In this chapter, you will learn
- —Understand and apply comparison by difference and division
- —Calculate percentage and apply it in practical situations
- —Work with ratios and proportions
- —Solve problems involving increase and decrease
- —Apply percentage in profit, loss, and discount
Comparison by Difference and Division
Comparison by Difference: Compares two quantities using subtraction to find how much more or less one is than the other.
Example: If A has 50 books and B has 30 books, then A has 50 - 30 = 20 more books than B.
Limitation: This doesn't show the relative comparison (B's books are 60% of A's books).
Comparison by Division (Ratio): Expresses one quantity as a fraction or ratio of another to show relative size.
Formula: Ratio = A : B = A/B
Example: If A = 50 and B = 30, Ratio A:B = 50:30 = 5:3. This means for every 5 books A has, B has 3.
When to use which:
- Difference: When absolute change matters (e.g., "5 kg heavier")
- Division: When relative comparison matters (e.g., "twice as heavy")
Exam Tip
Problems often test your understanding of when to use difference vs. ratio. Ratio is always more informative.
Common Mistake
Using only difference to compare without considering the base quantity. Always ask: 'Compared to what?'
Understanding Percentage
Percentage: A fraction expressed with denominator 100. It shows comparison per 100 units.
Percentage = (Part / Whole) × 100
Key Conversions:
- Percentage to Decimal: Divide by 100. Example: 25% = 0.25
- Decimal to Percentage: Multiply by 100. Example: 0.75 = 75%
- Fraction to Percentage: (Numerator/Denominator) × 100. Example: 3/4 = 75%
- Percentage to Fraction: Divide by 100. Example: 40% = 40/100 = 2/5
Practical Example: In a class of 40 students, 30 passed.
Pass percentage = (30/40) × 100 = 75%
Exam Tip
Always identify 'Part' and 'Whole' correctly. Part is always the quantity being compared.
Common Mistake
Confusing 'increase by 20%' with 'becomes 20 units'. Increase by 20% means multiply by 1.2.
Percentage Change: Increase and Decrease
Percentage Increase: Shows growth relative to original value.
Formula: Percentage Increase = [(New Value - Old Value) / Old Value] × 100
Or: New Value = Old Value × (1 + Percentage Increase/100)
Percentage Decrease: Shows reduction relative to original value.
Formula: Percentage Decrease = [(Old Value - New Value) / Old Value] × 100
Or: New Value = Old Value × (1 - Percentage Decrease/100)
Examples:
- Price increases from Rs. 100 to Rs. 120: Increase = (20/100) × 100 = 20%
- Price decreases from Rs. 100 to Rs. 80: Decrease = (20/100) × 100 = 20%
Exam Tip
Always divide by the ORIGINAL value, not the new value. This is the most common error.
Common Mistake
Calculating percentage decrease based on new value instead of old value. Wrong: (20/80)×100 = 25%
Profit, Loss, and Discount
Profit and Loss Concepts:
- Cost Price (CP): Amount paid to buy an item
- Selling Price (SP): Amount received when selling
- Profit: SP - CP (when SP > CP)
- Loss: CP - SP (when CP > SP)
- Profit %: (Profit / CP) × 100
- Loss %: (Loss / CP) × 100
Discount: Reduction offered on Marked Price.
- Marked Price (MP): Price printed on item
- Discount: MP - SP
- Discount %: (Discount / MP) × 100
- SP = MP - Discount or SP = MP × (1 - Discount%/100)
Example: MP = Rs. 500, Discount = 20%. SP = 500 × 0.8 = Rs. 400
Exam Tip
Profit/Loss % is always calculated on Cost Price. Discount % is always on Marked Price.
Common Mistake
Calculating profit % on Selling Price. Wrong: (Profit/SP)×100. Correct: (Profit/CP)×100
Ratio and Proportion
Ratio: Comparison of two quantities of the same kind expressed as a : b.
Properties of Ratios:
- Ratio has no unit. Example: 5 kg : 10 kg = 1 : 2
- Order matters. 3:5 is different from 5:3
- Ratios can be simplified. 12:18 = 2:3 (divide by GCD)
- Equal ratios form a proportion. 2:3 = 4:6 = 6:9
Proportion: Statement that two ratios are equal. Written as a:b = c:d or a:b::c:d
Fundamental Property: If a:b = c:d, then a × d = b × c (Cross Multiplication)
Example: 2:3 = x:9. Then 2 × 9 = 3 × x, so x = 6
Unitary Method: Find value of one unit, then multiply by required quantity.
- If 5 kg sugar costs Rs. 200, what is cost of 12 kg? Cost per kg = 200/5 = Rs. 40. Cost of 12 kg = 40 × 12 = Rs. 480
Exam Tip
Cross multiplication is the fastest method to check if two ratios are equal or find unknown in proportions.
Common Mistake
Not simplifying ratios to lowest terms. Always reduce 12:18 to 2:3 for clarity.
Chapter Summary
Comparing Quantities involves understanding relative differences between values:
- Comparison by Difference: Absolute change (A - B)
- Comparison by Division: Relative ratio (A/B)
- Percentage: Per hundred comparison (Part/Whole × 100)
- Percentage Change: Increase/Decrease relative to original
- Profit/Loss/Discount: Applications of percentage in commerce
- Ratio and Proportion: Comparing quantities and finding unknowns
Exam Focus: Percentage calculations, profit-loss-discount problems, solving proportions, real-life applications like simple interest approximations.