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Percentages, Profit & Loss, and Discounts - Aptitude Mastery Guide

Section titled “Percentages, Profit & Loss, and Discounts - Aptitude Mastery Guide”

Category: Quantitative Aptitude
Generated on: 2025-07-15 09:15:51
Source: Aptitude Mastery Guide Generator

Master Guide: Percentages, Profit & Loss, and Discounts

Section titled “Master Guide: Percentages, Profit & Loss, and Discounts”

This comprehensive guide will equip you with the knowledge and skills needed to master percentages, profit & loss, and discounts – crucial topics for various competitive exams and placement tests. We’ll cover foundational concepts, powerful tricks, essential formulas, solved examples, and challenging practice problems.

1. Foundational Concepts

Section titled “1. Foundational Concepts”

A. Percentages:

  • Definition: A percentage is a way of expressing a number as a fraction of 100. The word “percent” means “per hundred” or “out of one hundred.” The symbol for percent is %.

  • Why Percentages Matter: Percentages provide a standardized way to compare different quantities. Instead of saying “3 out of 10” and “6 out of 20,” we can express both as 30%, making the comparison immediate.

  • Understanding the Base: The key to percentage problems is identifying the base value. The base is the whole amount to which the percentage is applied. For example, in the statement “20% of 50,” 50 is the base.

B. Profit & Loss:

  • Cost Price (CP): The price at which an article is purchased.

  • Selling Price (SP): The price at which an article is sold.

  • Profit: Occurs when SP > CP. Profit = SP - CP

  • Loss: Occurs when SP < CP. Loss = CP - SP

  • Why Understanding Profit & Loss is Important: These concepts are fundamental to business and finance. They help determine the profitability of transactions and are essential for pricing strategies.

C. Discounts:

  • Marked Price (MP) or List Price: The price printed or displayed on an article.

  • Discount: A reduction in the marked price.

  • Selling Price (after discount): SP = MP - Discount

  • Why Discounts are Used: Discounts are used to attract customers, clear out inventory, and increase sales volume. Understanding discount calculations is crucial for both businesses and consumers.

2. Key Tricks & Shortcuts

Section titled “2. Key Tricks & Shortcuts”

  • Trick 1: Percentage to Fraction Conversion:

    • Explanation: Converting percentages to fractions and vice-versa simplifies calculations. Remember that ”%” means “divided by 100.”
    • How to Use: For example, 25% = 25/100 = 1/4. This means finding 25% of a number is the same as finding 1/4 of that number.
    • Example: What is 25% of 80? Instead of (25/100)*80, calculate (1/4)*80 = 20.
    • Common Conversions to Memorize:
      • 10% = 1/10
      • 20% = 1/5
      • 25% = 1/4
      • 33.33% (or 33 1/3%) = 1/3
      • 50% = 1/2
      • 66.66% (or 66 2/3%) = 2/3
      • 75% = 3/4
      • 100% = 1

  • Trick 2: The ‘x% of y’ is the same as ‘y% of x’ Rule:

    • Explanation: This is a powerful shortcut for simplifying calculations, especially when one of the numbers is easier to work with as a percentage.
    • How to Use: If you need to find 17% of 50, it’s easier to calculate 50% of 17, which is simply 17/2 = 8.5.
    • Example: Find 12% of 25. It’s the same as 25% of 12, which is (1/4)*12 = 3.

  • Trick 3: Percentage Increase/Decrease Shortcut:

    • Explanation: A quick way to calculate the new value after a percentage increase or decrease.
    • How to Use:
      • Increase: New Value = Original Value * (1 + (Percentage Increase/100))
      • Decrease: New Value = Original Value * (1 - (Percentage Decrease/100))
    • Example: Increase 200 by 15%. New Value = 200 * (1 + (15/100)) = 200 * 1.15 = 230.

  • Trick 4: Successive Percentage Change:

    • Explanation: Used when a value is changed by a percentage, and then the new value is changed by another percentage.
    • How to Use: Let the original value be ‘x’. If the value is increased by ‘a%’ and then decreased by ‘b%’, the final value is:
      • Final Value = x * (1 + a/100) * (1 - b/100)
    • Simplified Formula for Overall Percentage Change: If a value is increased by x% and then increased by y%, the overall percentage increase is:
      • Overall % Increase = x + y + (xy/100)
    • If a value is increased by x% and then decreased by y%, the overall percentage change is:
      • Overall % Change = x - y - (xy/100)
      • Note: If the result is positive, it’s an increase; if negative, it’s a decrease.
    • Example: A price is increased by 10% and then decreased by 5%. Overall % Change = 10 - 5 - (10*5/100) = 5 - 0.5 = 4.5%. Therefore, the price increased by 4.5%.

  • Trick 5: Vedic Maths - Base Method for Multiplication (Applicable to Percentage Calculations):

    • Explanation: This Vedic Maths trick helps in faster multiplication, which is very useful when dealing with percentage calculations that involve slightly more complex multiplications.
    • How to Use: Let’s say you want to calculate 104% of 106. You can rewrite this as 104 * 106 / 100.
      1. Choose a base near the numbers (in this case, 100).
      2. Find the difference of each number from the base: 104 - 100 = +4, 106 - 100 = +6
      3. Cross-add one of the differences to the other number: 104 + 6 = 110 (or 106 + 4 = 110)
      4. Multiply the differences: 4 * 6 = 24
      5. Combine the results: 110 | 24 = 11024
      6. Since we divided by 100 initially, divide this by 100 now to get the answer: 110.24
    • Example: Calculate 108% of 112.
      1. Base = 100
      2. Differences: +8, +12
      3. Cross-add: 108 + 12 = 120
      4. Multiply differences: 8 * 12 = 96
      5. Combine: 120 | 96 = 12096
      6. Divide by 100: 120.96

  • Trick 6: Assumption Method for Profit & Loss:

    • Explanation: When dealing with problems involving ratios of CP, SP, etc., assume a convenient value (often 100 or a multiple of the given ratios) to simplify calculations.
    • How to Use: If the ratio of CP to SP is 5:6, assume CP = 500 and SP = 600. This makes calculating profit or loss percentage much easier.
    • Example: If the cost price of an item is 20% less than its selling price, find the profit percentage.
      • Assume SP = 100. Then, CP = 100 - 20% of 100 = 80.
      • Profit = SP - CP = 100 - 80 = 20.
      • Profit % = (Profit/CP) * 100 = (20/80) * 100 = 25%.

  • Trick 7: Discount Series:

    • Explanation: When multiple discounts are given successively, don’t simply add them up. Calculate the effective discount.
    • How to Use: If discounts of x% and y% are given successively, the effective discount is:
      • Effective Discount = x + y - (xy/100)%
    • Example: A shopkeeper offers successive discounts of 10% and 20%. The effective discount is: 10 + 20 - (10*20/100) = 30 - 2 = 28%.

3. Essential Formulas & Rules

Section titled “3. Essential Formulas & Rules”
Formula/RuleDescription
Percentage = (Part/Whole) * 100Calculating a percentage from a part and a whole.
Amount = Percentage * BaseFinding the amount when given a percentage and a base.
Percentage Change = ((New Value - Old Value)/Old Value) * 100Calculating the percentage change (increase or decrease).
Profit = SP - CPCalculating the profit.
Loss = CP - SPCalculating the loss.
Profit % = (Profit/CP) * 100Calculating the profit percentage.
Loss % = (Loss/CP) * 100Calculating the loss percentage.
SP = CP * (1 + (Profit%/100))Calculating the selling price when profit percentage is given.
SP = CP * (1 - (Loss%/100))Calculating the selling price when loss percentage is given.
Discount = MP - SPCalculating the discount amount.
Discount % = (Discount/MP) * 100Calculating the discount percentage.
SP = MP * (1 - (Discount%/100))Calculating the selling price after a discount.
CP = SP / (1 + (Profit%/100))Calculating the cost price when the selling price and profit percentage are given.
CP = SP / (1 - (Loss%/100))Calculating the cost price when the selling price and loss percentage are given.
Effective Discount (Successive Discounts x% and y%) = x + y - (xy/100)%Calculating the overall discount after two successive discounts.

4. Detailed Solved Examples

Section titled “4. Detailed Solved Examples”

Example 1: Basic Percentage Calculation [Using Trick 1 & Trick 2]

  • Problem: A student scored 80 marks out of 120. What is the percentage of marks scored by the student?

  • Solution:

    1. Percentage = (Marks Scored / Total Marks) * 100
    2. Percentage = (80/120) * 100
    3. Simplify the fraction: 80/120 = 2/3
    4. Percentage = (2/3) * 100
    5. Trick 1: Recognize that 1/3 = 33.33%, therefore 2/3 = 2 * 33.33% = 66.66%
    6. Answer: The student scored 66.66%.

Example 2: Profit and Loss Calculation [Using Assumption Method - Trick 6]

  • Problem: A shopkeeper sells an article at a profit of 20%. If the cost price is increased by 10% and the selling price remains the same, find the new profit percentage.

  • Solution:

    1. Trick 6 (Assumption Method): Assume the initial CP = 100.
    2. Profit = 20% of 100 = 20.
    3. Initial SP = CP + Profit = 100 + 20 = 120.
    4. New CP = Initial CP + 10% of Initial CP = 100 + 10 = 110.
    5. The SP remains the same, so New SP = 120.
    6. New Profit = New SP - New CP = 120 - 110 = 10.
    7. New Profit % = (New Profit / New CP) * 100 = (10/110) * 100 = 9.09%.
    8. Answer: The new profit percentage is 9.09%.

Example 3: Successive Discounts [Using Trick 7]

  • Problem: A retailer offers a discount of 15% on all items. He offers a further discount of 5% to customers who pay in cash. What is the net selling price of an item with a marked price of Rs. 1000 if a customer pays in cash?

  • Solution:

    1. Trick 7 (Effective Discount): First, find the effective discount.
    2. Effective Discount = 15 + 5 - (15 * 5 / 100) = 20 - 0.75 = 19.25%
    3. Discount Amount = 19.25% of 1000 = (19.25/100) * 1000 = Rs. 192.50
    4. Net Selling Price = Marked Price - Discount Amount = 1000 - 192.50 = Rs. 807.50.
    5. Answer: The net selling price is Rs. 807.50.

Example 4: Reverse Percentage Problem

  • Problem: After a 20% price reduction, a shirt sells for Rs. 800. What was the original price of the shirt?

  • Solution:

    1. Let the original price be ‘x’.
    2. The shirt sells for Rs. 800 after a 20% reduction, meaning Rs. 800 represents 80% (100% - 20%) of the original price.
    3. Therefore, 0.80x = 800
    4. x = 800 / 0.80
    5. x = 1000
    6. Answer: The original price of the shirt was Rs. 1000.

5. Practice Problems (Graded Difficulty)

Section titled “5. Practice Problems (Graded Difficulty)”

[Easy]

  1. What is 35% of 200?
  2. A shopkeeper buys an item for Rs. 500 and sells it for Rs. 600. What is the profit percentage?

[Medium]

  1. A number is increased by 20% and then decreased by 20%. What is the net percentage change?
  2. The marked price of a book is Rs. 400. A shopkeeper sells it at a discount of 10%. What is the selling price?

[Hard]

  1. A trader marks up his goods by 20% above the cost price. He allows a discount of 10% on the marked price. What is his profit percentage?
  2. A shopkeeper sells two articles for Rs. 99 each. On one, he gains 10%, and on the other, he loses 10%. Find his overall gain or loss percentage.
  3. The population of a town increases by 10% every year. If the present population is 214,000, what was the population 2 years ago?

6. Advanced/Case-Based Question

Section titled “6. Advanced/Case-Based Question”

A clothing store is running a promotion. They offer a “Buy One Get One 50% Off” deal on all shirts. In addition, customers who spend over Rs. 2000 receive a further 10% discount on their entire purchase (including the discounted shirts). A customer selects two shirts priced at Rs. 1200 each and a pair of pants priced at Rs. 900.

  1. Calculate the price of the two shirts after the “Buy One Get One 50% Off” discount.
  2. Calculate the total cost of the shirts and pants before the final discount.
  3. Does the customer qualify for the 10% discount on the entire purchase?
  4. If yes, calculate the final amount the customer needs to pay.

This comprehensive guide should provide you with a solid foundation in percentages, profit & loss, and discounts. Practice consistently, and you’ll be well-prepared for any exam! Good luck!