Sunday, July 14, 2013

# Percentage Increase and Decrease Problems on the GRE

2:36 AM

### Remembering Percentage

In math, 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, " $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \%$". For example  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 45\%$ (read as "forty-five percent") is equal to $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \frac{45}{100}$  or $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 0.45$. Similarly $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 45\%$ of the total number of students in a class = $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \frac{45}{100} \times \text{Total Number of Students in a Class}$

### Percentage Increase or Decrease

Percentages are often used to express the increase or decrease of a measure relative to its original value. Some real life examples where you come across the changes expressed as percentages are:
Price of gasoline increased from  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \2$ to  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \2.50$ representing a  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 25\%$ increase in price
Population of a town declined from  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 100,000$  to $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 90,000$ in a decade representing a  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 10\%$ decline
And finally some of you may weigh twice as much as you were 10 years ago representing a 100% increase in weight!!
Here is a simple three step process to compute percentage changes:
1. Identify the change: First step is to find out what is the measure for which the percentage change needs to be calculated. You should identify the final value (i.e. the value after the change has occurred) and the original value (i.e. the value before the increase or decrease). If we are talking about an increase in price from $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \2$ to $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \2.50$, our original value is $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \2$ and the final value is $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \2.50$.
2. Find the change in value: This is the difference between final value and the original value. If we are talking about an increase in price from $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \2$ to $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \2.50$, the change in value is $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \2.50-\2.00=\0.50$
3. Express the change as Percentage: This can be easily done by dividing the change in value by the original value and multiplying by $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 100$: In our example this is equivalent to:
Percentage change =   $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \frac{0.50}{2.00} \times 100 = 25\%$
Percentage change problems are very common on the test. You may find percentage problems in the following situations:
1. Two step percentage change problems: In these types of problems you are first expected to calculate the original or final value based on the information provided. The context could be geometry or simple arithmetic. Once you have the original and final value, the question may ask you to calculate percentage changes
2. Find the original or final value based on percentage changes: In such problems you are expected to calculate the original value or the final value based on percentage change information.
3. Data interpretation: These problems expect you to obtain the final value and the original from graphs and expect you to calculate the percentage changes

Example:
The regular price for a television is \$1900. At a sale, George bought the television at a discount of 10%. What was the amount of discount that George got for the television?
A) \$19
B) \$150
C) \$180
D) \$190
E) \$200
Solution:

Step 1: In this problem we are told that George is getting a discount of 10% and that means there is a percentage decrease of 10%. We are expected to calculate the change in value based on the percentage decrease. Let’s summarize what we know:
To calculate the discount amount, we need to calculate 10% of the regular price i.e. \$1900.The discount amount = $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 1900 \times \frac{10}{100} = \190$

Remember, if a quantity is increased or decreased more than once, you cannot simply add or subtract the percentages. You have to work out each increase or decrease step by step.
In order to solve percentage change problems quickly, it’s a good idea to remember some common percentage change values. The following tables show Percentage Increases and Percentage decreases for some commonly used numbers.

 Original value 100: Percentage change - Final values Percentage Change 10 20 30 40 50 60 70 80 90 100 150 200 250 400 Increase - Final value 110 120 130 140 150 160 170 180 190 200 250 300 350 500 Decrease - Final value 90 80 70 60 50 40 30 20 10 0 -50 -100 -150 -300

 Original value 50: Percentage change - Final values Percentage Change 10 20 30 40 50 60 70 80 90 100 150 200 250 400 Increase - Final value 55 60 65 70 75 80 85 90 95 100 125 150 175 250 Decrease - Final value 45 40 35 30 25 20 15 10 5 0 -25 -50 -75 -150