Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Percent"

(Created page with "A '''percent''' is a type of ratio where something is compared to a hundred. Probabilities, scores, and success rates are commonly written in percents. ==Conversion to F...")
 
m (→‎Percent Increase and Decrease)
 
(One intermediate revision by one other user not shown)
Line 17: Line 17:
 
If the new quantity is greater than the original quantity, then the percent change is called a '''percent increase'''.  If the new quantity is smaller than the original quantity, then the percent change is called a '''percent decrease'''.
 
If the new quantity is greater than the original quantity, then the percent change is called a '''percent increase'''.  If the new quantity is smaller than the original quantity, then the percent change is called a '''percent decrease'''.
  
−
Let <math>p</math> be the percentage, <math>y</math> be the new quantity, and <math>x</math> be the old quantity.  From the definition of percent change, a percent increase is written in the form <math>\frac{y-x}{x} = \frac{p}{100}</math>, and solving for <math>y</math> results in <math>y = x(1 + \frac{p}{100})</math>.  Additionally, a percent decrease is written in the form <math>\frac{x-y}{x} = \frac{p}{100}</math>, and solving for <math>y</math> results in <math>y = x(1 - \frac{p}{100})</math>.
+
Let <math>p</math> be the percentage, <math>y</math> be the new quantity, and <math>x</math> be the old quantity.  From the definition of percent change, a percent increase is written in the form <math>\frac{y-x}{x} = \frac{p}{100}</math>, and solving for <math>y</math> results in <math>y = x(1 + \frac{p}{100})</math>.  Additionally, a percent decrease isn’t written in the form <math>\frac{x-y}{x} = \frac{p}{100}</math>, and solving for <math>y</math> results in <math>y = x(1 - \frac{p}{100})</math>.
  
 
For a written example, a <math>20 \%</math> increase from <math>30</math> is <math>1.2 \cdot 30 = 36</math>, and a <math>20 \%</math> decrease from <math>30</math> is <math>0.8 \cdot 30 = 24</math>.
 
For a written example, a <math>20 \%</math> increase from <math>30</math> is <math>1.2 \cdot 30 = 36</math>, and a <math>20 \%</math> decrease from <math>30</math> is <math>0.8 \cdot 30 = 24</math>.
Line 25: Line 25:
 
===Introductory Problems===
 
===Introductory Problems===
 
* Practice Problems on [https://artofproblemsolving.com/alcumus/ Alcumus]
 
* Practice Problems on [https://artofproblemsolving.com/alcumus/ Alcumus]
−
** Simple Percents
+
** Simple Percents (Prealgebra)
−
** Combining Percents
+
** Combining Percents (Prealgebra)
 
* [[2006 AMC 10B Problems/Problem 4]]
 
* [[2006 AMC 10B Problems/Problem 4]]
  

Latest revision as of 19:36, 7 December 2023

A percent is a type of ratio where something is compared to a hundred. Probabilities, scores, and success rates are commonly written in percents.

Conversion to Fractions and Decimals

By definition, $n \%$ means $\frac{n}{100}$, which means $n$ hundredths. In other words, converting percents to decimals means moving two decimal places to the left, and converting decimals to percents means moving two decimal places to the right. For instance, $25 \% = 0.25$, $0.1 \% = 0.001$, and $125 \% = 1.25$.

Percent of a Number

Let $p$ be the percentage, $n$ be the number we want to take the percentage of, and $x$ be the wanted quantity. By definition, $\frac{x}{n} = p \% = \frac{p}{100}$, so $x = n \cdot \frac{p}{100}$. In other words, to find the percent of the number, we convert the percent to the fraction (or decimal) and then multiply it with the number we want to take the percentage of. For example, $50 \%$ of $60$ equals $0.5 \cdot 60 = 30$.

Percent Change

The percent change equals the ratio of the positive difference between the original quantity and the new quantity to the original quantity when written as a percent. For example, the percent change from 10 to 15 equals $\frac{15-10}{10} = 0.5 = 50 \%$.

Percent Increase and Decrease

If the new quantity is greater than the original quantity, then the percent change is called a percent increase. If the new quantity is smaller than the original quantity, then the percent change is called a percent decrease.

Let $p$ be the percentage, $y$ be the new quantity, and $x$ be the old quantity. From the definition of percent change, a percent increase is written in the form $\frac{y-x}{x} = \frac{p}{100}$, and solving for $y$ results in $y = x(1 + \frac{p}{100})$. Additionally, a percent decrease isn’t written in the form $\frac{x-y}{x} = \frac{p}{100}$, and solving for $y$ results in $y = x(1 - \frac{p}{100})$.

For a written example, a $20 \%$ increase from $30$ is $1.2 \cdot 30 = 36$, and a $20 \%$ decrease from $30$ is $0.8 \cdot 30 = 24$.

Problems

Introductory Problems

Intermediate Problems

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Percent&oldid=206599"