🍕 The Pizza Party Guide to Ratios & Percentages
Imagine you’re at the world’s best pizza party. Everything you need to know about ratios and percentages is right here—served hot and delicious!
🎯 The Big Picture
Ratios compare things. Percentages are just ratios out of 100.
That’s it. Everything else is just practice!
📌 Part 1: Percentage Calculations
What is a Percentage?
Think of 100 tiny pizza slices. If you eat 25 of them, you ate 25% (25 out of 100).
Percentage = (Part ÷ Whole) × 100
🍕 Simple Example
You have 20 candies. You give away 5.
Percentage given = (5 ÷ 20) × 100
= 0.25 × 100
= 25%
You gave away 25% of your candies!
Finding a Percentage of a Number
Question: What is 30% of 50?
30% of 50 = (30/100) × 50
= 0.30 × 50
= 15
Magic Trick: Move the decimal two places left, then multiply!
📌 Part 2: Percentage Change
The Story
Your favorite toy cost ₹100 last year. Now it costs ₹120.
How much did it change?
Percentage Change = [(New - Old) ÷ Old] × 100
Change = (120 - 100) ÷ 100 × 100
= 20 ÷ 100 × 100
= 20%
The price increased by 20%!
📈 Increase vs 📉 Decrease
| Type | Formula |
|---|---|
| Increase | New = Old × (1 + rate/100) |
| Decrease | New = Old × (1 - rate/100) |
Quick Example
Price drops from ₹80 to ₹60
Change = (60 - 80) ÷ 80 × 100
= -20 ÷ 80 × 100
= -25%
25% decrease! (The minus sign means “down”)
📌 Part 3: Successive Percentage Changes
The Tricky Part (Made Easy!)
Scenario: A shirt costs ₹100.
- First, price goes UP by 10%
- Then, price goes DOWN by 10%
Is the final price ₹100?
NO! Let’s see why.
After 10% increase: ₹100 × 1.10 = ₹110
After 10% decrease: ₹110 × 0.90 = ₹99
Final price is ₹99, not ₹100!
The Golden Formula
For two successive changes of a% and b%:
Net Change = a + b + (a×b)/100
Example
10% up, then 10% down:
Net = 10 + (-10) + (10 × -10)/100
= 0 + (-100)/100
= -1%
Overall: 1% decrease!
graph TD A["Original ₹100"] --> B["+10% = ₹110"] B --> C["-10% = ₹99"] C --> D["Net: -1%"]
📌 Part 4: Percentage Applications
🛒 Profit and Loss
| Term | Formula |
|---|---|
| Profit | SP - CP (when SP > CP) |
| Loss | CP - SP (when CP > SP) |
| Profit % | (Profit/CP) × 100 |
| Loss % | (Loss/CP) × 100 |
CP = Cost Price (what you paid) SP = Selling Price (what you sold for)
Example
Bought a book for ₹40, sold for ₹50.
Profit = 50 - 40 = ₹10
Profit % = (10/40) × 100 = 25%
💰 Simple Interest
SI = (Principal × Rate × Time) / 100
Example: ₹1000 at 5% for 2 years
SI = (1000 × 5 × 2) / 100
= 10000 / 100
= ₹100
💹 Compound Interest
Amount = P × (1 + R/100)^n
Money grows on money—interest earns interest!
🏷️ Discount
Discount % = (Discount Amount / Marked Price) × 100
Example: Shirt marked ₹500, sold at ₹400
Discount = 500 - 400 = ₹100
Discount % = (100/500) × 100 = 20%
📌 Part 5: Ratio Fundamentals
What is a Ratio?
A ratio compares two or more things.
You have 3 apples and 2 oranges.
- Ratio of apples to oranges = 3:2
- Read as “3 to 2”
The Pizza Analogy 🍕
8-slice pizza shared by you and your friend:
- You get 5 slices, friend gets 3
- Ratio = 5:3
Key Rules
- Order matters: 3:2 is NOT the same as 2:3
- Simplify: 6:4 = 3:2 (divide both by 2)
- Same units: Always compare like with like
Example: Simplifying
Ratio 24:36
GCD of 24 and 36 = 12
24 ÷ 12 = 2
36 ÷ 12 = 3
Simplified ratio = 2:3
📌 Part 6: Proportion and Variation
Proportion = Equal Ratios
If a:b = c:d, then a, b, c, d are in proportion.
Cross multiply: a × d = b × c
Example
Is 2:3 = 4:6?
2 × 6 = 12
3 × 4 = 12
12 = 12 ✓ Yes, they're proportional!
Direct Variation
More of one = More of the other
More hours worked → More money earned
If y varies directly with x: y = kx
Example
5 chocolates cost ₹25. What do 8 cost?
5/25 = 8/x
5x = 200
x = ₹40
Inverse Variation
More of one = Less of the other
More workers → Less time to finish
If y varies inversely with x: y = k/x
Example
4 workers finish in 6 days. How long for 8 workers?
Workers × Days = Constant
4 × 6 = 8 × x
24 = 8x
x = 3 days
graph TD A["Variation Types"] --> B["Direct"] A --> C["Inverse"] B --> D["↑ x means ↑ y"] C --> E["↑ x means ↓ y"]
📌 Part 7: Dividing Quantities in Ratios
The Fair Share Problem
Divide ₹500 between A and B in ratio 2:3
Step-by-Step
- Total parts = 2 + 3 = 5
- Value of 1 part = 500 ÷ 5 = ₹100
- A’s share = 2 × 100 = ₹200
- B’s share = 3 × 100 = ₹300
Quick Check
₹200 + ₹300 = ₹500 ✓
Three-Way Split Example
Divide 180 marbles among X, Y, Z in ratio 2:3:4
Total parts = 2 + 3 + 4 = 9
Each part = 180 ÷ 9 = 20
X = 2 × 20 = 40 marbles
Y = 3 × 20 = 60 marbles
Z = 4 × 20 = 80 marbles
Check: 40 + 60 + 80 = 180 ✓
🧠 Quick Memory Tricks
| Concept | Remember This |
|---|---|
| Percentage | “Per cent” = “Per 100” |
| Ratio | “A to B” = A:B |
| Proportion | Cross multiply to check |
| Successive % | Never just add them! |
| Division | Total parts first! |
🎯 The One Formula to Rule Them All
Almost everything comes down to:
Part/Whole × 100 = Percentage
Or flip it:
Whole × (Percentage/100) = Part
🏆 You Did It!
You now understand:
- ✅ Percentage calculations
- ✅ Percentage changes (up and down)
- ✅ Successive changes (the sneaky trap!)
- ✅ Real-world applications (profit, loss, interest, discount)
- ✅ Ratios and how they compare things
- ✅ Proportions and variations
- ✅ Dividing things fairly
Remember: Every big problem is just small pieces of pizza. Take it one slice at a time! 🍕
“Mathematics is not about numbers, equations, or algorithms: it is about understanding.” — William Paul Thurston