Chapter 12 - Comparing Quantities Revision Notes - Class 7 Mathematics

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📌 Key Points

Comparison by difference: A - B shows how much more or less A is than B.
Comparison by division (ratio): A/B or A:B shows relative comparison between quantities.
Percentage expresses a quantity per hundred: (Part/Whole) × 100
To find x% of a number: Multiply the number by x/100
To find what percent one number is of another: (Number/Total) × 100
Percentage increase = [(New - Old)/Old] × 100; New Value = Old × (1 + Increase%/100)
Percentage decrease = [(Old - New)/Old] × 100; New Value = Old × (1 - Decrease%/100)
Profit = SP - CP; Profit % = (Profit/CP) × 100; Note: Always divide by CP, not SP
Loss = CP - SP; Loss % = (Loss/CP) × 100; Note: Always divide by CP, not SP
Discount is calculated on Marked Price (MP): Discount % = (Discount/MP) × 100
SP = MP - Discount; Also SP = MP × (1 - Discount%/100)
Ratio a:b compares two quantities; Order matters; Simplify by dividing by GCD
Proportion is equality of two ratios: a:b = c:d means a×d = b×c (cross multiplication)
Unitary method: Find value of 1 unit, then multiply by required quantity
In all percentage calculations: Always identify what is the Part and what is the Whole

📘 Important Definitions

Percentage

A fraction expressed with denominator 100. Example: 25% = 25/100. It represents a value per hundred units.

Cost Price (CP)

The amount of money paid to buy or manufacture an item. Also called purchase price.

Selling Price (SP)

The amount of money received when selling an item. This may be more or less than cost price.

Profit and Loss

Profit: SP > CP, so Profit = SP - CP. Loss: SP < CP, so Loss = CP - SP.

Marked Price (MP)

The price printed on an item or the original price before offering any discount.

Discount

A reduction offered on marked price. Discount = MP - SP. Usually expressed as percentage of MP.

Ratio

A comparison of two quantities of the same kind. Written as a:b or a/b. Has no unit.

Proportion

A statement that two ratios are equal. Written as a:b = c:d. Can be solved using cross multiplication.

🔢 Formulas & Laws

Percentage Formula

Percentage = (Part / Whole) × 100

Also: Part = (Percentage/100) × Whole; Whole = Part × (100/Percentage)

Percentage Increase

% Increase = [(New - Old) / Old] × 100; New = Old × (1 + %/100)

New value is always larger than original

Percentage Decrease

% Decrease = [(Old - New) / Old] × 100; New = Old × (1 - %/100)

New value is always smaller than original

Profit and Loss

Profit % = (Profit/CP) × 100; Loss % = (Loss/CP) × 100; Note: Always divide by CP

SP = CP + Profit; SP = CP - Loss

Discount

Discount % = (Discount/MP) × 100; SP = MP - Discount = MP × (1 - Discount%/100)

Discount is always on Marked Price, not on Cost Price

⚠️ Common Mistakes

✗ Wrong: Calculating profit % on SP instead of CP. Wrong: (Profit/SP) × 100

✓ Correct: Correct: Always use CP. Profit % = (Profit/CP) × 100

✗ Wrong: Calculating percentage increase/decrease on new value. Wrong: (Change/New) × 100

✓ Correct: Correct: Always use original value. % Change = (Change/Original) × 100

✗ Wrong: Calculating discount % on SP instead of MP. Wrong: (Discount/SP) × 100

✓ Correct: Correct: Discount % = (Discount/MP) × 100 (always on Marked Price)

✗ Wrong: Not simplifying ratios. Writing 12:18 instead of 2:3

✓ Correct: Correct: Always simplify ratios to lowest terms by dividing by GCD

✗ Wrong: Confusing percentage increase multiplier. 25% increase = 1.25, not 0.25

✓ Correct: Correct: x% increase multiplies by (1 + x/100). For 25%: multiply by 1.25

📝 Exam Focus

These questions are frequently asked in CBSE exams:

Percentage calculations and conversions between percentage, fraction, and decimal

2m★★★

Profit, loss, and discount problems with percentage calculations

3m★★★

Finding original value after percentage change (reverse percentage)

3m★★

Ratio and proportion problems, simplifying ratios, unitary method

2m★★

Word problems involving multiple percentage changes or mixed profit/loss

5m★★

Practical applications like compound percentage changes, commission, multiple discounts

🎯 Last-Minute Recall