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Math: Number Theory - Percents  [#permalink]

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57
PERCENTS This post is a part of [GMAT MATH BOOK]

created by: Bunuel
edited by: bb, walker

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Definition

A percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, "%", or the abbreviation "pct". Since a percent is an amount per 100, percents can be represented as fractions with a denominator of 100. For example, 25% means 25 per 100, 25/100 and 350% means 350 per 100, 350/100.

• A percent can be represented as a decimal. The following relationship characterizes how percents and decimals interact. Percent Form / 100 = Decimal Form

For example: What is 2% represented as a decimal?
Percent Form / 100 = Decimal Form: 2%/100=0.02

Percent change

General formula for percent increase or decrease, (percent change):

$$Percent=\frac{Change}{Original}*100$$

Example: A company received $2 million in royalties on the first$10 million in sales and then $8 million in royalties on the next$100 million in sales. By what percent did the ratio of royalties to sales decrease from the first $10 million in sales to the next$100 million in sales?

Solution: Percent decrease can be calculated by the formula above:
$$Percent=\frac{Change}{Original}*100=$$
$$=\frac{\frac{2}{10}-\frac{8}{100}}{\frac{2}{10}}*100=60%$$, so the royalties decreased by 60%.

Simple Interest

Simple interest = principal * interest rate * time, where "principal" is the starting amount and "rate" is the interest rate at which the money grows per a given period of time (note: express the rate as a decimal in the formula). Time must be expressed in the same units used for time in the Rate.

Example: If $15,000 is invested at 10% simple annual interest, how much interest is earned after 9 months? Solution:$15,000*0.1*9/12 = $1125 Compound Interest $$Balance(final)=$$ $$=principal*(1+\frac{interest}{C})^{time*C}$$, where C = the number of times compounded annually. If C=1, meaning that interest is compounded once a year, then the formula will be: $$Balance(final)=$$ $$principal*(1+interest)^{time}$$, where time is number of years. Example: If$20,000 is invested at 12% annual interest, compounded quarterly, what is the balance after 2 year?
Solution: $$Balance=20,000*(1+\frac{0.12}{4})^{2*4}=$$
$$=20,000*(1.03)^8=25,335.4$$

Percentile

If someone's grade is in $$x_{th}$$ percentile of the $$n$$ grades, this means that $$x%$$ of people out of $$n$$ has the grades less than this person.

Example: Lena’s grade was in the 80th percentile out of 120 grades in her class. In another class of 200 students there were 24 grades higher than Lena’s. If nobody had Lena’s grade, then Lena was what percentile of the two classes combined?

Solution:
Being in 80th percentile out of 120 grades means Lena outscored $$120*0.8=96$$ classmates.

In another class she would outscored $$200-24=176$$ students.

So, in combined classes she outscored $$96+176=272$$. As there are total of $$120+200=320$$ students, so Lena is in $$\frac{272}{320}=0.85=85%$$, or in 85th percentile.

Official GMAC Books:

The Official Guide, 12th Edition: PS #10; PS #17; PS #19; PS #47; PS #55; PS #60; PS #64; PS #78; PS #92; PS #94; PS #109; PS #111; PS #115; PS #124; PS #128; PS #131; PS #151; PS #156; PS #166; PS #187; PS #193; PS #200; PS #202; PS #220; DS #2; DS #7; DS #21; DS #37; DS #48; DS #55; DS #61; DS #63; DS #78; DS #88; DS #92; DS #120; DS #138; DS #142; DS #143;

Generated from [GMAT ToolKit]

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Attachment: Math_icon_percents.png [ 2.3 KiB | Viewed 88350 times ]

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Originally posted by Bunuel on 22 Mar 2010, 15:24.
Last edited by walker on 01 Apr 2010, 12:15, edited 3 times in total.
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GMAT 1: 680 Q48 V34 Re: Math: Number Theory - Percents  [#permalink]

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Great post Bunuel!!!

Kudoos!!!
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Re: Math: Number Theory - Percents  [#permalink]

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2
In the Percent Section for the specified example, why is it 2/10 - 10/100 / 2/10 ? Shouldn't it be 2/10 - 8/100 / 2/10 ? Where did the 10/100 come from?
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Re: Math: Number Theory - Percents  [#permalink]

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Yet another great work by Bunuel Thank you for sharing.
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Re: Math: Number Theory - Percents  [#permalink]

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I am with ThisisGeero, can you please explain how you came up with the 10/100 in the fraction instead of 8/100?
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Re: Math: Number Theory - Percents  [#permalink]

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Thanks a lot!

That's just a typo perhaps?
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Re: Math: Number Theory - Percents  [#permalink]

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Great work Bunuel! Thank you for your efforts! CEO  B
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GMAT 1: 750 Q50 V40 Re: Math: Number Theory - Percents  [#permalink]

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I've added the topic to GMAT ToolKit 1.3.3 and references from the OG12 book.
Thanks, Bunuel!
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Re: Math: Number Theory - Percents  [#permalink]

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ThisisGeero wrote:
In the Percent Section for the specified example, why is it 2/10 - 10/100 / 2/10 ? Shouldn't it be 2/10 - 8/100 / 2/10 ? Where did the 10/100 come from?

There was a typo - edited. Thanks.
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Re: Math: Number Theory - Percents  [#permalink]

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Thanks its a big help!!

Kudos!!!

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GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Re: Math: Number Theory - Percents  [#permalink]

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1
1
Bunuel wrote:
PERCENTS

Example: Lena’s grade was in the 80th percentile out of 120 grades in her class. In another class of 200 students there were 24 grades higher than Lena’s. If nobody had Lena’s grade, then Lena was what percentile of the two classes combined?

Solution:
Being in 80th percentile out of 120 grades means Lena outscored $$120*0.8=96$$ classmates.

In another class she would outscored $$200-24=176$$ students.

So, in combined classes she outscored $$96+176=272$$. As there are total of $$120+200=320$$ students, so Lena is in $$\frac{272}{320}=0.85=85%$$, or in 85th percentile.

Attachment:
Math_icon_percents.png

In another class she would outscored $$200-24=176$$ students.
I think it should be 200-24-1 = 175 as 24 were higher than Lena , thus 24+1 are lower than her, we need to count her as well
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Re: Math: Number Theory - Percents  [#permalink]

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gurpreetsingh wrote:
Bunuel wrote:
PERCENTS

Example: Lena’s grade was in the 80th percentile out of 120 grades in her class. In another class of 200 students there were 24 grades higher than Lena’s. If nobody had Lena’s grade, then Lena was what percentile of the two classes combined?

Solution:
Being in 80th percentile out of 120 grades means Lena outscored $$120*0.8=96$$ classmates.

In another class she would outscored $$200-24=176$$ students.

So, in combined classes she outscored $$96+176=272$$. As there are total of $$120+200=320$$ students, so Lena is in $$\frac{272}{320}=0.85=85%$$, or in 85th percentile.

Attachment:
Math_icon_percents.png

In another class she would outscored $$200-24=176$$ students.
I think it should be 200-24-1 = 175 as 24 were higher than Lena , thus 24+1 are lower than her, we need to count her as well

I was going to post it, but you already did.
I completely agree with you.
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Re: Math: Number Theory - Percents  [#permalink]

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gurpreetsingh wrote:
Bunuel wrote:
PERCENTS

Example: Lena’s grade was in the 80th percentile out of 120 grades in her class. In another class of 200 students there were 24 grades higher than Lena’s. If nobody had Lena’s grade, then Lena was what percentile of the two classes combined?

Solution:
Being in 80th percentile out of 120 grades means Lena outscored $$120*0.8=96$$ classmates.

In another class she would outscored $$200-24=176$$ students.

So, in combined classes she outscored $$96+176=272$$. As there are total of $$120+200=320$$ students, so Lena is in $$\frac{272}{320}=0.85=85%$$, or in 85th percentile.

Attachment:
Math_icon_percents.png

In another class she would outscored $$200-24=176$$ students.
I think it should be 200-24-1 = 175 as 24 were higher than Lena , thus 24+1 are lower than her, we need to count her as well

The point here is that Lena herself is not in the other class. So in another class she outscored 200-24=176 not 175.

Hope it's clear.
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GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Re: Math: Number Theory - Percents  [#permalink]

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yes you are right. thanks
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Re: Math: Number Theory - Percents  [#permalink]

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being new to this forum i found such posts amazingly helpful.. thanks
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Re: Math: Number Theory - Percents  [#permalink]

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[quote="Bunuel"]PERCENTS

The Official Guide, 12th Edition: PS #10; PS #17; PS #19; PS #47; PS #55; PS #60; PS #64; PS #78; PS #92; PS #94; PS #109; PS #111; PS #115; PS #124; PS #128; PS #131; PS #151; PS #156; PS #166; PS #187; PS #193; PS #200; PS #202; PS #220; DS #2; DS #7; DS #21; DS #37; DS #48; DS #55; DS #61; DS #63; DS #78; DS #88; DS #92; DS #120; DS #138; DS #142; DS #143;

Dear bunuel,

can you please put inthe corresponding questions in the 11 the edition as well... so that once after going throuhg your notes we can straight away do some quant work on the same topics???

regards,
kyle
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Re: Math: Number Theory - Percents  [#permalink]

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Excellent resources!
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Re: Math: Number Theory - Percents  [#permalink]

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Thanks for the effort and good work! Kudos.
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Re: Math: Number Theory - Percents  [#permalink]

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thanks for sharing!
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Re: Math: Number Theory - Percents  [#permalink]

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Hello Bunnel,

This is indeed very helpful. Can you please also add a few examples containing cost price, marked price and selling price scenarios.
Thanks Re: Math: Number Theory - Percents   [#permalink] 11 Jul 2010, 02:48

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