Comparing Quantities — Class 8

Complete notes on ratio, proportion, percentage, percentage change, profit & loss basics, and application problems — mobile-friendly & SEO-optimized.

1. Ratio

A ratio compares two quantities of the same kind. Written as a : b or a/b. It shows how many times one quantity is of another.

If a : b = c : d, then a/b = c/d (provided b,d ≠ 0).

Simplifying Ratios

Divide both terms by their GCD to get the ratio in lowest form. Example: 18 : 24 = 3 : 4 (divide by 6).

2. Proportion

Two ratios are in proportion if they are equal: a : b = c : d. If a/b = c/d then a×d = b×c (cross-multiplication).

Direct proportion: y = kx (y increases as x increases). Inverse proportion: y = k/x (y decreases as x increases).

3. Percentage

Percentage means 'per hundred'. 1% = 1/100. To convert a number to percent: multiply by 100 and add the % sign. To convert percent to fraction: divide by 100.

Example: 45% = 45/100 = 9/20. To find p% of x → (p/100) × x.

4. Increase & Decrease by Percentage

To increase x by p%: new value = x × (1 + p/100). To decrease x by p%: new value = x × (1 − p/100).

Example: Increase 500 by 12% → 500 × 1.12 = 560.

Successive percentage changes

Apply one change after another using multiplication. Example: Increase by 10% then decrease by 20% → multiply by 1.10 × 0.80 = 0.88 overall.

5. Profit & Loss (Basics)

Cost Price (CP): price paid. Selling Price (SP): price received. Profit = SP − CP (if SP > CP). Loss = CP − SP (if SP < CP).

Profit % = (Profit / CP) × 100. Loss % = (Loss / CP) × 100.

Example: CP = ₹200, SP = ₹250 → Profit = ₹50 → Profit% = (50/200)×100 = 25%.

6. Solved Examples

Example 1 (Ratio): Split ₹420 among A:B:C as 3:4:5. A's share = (3/12)×420 = ₹105.
Example 2 (Percentage): Find 15% of 360 → (15/100)×360 = 54.
Example 3 (Increase/Decrease): A shirt priced ₹800 is marked 25% off → new price = 800×(1−0.25)=₹600.
Example 4 (Profit): CP = ₹480, SP = ₹600 → Profit% = ((600−480)/480)×100 = 25%.

7. Practice Questions (with answers)

  1. Reduce the ratio 45:75 to lowest form.
    Answer
    Divide by GCD 15 → 3:5
  2. Find 12.5% of 240.
    Answer
    12.5% = 1/8 → 240/8 = 30
  3. Price increases from ₹400 to ₹460. Find percentage increase.
    Answer
    Increase = 60 → (60/400)×100 = 15%
  4. A shopkeeper sold an item at 10% profit. CP = ₹150. Find SP.
    Answer
    Profit = 15 → SP = 165

Show intermediate steps for percent-change and profit/loss problems — full marks depend on clear working.