Percentages are a mathematical way to describe parts of a whole. The term "percent" means "out of one hundred" and is represented by the symbol %. Understanding percentages is vital as they are commonly used in daily life—sales, statistics, finance, and more. Here's a comprehensive guide to grasping percentages:
What are Percentages?
- Definition: Percentages divide a whole into 100 equal parts.
- Example: 1% = 1 part out of 100.
- Visualization:
- A 100-cell grid can represent percentages.
- 1 cell = 1%.
- 50 cells = 50% (half).
- 25 cells = 25% (one-quarter).
- Key Use: Percentages simplify comparisons and calculations across fractions, decimals, and whole numbers.
Calculating Percentages
1. Finding a Percentage of a Whole
Rule: Find 1%, then multiply by the required percentage.
Example: A laptop costs $500 with a 20% discount.
1. Calculate 1% of $500:
( 500 \div 100 = $5 )
2. Multiply for 20%:
( $5 * 20 = $100 )
3. Subtract the discount:
( $500 - $100 = $400 ).
2. Adding a Percentage
Example: George gets a 5% pay rise on a $24,000 salary.
1. Find 1% of $24,000:
( 24,000 \div 100 = $240 ).
2. Multiply for 5%:
( $240 * 5 = $1,200 ).
3. Add the pay rise:
( $24,000 + $1,200 = $25,200 ).
3. Mental Maths Hacks
- Find 10% first and scale up: For 20% of $500, calculate 10% ($50) and double it ($100).
- Reversible Rule: ( 16\% { of } 25 = 25\% { of } 16 ).
Converting Between Percentages, Decimals, and Fractions
To Decimal: Divide by 100.
- ( 50\% = 0.5 ), ( 1\% = 0.01 ).
To Fraction: Write as a fraction over 100, simplify if possible.
- ( 50\% = \frac{50}{100} = \frac{1}{2} ).
To Percentage from Decimal: Multiply by 100.
Percentages Greater than 100%
- Example: George’s new salary is 105% of his old salary.
- 105% = ( 1.05 * {Old Salary} ).
Calculating Percentage Proportions in a Group
Example: An organization employs 40 staff in these roles:
- Managers: 9
- Administrators: 12
- Accountants: 5
- HR: 3
- Cleaners: 7
- Catering: 4
Steps:
1. Find the total staff: ( 9 + 12 + 5 + 3 + 7 + 4 = 40 ).
2. Calculate each role’s percentage:
- Managers: ( \frac{9}{40} * 100 = 22.5\% ).
- Administrators: ( \frac{12}{40} * 100 = 30\% ).
Pie Chart Representation: Visualize the percentages for clarity.
Key Points to Remember
- Percentages always describe parts of a whole (divided into 100 parts).
- They simplify comparisons and calculations.
- Use mental math tricks to save time and verify your results by ensuring the total equals 100%.