Class 8 Maths Comparing Quantities

Comparing Quantities – Quick Notes (Class 8 NCERT)

Definition:

Comparing quantities means analyzing how one quantity relates to another using percentages, ratios, or proportions.

• Applications: Profit & Loss, Simple Interest, Discount, Tax, Commission, CPI, etc.

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Key Concepts

1. Ratio
   • A ratio compares two quantities of the same kind.
   • Example: If 5 pens cost ₹20 and 10 pens cost ₹40 → ratio of pens = 5:10 = 1:2
2. Proportion
   • Two ratios are equal → they are in proportion.
   • Example: 2:3 = 4:6 ✅
3. Percentage
   • A fraction expressed out of 100.

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$Percentage=WholePart​×100

4. Profit and Loss
   • Profit: Selling Price (SP) > Cost Price (CP) → Profit = SP − CP
   • Loss: CP > SP → Loss = CP − SP
   • Percentage:

$$\text{Profit\%} = \frac{\text{Profit}}{\text{CP}} \times 100, \quad \text{Loss\%} = \frac{\text{Loss}}{\text{CP}} \times 100$$Profit%=CPProfit​×100,Loss%=CPLoss​×100

5. Simple Interest (SI)

$$SI = \frac{P \cdot R \cdot T}{100}$$SI=100P⋅R⋅T​

Where P = principal, R = rate per year, T = time in years

6. Compound Interest (CI)

$$CI = P \left(1 + \frac{R}{100}\right)^T – P$$CI=P(1+100R​)T−P

7. Discount

$$\text{Discount} = \text{Marked Price} – \text{Selling Price}$$Discount=Marked Price−Selling Price $$\text{Discount\%} = \frac{\text{Discount}}{\text{Marked Price}} \times 100$$Discount%=Marked PriceDiscount​×100

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Tips / Tricks:

• Always check whether % is on CP, SP, or total.
• Use unitary method for ratio and proportion problems.
• Keep profit and loss formulas separate in mind for quick calculation.

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Comparing Quantities – MCQ Q&A

1. Q: 50 is what % of 200?
   A: $50/200 \times 100 = 25\%$50/200×100=25%
2. Q: Ratio of 20:50 in simplest form?
   A: 2:5 ✅
3. Q: If SP = 120 and CP = 100, profit = ?
   A: Profit = SP − CP = 120 − 100 = 20
4. Q: Profit % if SP = 150, CP = 120?
   A: Profit = 30, Profit% = 30/120 × 100 = 25%
5. Q: Loss % if CP = 200, SP = 180?
   A: Loss = 20, Loss% = 20/200 × 100 = 10%
6. Q: Simple Interest for ₹500 at 5% per year for 2 years?
   A: SI = (500 × 5 × 2)/100 = ₹50
7. Q: Compound Interest for ₹1000 at 10% per year for 2 years?
   A: CI = 1000(1 + 0.1)^2 − 1000 = 1000 × 1.21 − 1000 = ₹210
8. Q: Marked Price = ₹500, SP = ₹450. Discount = ?
   A: Discount = 500 − 450 = ₹50
9. Q: Discount % if MP = 800, SP = 720?
   A: 80/800 × 100 = 10%
10. Q: Two ratios are equal. What is this called?
    A: Proportion ✅
11. Q: Ratio of 0.5 : 0.25
    A: 2:1
12. Q: 25% of 400?
    A: 100
13. Q: If CP = SP, profit or loss?
    A: Neither, 0%
14. Q: Profit = 15, CP = 60 → Profit %?
    A: 15/60 × 100 = 25%
15. Q: SP = 250, Profit = 50 → CP?
    A: CP = SP − Profit = 250 − 50 = 200
16. Q: SI = 120, P = 600, R = 10% → Time?
    A: T = SI × 100 / (P × R) = 120 × 100 / (600 × 10) = 2 years
17. Q: MP = 1000, Discount = 5% → SP?
    A: SP = 1000 − 5% of 1000 = 1000 − 50 = 950
18. Q: 3:4 = ?
    A: Ratio remains 3:4 ✅
19. Q: 30% of 50?
    A: 15
20. Q: A man buys an article at ₹80 and sells at ₹100 → Profit %?
    A: Profit = 100 − 80 = 20 → Profit% = 20/80 × 100 = 25%