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('quantitative-techniques', 'Quantitative Techniques') 3% exam weight

Topic 2

Part of the GAT Pakistan study roadmap. ('quantitative-techniques', 'Quantitative Techniques') topic quanti-002 of ('quantitative-techniques', 'Quantitative Techniques').

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Percentage

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Percentage — Key Facts for GAT Pakistan

Basic Formula: $$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$

Conversions:

FractionDecimalPercentage
1/20.550%
1/40.2525%
1/50.220%
1/80.12512.5%
1/100.110%
1/200.055%
1/30.333…33.33%
2/30.666…66.67%
3/40.7575%
1/1000.011%

Quick Percentage Calculations:

  • To find 10% of a number → Divide by 10
  • To find 5% → Find 10%, then divide by 2
  • To find 25% → Divide by 4
  • To find 50% → Divide by 2
  • To find 1% → Divide by 100

⚡ GAT Exam Tip: For percentage questions, convert percentages to fractions first - it makes calculations much faster.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Percentage — Detailed Study Guide

Percentage Increase and Decrease

Increase Formula: $$\text{Increased Value} = \text{Original} \times \left(1 + \frac{r}{100}\right)$$

Decrease Formula: $$\text{Decreased Value} = \text{Original} \times \left(1 - \frac{r}{100}\right)$$

Successive Percentage Change: When two percentage changes occur successively: $$\text{Final Value} = \text{Original} \times \left(1 + \frac{a}{100}\right) \times \left(1 + \frac{b}{100}\right)$$

Worked Examples:

Example 1: Price increased from Rs. 80 to Rs. 100. Find percentage increase.
Solution:
Increase = 100 - 80 = 20
Percentage increase = (20/80) × 100 = 25%

Example 2: A shopkeeper increases price by 20%, then decreases by 10%.
If original price is Rs. 500, find final price.

Solution:
After 20% increase: 500 × 1.20 = Rs. 600
After 10% decrease: 600 × 0.90 = Rs. 540

Net effect: 540 - 500 = Rs. 40 increase
Net percentage: (40/500) × 100 = 8% increase

⚡ Common Mistake: Students often add percentages directly (20% + 10% = 30%) when successive changes occur. But the actual net effect is 8%, not 30%!

Population and Depreciation Problems

Population Growth Formula: $$\text{Population after n years} = P \times \left(1 + \frac{r}{100}\right)^n$$

Depreciation Formula: $$\text{Value after n years} = P \times \left(1 - \frac{r}{100}\right)^n$$

Worked Examples:

Example 1: Population of a city is 2,00,000. It increases at 10% per year.
What will be the population after 3 years?

Solution:
P = 2,00,000, r = 10%, n = 3

Population = 2,00,000 × (1 + 10/100)³
           = 2,00,000 × (1.1)³
           = 2,00,000 × 1.331
           = 2,66,200

Example 2: A car costs Rs. 10,00,000. It depreciates at 20% per year.
What will be its value after 2 years?

Solution:
Value = 10,00,000 × (1 - 20/100)²
      = 10,00,000 × (0.8)²
      = 10,00,000 × 0.64
      = Rs. 6,40,000

⚡ GAT PYQ: “The population of a town is 10,000. It increases at 5% per annum. What will be the population after 2 years?” Answer: 11,025

Percentage in Comparison

Original vs New Comparison:

Question: If A's salary is 20% more than B's, by what percentage is B's
salary less than A's?

Solution:
Let B's salary = Rs. 100
A's salary = Rs. 120

Difference = Rs. 20
Percentage less = (20/120) × 100 = 16.67%

Answer: B's salary is 16.67% less than A's

Worked Comparison Example:

Question: In an election, candidate A got 40% votes and won by 500 votes.
How many total votes were cast?

Solution:
Candidate B got: 100% - 40% = 60%
Margin = 60% - 40% = 20%
20% of total = 500
Total votes = 500 × 100/20 = 2500

Answer: 2500 votes

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Percentage — Complete Notes for GAT

Product Consistency and Market Share

Market Share Problems:

Example: Company A's market share increased from 20% to 25% while
market size remained constant at Rs. 100 crore. Find actual increase
in revenue for Company A.

Solution:
Original revenue = 20% of 100 = Rs. 20 crore
New revenue = 25% of 100 = Rs. 25 crore
Actual increase = Rs. 5 crore

Percentage increase = (5/20) × 100 = 25%
(Note: Market share increased by 5 percentage points,
      but revenue increased by 25%!)

⚡ Important for GAT: Don’t confuse “percentage points” with “percentage increase”!

Mixture and Alligation

Alligation Method: When mixing two ingredients at different prices:

$$\text{Mean Price} = \frac{(\text{Qty}_1 \times \text{Price}_1) + (\text{Qty}_2 \times \text{Price}_2)}{\text{Qty}_1 + \text{Qty}_2}$$

Shortcut for Alligation:

Question: In what ratio must rice at Rs. 40/kg be mixed with rice at
Rs. 60/kg to get a mixture at Rs. 50/kg?

Solution using alligation:
Price 1: Rs. 40    Mean: Rs. 50    Price 2: Rs. 60
         |----50-60=10----|  |----40-50=10---|
Ratio = 10:10 = 1:1

Worked Mixture Example:

Example: How much water must be added to 20 litres of milk containing
5% water to make water content 10%?

Solution:
Water in 20L milk = 5% of 20 = 1 litre
Milk = 19 litres (95%)

For 10% water, water should be (1/9) of total
If water = x litres, total = 19 + x
x/(19+x) = 0.10
x = 1.9 + 0.1x
0.9x = 1.9
x = 2.11 litres

Percentage Error

Absolute Error and Percentage Error:

Example: A student measures a length as 5.2 cm, but actual is 5.0 cm.
Find percentage error.

Solution:
Absolute error = 5.2 - 5.0 = 0.2 cm
Percentage error = (0.2/5.0) × 100 = 4%

GAT-Style Practice Questions

1. If 30% of a number is 45, what is 80% of that number?
   (a) 100 (b) 120 (c) 135 (d) 150

   Answer: (b) 120
   Solution: Let number = x
             30% of x = 45
             x = 45 × 100/30 = 150
             80% of 150 = 120

2. The price of an article is first increased by 25% and then
   decreased by 20%. The net change is:
   (a) +5% (b) -5% (c) 0% (d) +10%

   Answer: (a) +5%
   Solution: Let price = Rs. 100
             After 25% increase: Rs. 125
             After 20% decrease: 125 × 0.80 = Rs. 100
             Net change = 0%
             (Wait, that's wrong)
             Let price = 100
             After 25% increase: 125
             After 20% decrease of 125: 125 × 0.8 = 100
             Actually that's 0% change.
             Let me recalculate...

             Let original = 100
             After 25% increase = 125
             After 20% decrease on 125 = 125 × 0.8 = 100
             So it's back to 100... 

             Wait, the 20% decrease is on the NEW value (125)
             20% of 125 = 25
             125 - 25 = 100
             Net change = 0%

             Hmm, that can't be right either since (1.25 × 0.8 = 1.0)

             Actually: (1 + 25/100)(1 - 20/100) = 1.25 × 0.8 = 1.0 = 0% change
             Answer: (c) 0%

3. A shop offers two successive discounts of 20% and 15%.
   What is the equivalent single discount?
   (a) 35% (b) 32% (c) 30% (d) 28%

   Answer: (b) 32%
   Solution: Let MP = Rs. 100
             After 20% discount = Rs. 80
             After 15% discount on 80 = 80 × 0.85 = Rs. 68
             Equivalent single discount = (100-68)% = 32%

⚡ GAT Strategy: For successive percentage changes, ALWAYS multiply the factors, don’t add them!

Profit and Loss Percentage

Key Formulas:

TermFormula
Profit %(Profit/Cost Price) × 100
Loss %(Loss/Cost Price) × 100
Selling Price (Profit)CP × (1 + r/100)
Selling Price (Loss)CP × (1 - r/100)

Worked Examples:

Example 1: A merchant buys goods at Rs. 80 and sells at Rs. 100.
Find profit percentage.

Solution:
CP = Rs. 80, SP = Rs. 100
Profit = 100 - 80 = Rs. 20
Profit % = (20/80) × 100 = 25%

Example 2: A shopkeeper sells a shirt for Rs. 690 at a loss of 8%.
What was the cost price?

Solution:
SP = CP × (1 - 8/100)
690 = CP × 0.92
CP = 690/0.92 = Rs. 750

Example 3: A shopkeeper uses false weights. He uses 900g as 1kg.
If he sells at cost price, find his profit percentage.

Solution:
Actual weight for 1kg = 1000g
Given weight = 900g

CP of 900g goods = Cost of 900g at cost price
SP of 900g (sold as 1kg) = Cost of 1000g at cost price

Profit = 1000 - 900 = 100 on 900
Profit % = (100/900) × 100 = 11.11%

⚡ GAT PYQ: “A shopkeeper sells an article at 20% profit. If he had bought it at 10% less and sold it at Rs. 20 less, he would have made 25% profit. Find the cost price.” This requires setting up equations - practice such questions!

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