Percentage

📘 1. What is a Percentage?

"Percent" means "per hundred". A percentage is a fraction or ratio where the denominator is always 100.

$$\text{Percentage}=\left(\frac{\text{Value}}{\text{Total}}\right)\times 100$$

Example: 25% = 25 out of 100 

                           = 25/100 

                           = 0.25

🧠 2. Converting Forms

Form	Example	Convert to %
Fraction	1/4	(1/4 × 100)% = 25%
Decimal	0.6	0.6 × 100 = 60%
Percentage	30%	Already in %

🧮 3. Basic Applications

A. Finding a percentage of a number

What is 20% of 250?
→ (20/100) × 250 = 50

B. Finding what % one number is of another

What percent is 40 of 200?
→ (40/200) × 100 = 20%

🔁 4. Percentage Increase / Decrease

A. Percentage Increase

$$\text{Increase \%}=\left(\frac{\text{Increase}}{\text{Original}}\right)\times 100$$

Example: Price rises from ₹200 to ₹250
→ Increase = 50 → (50/200)×100 = 25%

B. Percentage Decrease

$$\text{Decrease \%}= \left(\frac{\text{Decrease}}{\text{Original}}\right)\times 100$$

Example: Price drops from ₹300 to ₹240
→ Decrease = 60 → (60/300)×100 = 20%

🔄 5. Successive Percentage Change

If a number increases by x% and then by y%, the net effect is:

$$\text{Net \% Change}=x+y+\frac{xy}{100}$$

Example: Increase of 20%, then 10%
→ 20 + 10 + (20×10)/100 = 32% total increase

If one is increase and another is decrease:

$$\text{Net Change}=x-y-\frac{xy}{100}$$

🧭 6. Reverse Percentage

To find original price after decrease or increase:

Example: Final price after 20% discount is ₹80
→ Original = 80 / (1 - 0.20) = ₹100

💰 7. Percentage in Profit & Loss, Interest, Discount

These all use the concept of percentage based on:

• Cost Price (CP)

• Selling Price (SP)

• Marked Price (MP)

Example:
Profit % = (Profit / CP) × 100
Discount % = (Discount / MP) × 100

🧩 8. Important Tricks & Facts

Situation	Shortcut
Increase by x%, then decrease by x%	Net change = x²/100 decrease
A is x% more than B	A = B × (1 + x/100)
A is x% less than B	A = B × (1 - x/100)
% Error in base value	$(\text{Error}\mathrm{/}\text{Correct})\times 100$%

🧪 Practice Questions with Answers & Explanations

🧠 1. What is 15% of 320?

A. 42
B. 45
C. 48
D. 52

✅ Answer: C. 48
📝 Explanation:
(15/100) × 320 = 48

🧠 2. A number is increased by 20% and then decreased by 20%. What is the net change?

A. 0%
B. 4% increase
C. 4% decrease
D. 8% decrease

✅ Answer: C. 4% decrease
📝 Explanation:
Net Change = x − y − (xy / 100)
= 20 − 20 − (20×20 / 100) = 0 − 4 = −4%
⇒ Decrease of 4%

🧠 3. A student scored 360 marks out of 450. What was his percentage?

A. 75%
B. 78%
C. 80%
D. 82%

✅ Answer: C. 80%
📝 Explanation:
(360/450) × 100 = 80%

🧠 4. The price of a product increased from ₹400 to ₹500. What is the percentage increase?

A. 25%
B. 20%
C. 18%
D. 30%

✅ Answer: A. 25%
📝 Explanation:
Increase = 500 − 400 = 100
% Increase = (100 / 400) × 100 = 25%

🧠 5. If 60% of the students passed an exam, and there were 80 students, how many failed?

A. 48
B. 32
C. 30
D. 28

✅ Answer: B. 32
📝 Explanation:
Passed = 60% of 80 = 48 → Failed = 80 − 48 = 32

🧩 Fill in the Blanks (with Explanation):

🧩 6. 25% of 800 is __________.
✅ Answer: 200
(25/100) × 800 = 200

🧩 7. If the final price after 20% discount is ₹240, then the marked price is __________.
✅ Answer: ₹300
240 = MP × (1 − 0.20) ⇒ MP = 240 / 0.8 = 300

🧩 8. A value is increased by 10% and then by another 20%. The overall percentage increase is __________.
✅ Answer: 32%
10 + 20 + (10×20)/100 = 32%

🧩 9. A is 25% more than B. If B is 120, A is __________.
✅ Answer: 150
A = 120 × (1 + 25/100) = 120 × 1.25 = 150

🧩 10. A number was mistakenly taken as 80 instead of 100. The percentage error is __________.
✅ Answer: 20%
Error = 100 − 80 = 20
% Error = (20 / 100) × 100 = 20%