Percentages

Master the concepts of percentage, fractions, and decimals with our comprehensive guide. Learn about conversions, calculations, and real-world applications.

Table of Contents

Simplified Quantitative Formulas: Percentages

  • Percentage: A way to express a number as a fraction of 100. x% = x/100
  • Converting: To convert a fraction to percent, multiply by 100. To convert percent to fraction, divide by 100.
  • Percentage Change: % Change = (Difference / Original) × 100
  • Successive Percentage Change: If a value is increased by a% and then by b%, net change = a + b + (ab/100)%
  • Successive Discounts: If two discounts a% and b% are given, net discount = a + b – (ab/100)%
  • Increase/Decrease to Reverse: If a value is increased by x%, to bring it back, decrease by [x/(100+x)] × 100%
  • Base Change: When base changes, adjust percentage accordingly. E.g., if a number increases from 200 to 250, the percent increase is (50/200)×100 = 25%.
  • Relation to Ratio: Percentages can be converted to ratios and vice versa. E.g., 20% = 1:4.
  • Profit, Loss, Discount: Often calculated using percentages. Profit% = (Profit/CP)×100, Discount% = (Discount/MP)×100
  • Variable Definitions: CP = Cost Price, SP = Selling Price, MP = Marked Price, Discount = MP – SP, Profit = SP – CP, Loss = CP – SP

What do these mean? (Super Simple Explanations & Examples)

  • Percentage: 25% means 25 out of 100. Example: 25% of 200 = (25/100)×200 = 50.
  • Converting: 3/4 as a percent = (3/4)×100 = 75%. 60% as a fraction = 60/100 = 3/5.
  • Percentage Change: If a price goes from 80 to 100, % change = (20/80)×100 = 25% increase.
  • Successive Percentage Change: Increase by 10%, then 20%: Net = 10 + 20 + (10×20/100) = 32% increase.
  • Successive Discounts: 10% then 20% off: Net = 10 + 20 – (10×20/100) = 28% discount.
  • Increase/Decrease to Reverse: If a value is increased by 25%, decrease by (25/125)×100 = 20% to return to original.
  • Base Change: If a number increases from 200 to 250, percent increase = (50/200)×100 = 25%.
  • Relation to Ratio: 20% = 1:4. Example: 40% = 2:5.
  • Profit, Loss, Discount: If CP = 100, SP = 120, Profit% = (20/100)×100 = 20%.
  • Variable Definitions: CP = Cost Price, SP = Selling Price, MP = Marked Price, Discount = MP – SP, Profit = SP – CP, Loss = CP – SP

1. Basic Concepts

(a) Understanding Percentage

A percentage is a way to express a number as a fraction of 100. The symbol % is used to denote percentage.

Key Points:

  • Percentage means "per hundred"
  • 100% represents the whole
  • Percentages can be greater than 100%
  • Percentages can be less than 1%

Quick Reference

Hover to see common percentage values

Common Values

  • 50% = ½ = 0.5
  • 25% = ¼ = 0.25
  • 75% = ¾ = 0.75
  • 10% = ⅒ = 0.1

2. Conversions

(a) Percentage to Fraction

To convert a percentage to a fraction, divide by 100 and simplify.

Steps:

  1. Write the percentage as a fraction with denominator 100
  2. Simplify the fraction to lowest terms

Example: Convert 75% to fraction

75% = 75/100

= (75÷25)/(100÷25)

= 3/4

(b) Fraction to Percentage Conversions (1/1 to 1/30)

Fraction Percent Decimal
1/1 100% 1.0
1/2 50% 0.5
1/3 33.33% 0.333...
1/4 25% 0.25
1/5 20% 0.2
1/6 16.67% 0.166...
1/7 14.29% 0.142857...
1/8 12.5% 0.125
1/9 11.11% 0.111...
1/10 10% 0.1
1/11 9.09% 0.0909...
1/12 8.33% 0.0833...
1/13 7.69% 0.0769...
1/14 7.14% 0.0714...
1/15 6.67% 0.0666...
1/16 6.25% 0.0625
1/17 5.88% 0.0588...
1/18 5.56% 0.0555...
1/19 5.26% 0.0526...
1/20 5% 0.05
1/21 4.76% 0.0476...
1/22 4.55% 0.0454...
1/23 4.35% 0.0435...
1/24 4.17% 0.0416...
1/25 4% 0.04
1/26 3.85% 0.0385...
1/27 3.70% 0.0370...
1/28 3.57% 0.0357...
1/29 3.45% 0.0345...
1/30 3.33% 0.0333...

(c) Decimal to Percentage

To convert a decimal to a percentage, multiply by 100.

Steps:

  1. Multiply the decimal by 100
  2. Add % symbol

Example: Convert 0.375 to percentage

0.375 × 100 = 37.5%

(b) Common Percentage Conversions

Percent Decimal Fraction
1% 0.01 1/100
5% 0.05 1/20
10% 0.1 1/10
12½% 0.125 1/8
20% 0.2 1/5
25% 0.25 1/4
33⅓% 0.333... 1/3
50% 0.5 1/2
75% 0.75 3/4
80% 0.8 4/5
90% 0.9 9/10
99% 0.99 99/100
100% 1 1
125% 1.25 5/4
150% 1.5 3/2
200% 2 2

3. Percentage Calculations

(a) Finding Percentage of a Number

To find what percentage one number is of another, divide the first number by the second and multiply by 100.

Formula:

Percentage = (Part/Whole) × 100

Example: What percentage is 25 of 200?

Percentage = (25/200) × 100

= 0.125 × 100

= 12.5%

(b) Finding a Number Given Percentage

To find a number when a percentage of it is known, divide the known part by the percentage and multiply by 100.

Formula:

Whole = (Part/Percentage) × 100

Example: 15 is 30% of what number?

Whole = (15/30) × 100

= 0.5 × 100

= 50

(c) Percentage Change

Percentage change measures the relative change between two values.

Formula:

Percentage Change = ((New Value - Old Value)/Old Value) × 100

Example: Price increased from $50 to $65

Percentage Change = ((65 - 50)/50) × 100

= (15/50) × 100

= 30% increase

4. Real-world Applications

(a) Discounts and Markups

Understanding how to calculate discounts and markups is essential for shopping and business.

Formulas:

Discount Amount = Original Price × Discount Rate

Final Price = Original Price - Discount Amount

Markup Amount = Cost Price × Markup Rate

Selling Price = Cost Price + Markup Amount

Example: 20% discount on $100 item

Discount Amount = $100 × 0.20 = $20

Final Price = $100 - $20 = $80

(b) Interest Calculations

Simple and compound interest calculations use percentages.

Simple Interest:

I = P × R × T

where:

I = Interest

P = Principal

R = Rate (as decimal)

T = Time (in years)

Example: $1000 at 5% for 2 years

I = 1000 × 0.05 × 2

= $100

Total Amount = $1000 + $100 = $1100

(c) Tax Calculations

Understanding how to calculate taxes is important for financial planning.

Formula:

Tax Amount = Base Amount × Tax Rate

Total Amount = Base Amount + Tax Amount

Example: 8% tax on $50 purchase

Tax Amount = $50 × 0.08 = $4

Total Amount = $50 + $4 = $54

5. Shortcuts & Tricks

(a) Mental Math Tricks

Quick methods to calculate percentages mentally.

Common Tricks:

  • 10%: Move decimal point one place left
  • 5%: Half of 10%
  • 15%: 10% + 5%
  • 20%: Double 10%
  • 25%: Quarter of the number

Example: Find 15% of $80

10% of $80 = $8

5% of $80 = $4

15% of $80 = $8 + $4 = $12

(b) Successive Percentage Changes

When multiple percentage changes occur in sequence.

Formula:

Final Value = Initial Value × (1 ± p₁) × (1 ± p₂) × ...

where p₁, p₂, etc. are percentage changes as decimals

Example: Price changes by +10% then -20%

Final Value = 100 × 1.10 × 0.80

= 100 × 0.88

= 88

Net change = -12%

(c) Percentage Points vs Percentages

Understanding the difference between percentage points and percentage changes.

Key Points:

  • Percentage points are absolute differences
  • Percentage changes are relative differences
  • Important in statistics and data analysis

Example:

Interest rate increases from 5% to 7%

Change in percentage points = 2 points

Percentage change = (2/5) × 100 = 40% increase

Practice: Percentage Questions (Basic to Advanced)

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