place. So there are no further successive percents, and the final price = 110

Therefore:

88/100 * (1+P/100) * 100 = 110

=> 8800 + 88P = 11,000
=> 88P = 2200
=> P = 2200/88
=> P = 25%
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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post Updated on: 18 Dec 2016, 22:12
Hey,

The given question can be solved in a number of ways. Let's look at two most common ways to solve this question. We will share two more ways of solving this question tomorrow.


Method 1
:

    • Let us consider the price of the consumer good at the beginning of 2012 to be 100.
    • Let us also assume the price to be “C” at the beginning of 2013, after an increase of p%.
    • Since we know that with respect to the initial price, the price at the end of 2013 went up by 10%.
      o Therefore, the price at the end of 2013 = \(100 + (10\) % of \(100) = 110\)
    • Now we can write -
      o \(C – 12\) % of \(C = 110\)
      o \(C * (1 - \frac {12}{100}) = 110\)
      o \(C = 110 * \frac {25}{22} = 125\)

Therefore, the price at the beginning of 2013 is 125 and we got this value after p% increase over the initial value.
Thus, we can write –
    • \(100 + (p\) % of \(100) = 125\)
    • \(P = 25\) %


Method 2
:

Conventional method –


    • Let the price of consumer good at the beginning of 2012 be 100.
    • After an increase of p%, the price at the beginning of 2013 will be –
      o Price at the beginning of 2013 \(= 100 + (p\) % of \(100) = 100 + p\)
    Therefore the price at the beginning of 2013 is (100 + p)

    • After a decrease of 12%, the price at the end of 2013 will be –
      o Value at the beginning of 2013 \(* (1 – \frac{12}{100}) = (100 + p) * \frac {22}{25}\)……………..(i)

    • And we are also given that the overall increase in the price of consumer good is 10%.
    • Therefore, the value at the end of 2013 = \(100 + (10\)% of \(100)\) = 110………(ii)

    • Equating equation (i) and (ii) we get –
      o \((100 + p) * \frac {22}{25} = 110\)
      o \(100 + p = 125\)
Therefore, \(p = 25\)%

There are more innovative ways to solve this question. A few of them have not been discussed here. Can you all think of any other way to solve it? Would love to see a few other methods! :)

I will post two more ways to solve this question tomorrow. In the mean time, expecting some more responses with other ways to solve this question :)


Thanks,
Saquib
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Originally posted by EgmatQuantExpert on 15 Dec 2016, 23:26.
Last edited by EgmatQuantExpert on 18 Dec 2016, 22:12, edited 1 time in total.
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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 16 Dec 2016, 04:11
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joannaecohen wrote:
Lets start with a number for the original value. 100 is the easiest.

So we're looking for a value of 110 at the end of 2013.

Just by looking at the values we can get an idea of what to start testing. If we're increasing 100 by p% then decreasing it by 12% and the original value is still 10% higher we need a value much higher than 12. 25% is the easiest value to start with.

100+25%=125

125-12%=110

D


Hi,

Thanks for posting a different way of approaching this problem. In fact, the approach followed by you is very close to one of the innovative ways that we talked about in our official solution. The only difference being in your approach you have concluded that p should be much larger than 12. Going a step further, you can also conclude that p should be larger than 22% (12%+10%). Once you do so, you don't even need to pick any number. The only option which will satisfy it is D. 25%.

When we post our detailed solution using the two innovative methods tomorrow, we will explain how we can conclude that p should be greater than 22%.

Regards,
Saquib
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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 25 Dec 2016, 07:32
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As discussed, let's look at one of the innovative ways of solving the above question. It is one of the quickest ways to solve a question that involves successive percentage increase/decrease on the same value. Please take a note of this approach and apply it on some GMAT questions to master it.

So, let's quickly look at this smart approach.

When a number is increased successively by two percentage, let's assume, \(a\)% and \(b\)%, the net increase in the value of the number can be expressed by the formula,

Net increase \(=a+b+\frac {ab}{100}\)


Le's take a simple example to understand. If we increase a number, let's say, X successively by 10% and 20% respectively, the net increase according to the above formula should be,

Net increase \(=10+20+\frac {10*20}{100}=10+20+\frac {200}{100} = 10+20+2 = 32\)%


Isn't that quick!! A nice method to keep in your arsenal to solve Percent question involving successive increase quickly.

One good thing about the above formula is that you can use it to calculate the net decrease in case of successive decrease too. All you need to do is in case of decrease represent the percent as negative. Easy isn't it . Let's see an application quickly.

If we decrease a number, let's say, X successively by 10% and 20% respectively, the net increase according to the above formula should be,

Net increase \(=(-10)+(-20)+\frac {(-10)*(-20)}{100}=-10-20+\frac {200}{100} = -10-20+2 = (-28)\)%


Notice carefully, the sign of the net increase is negative, clearly indicating the after the successive decrease the value of the original number, decreased instead of increasing. And what was the magnitude??? Right 28%. The net decrease is 28%.

So, before we use this approach to give you an official answer for the above question, would you like to have a quick stab at it. Remember, you need to be careful about the sign of the change. Increase is represented by positive and decrease is represented by negative. All the best.

We will post the detailed solution tomorrow and then we will show another innovative method of solving this question.

Regards,
Saquib
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The price of a consumer good increased by p%. . .  [#permalink]

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New post Updated on: 07 Aug 2018, 07:11
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Alright, so let's look at the official solution to the above questions using the innovative formula on Net increase discussed in the last post.

We know that the price of the consumer good increased by \(p\)% and then decreased by \(12\)%. Hence, using the formula for net increase we can say,

Net increase \(=p+(-12)+\frac {p*(-12)}{100}=p-12-\frac {3p}{25} = (\frac {22p}{25} - 12)\)%


It is given in the question that the net increase finally is \(10\)%. Hence, we can equate the two values.

\((\frac {22p}{25} - 12)\)% = \(10\)%

or,
\(\frac {22p}{25} = 10+12 = 22\)%

or,
\(p = 25\)%


Hence, the answer is 25%.

Now, with this understanding try to solve this question in an even better way. Give it a try and we will post the official solution in another innovative way soon.

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Originally posted by EgmatQuantExpert on 27 Dec 2016, 00:45.
Last edited by EgmatQuantExpert on 07 Aug 2018, 07:11, edited 1 time in total.
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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 27 Dec 2016, 07:56
EgmatQuantExpert 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


\(p - 12 - \frac{12p}{100} = 10\)

\(100p - 1200 -12p = 1000\)

\(88p = 2200\)
\(p = 25\)

Hence, the correct answer must be (D) 25

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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 01 Feb 2019, 18:59
EgmatQuantExpert 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


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
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Re: The price of a consumer good increased by p%. . .  [#permalink]

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New post 01 Feb 2019, 21:46
EgmatQuantExpert 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



10 % will be a successive percentage change, which is calculated by

a + b + ab/100 = Percentage change

p - 12 - 12p/100 = 10

100p -1200 - 12p = 1000
p = 2200/88

p = 25 %

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Re: The price of a consumer good increased by p%. . .   [#permalink] 01 Feb 2019, 21:46
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