__2015 OG Quant Review PS #10.__ If there are 664,579 primes among the first 10 million positive integers, approximately what percent of the first ten million positive integers are prime numbers?

__Math Lessons__: (1) Desired/Σ is the Rate or Probability that something happens (here prime numbers) and can be expressed as a Percentage; (2)** Percent** is a derivation of the Latin ‘Per Centum’, meaning ‘per 100’. To find a percentage, multiply the numerator and denominator by the same quantity so that the resulting denominator equals 100 = 10². Because the original denominator was 10 to the seventh, we multiply both numerator and denominator by 10 to the negative 5^{th}; (3) To be efficient, first write the fraction 664,579/10,000,000 and then multiply numerator and denominator by 10 to the negative 5^{th}; (4) Know that multiplying by 10 to the negative 5^{th} moves the decimal point 5 places to the left, making each the numerator and denominator smaller by 5 **Orders of Magnitude**; and (5) Because we use a base 10 system (due to our 10 fingers) increasing by an order of magnitude means multiplying by 10 (moving the decimal point one place to the right). Similarly, decreasing by an order of magnitude means dividing by 10 (moving decimal point one place to the left).

__Character count__: The OG writes 5 lines and 163 characters; Shawn Berry writes 1 line and 51 characters. The OG writes 4 more lines and 320% as many characters. You don’t have extra time or energy for long Quant solutions. Even if you could perform long solutions well, you’d be so tired from the Quant that your subsequent Verbal score would suffer.

__If the original denominator is not a power of 10__: Tackle percent questions by setting the denominator to 100. If the original denominator is not a power of 10, multiply the original numerator and denominator by 100 divided by the original denominator. This will force the new denominator to equal 100.

__Shawn Berry (550 level).__ By volume, the composition of carbon dioxide in the Earth’s atmosphere in 2014 was 400 parts per million. Expressed as a percentage, what portion of the Earth’s atmosphere was composed of carbon dioxide in 2014?

A. 0.004%

B. 0.04%

C. 0.4%

D. 4.0%

E. 40%

__Shawn Berry (650 level).__ There are 168 prime numbers among the first 1,000 positive integers and 664,579 prime numbers among the first 10,000,000 positive integers. By approximately what percentage does the rate of prime numbers fall from the first 1,000 positive integers to the first 10,000,000 positive integers?

A. 50%

B. 60%

C. 70%

D. 80%

E. 90%

__Shawn Berry (700 level).__ Given that an order of magnitude larger means move the decimal point one place to the right and an order of magnitude smaller means move the decimal point one place to the left, for positive integer n, 10 raised to the n^{th} power is how many fewer orders of magnitude than 100 raised to the n^{th} power?

A. n

B. 2n

C. 10

D. 10n

E. 90

__Shawn Berry (700 level).__ For N equal to an integer, the product of all prime numbers is

A. 4N

B. 4N + 1

C. 4N + 2

D. 4N + 3

E. Cannot be determined

__Shawn Berry (750 level)__ If y equals 50x percent of x, the x equals what percent of y?

A. 50

B. 50/x

C. 100

D. 200

E. 200/x

Legal Note: “The Graduate Management Admissions Council (GMAC) firmly believes that the Official Guide for GMAT Review is all that you need to perform your best on the GMAT … and that no additional techniques or strategies are needed to do well.” I, Shawn Berry, know better. I have twice earned a perfect 800 on the GMAT-CAT. I document that the Official Guide writes inconsistent, inefficient, and downright confusing solutions that take longer than the allotted 2 minutes/question. Herein I make fair use of GMAC copyrighted material – mostly its confusing solutions – for the transformative educational purpose of teaching students the clear, consistent, and efficient Mathematics, Grammar, and Logic needed to answer GMAT questions in less than 2 minutes.