Math: Number Theory - Percents

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;



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