Arithmetic
Percentage, Ratio-Proportion, Profit-Loss and Interest
Overview
Arithmetic forms the backbone of quantitative reasoning in KTET Mathematics. This topic covers four interconnected concepts—percentage, ratio-proportion, profit-loss, and interest—that appear both as direct calculation questions and as word problems requiring multi-step reasoning.
Mastery here also strengthens your ability to teach these topics effectively to primary and upper-primary students, which aligns with KTET's dual focus on content knowledge and pedagogical competence.
Students must build fluency in quick conversions (fractions to percentages, ratios to fractions) and develop mental math shortcuts. These skills transfer directly to data interpretation and mensuration problems as well.
Key Concepts
- Percentage means "per hundred": 25% = 25/100 = 0.25 = 1/4. Always convert to the most convenient form for calculation.
- Ratio compares two quantities of the same kind: Written as a:b or a/b. Ratios have no units—they are pure numbers.
- Proportion states that two ratios are equal: If a:b = c:d, then a×d = b×c (cross-multiplication property).
- Cost Price (CP) and Selling Price (SP) determine profit or loss: Profit = SP − CP when SP > CP; Loss = CP − SP when CP > SP.
- Profit% and Loss% are always calculated on CP: This is a standard convention that students often forget.
- Simple Interest (SI) remains constant each year: Interest is calculated only on the principal amount.
- Compound Interest (CI) grows because interest earns interest: Each year's interest is added to principal before calculating next year's interest.
- Marked Price and Discount: Discount is calculated on Marked Price (MP), not on CP. SP = MP − Discount.
Formulas / Key Facts
Percentage
- Percentage = (Value / Total) × 100
- To find x% of y: (x/100) × y
- Percentage increase = [(New − Original) / Original] × 100
- Percentage decrease = [(Original − New) / Original] × 100
Ratio and Proportion
- If a:b = c:d, then ad = bc (product of extremes = product of means)
- To divide quantity Q in ratio a:b: First part = Q × a/(a+b), Second part = Q × b/(a+b)
- Duplicate ratio of a:b is a²:b²
- Sub-duplicate ratio of a:b is √a:√b
Profit and Loss
- Profit% = (Profit / CP) × 100
- Loss% = (Loss / CP) × 100
- SP = CP × (100 + Profit%) / 100
- SP = CP × (100 − Loss%) / 100
- If SP = CP, then no profit, no loss
Interest
- Simple Interest: SI = (P × R × T) / 100, where P = Principal, R = Rate%, T = Time in years
- Amount = P + SI
- Compound Interest: A = P(1 + R/100)^T, CI = A − P
- For 2 years: CI − SI = P(R/100)²
Worked Examples
Example 1: Percentage A school has 800 students. If 35% are girls, how many boys are in the school?
Solution:
- Girls = 35% of 800 = (35/100) × 800 = 280
- Boys = 800 − 280 = 520 boys
Quick method: Boys = 65% of 800 = (65 × 8) = 520
Example 2: Ratio and Proportion Divide ₹1,800 between A and B in the ratio 2:3.
Solution:
- Total parts = 2 + 3 = 5
- A's share = 1800 × (2/5) = ₹720
- B's share = 1800 × (3/5) = ₹1,080
Verification: 720 + 1080 = 1800 ✓
Example 3: Profit and Loss A shopkeeper buys an article for ₹400 and sells it for ₹460. Find the profit percentage.
Solution:
- Profit = SP − CP = 460 − 400 = ₹60
- Profit% = (60/400) × 100 = 15%
Answer: 15% profit
Example 4: Simple and Compound Interest Find SI and CI on ₹5,000 at 10% per annum for 2 years.
Solution:
- SI = (5000 × 10 × 2) / 100 = ₹1,000
- CI: A = 5000(1 + 10/100)² = 5000 × 1.21 = ₹6,050
- CI = 6050 − 5000 = ₹1,050
Difference: CI − SI = 1050 − 1000 = ₹50
Shortcut verification: CI − SI for 2 years = P(R/100)² = 5000 × (0.1)² = 5000 × 0.01 = ₹50 ✓
Common Mistakes
| Wrong Thinking | Correct Approach |
|---|---|
| Calculating profit% on SP instead of CP | Profit% is always on Cost Price. Remember: "Profit is earned on investment (CP)." |
| Adding percentages directly in successive changes | Use multiplication: 10% increase then 20% increase = 1.1 × 1.2 = 1.32, i.e., 32% total increase, not 30%. |
| Confusing ratio with fraction | Ratio a:b means a/b only when comparing first to second. For part-to-whole, use a/(a+b). |
| Using wrong time units in interest | If rate is per annum but time is in months, convert: 6 months = 0.5 years. |
| Forgetting that discount is on Marked Price | SP = MP − Discount, not CP − Discount. The shopkeeper marks up from CP, then offers discount on MP. |
Quick Reference
- Percentage to fraction: 25% = 1/4, 33.33% = 1/3, 12.5% = 1/8, 20% = 1/5
- Profit% on CP, Discount% on MP—never mix these bases
- Ratio a:b:c means total parts = a+b+c when dividing a quantity
- SI formula: Principal × Rate × Time ÷ 100
- CI shortcut for 2 years: CI = SI + (SI × R/100) or CI − SI = P(R/100)²
- Cross-multiply to test proportion: If a/b = c/d, then ad = bc