Sunday, July 14, 2013

# Be careful solving percentage of a percentage problems on the GRE

2:29 AM
Some GRE problems may involve percentage of a percentage. Usually these problems are word problems such as "What is 20% of 50% of 10". To calculate a percentage of a percentage is simple. You need to convert both percentages to fractions of 100, or to decimals, and multiply them. For example, 50% of 40% is:
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \frac{50}{100} \times \frac{40}{100} = 0.5 \times 0.4 = 0.2 = \frac{20}{100}=20 \%$

Note: It is not correct to divide by 100 and use the percent sign at the same time.
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 25\%=\frac{25}{100}=0.25 \ne \frac{25\%}{100}$
GRE problems will rarely ask you to compute percentages. Normally the percentage problems will be described in words. Whenever we talk about a percentage, it is important to specify what it is relative to, i.e. what is the total that corresponds to 100%. The following problem illustrates this point.
Example:
In a school 60% of all students are female, and 10% of all students play football. If 5% of female students play football, what percentage of football players are female?
A) 10%
B) 20%
C) 25%
D) 30%
E) 50%
Solution:

In this problem, we are asked to compute the ratio of female football players to all football players expressed as percentage. To compute the ratio, we need to know the number of football players and number of female players. However all we see in the problem are some percentages. The trick to answering this question is that the answer is also expressed in percentage terms and therefore you do not need to know the exact numbers. In fact this problem can be easily solved by assuming the total number of students to be 100. Why 100? Because it’s a nice round number and easy to perform percentage calculations.
Step 1: We know that 10% of all students play football. Therefore assuming 100 students, we know 10 students play football.
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \frac{10}{100} \times 100 = 10$
Step 2: We know that 60% of all students are female, and among these 5% play football. Therefore we can conclude that number of all students who are female and who play football is:
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \frac{60}{100} \times \frac{5}{100} \times 100=\frac{3}{100}=3$
Step 3: Next we compute the ratio of female football players to all football players expressed as percentage i.e.
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \frac{3}{10} \times 100 = 30\%$