Arithmetic
Percentage, Ratio-Proportion, Profit-Loss and Interest
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Overview
Arithmetic forms the backbone of the quantitative section in GTET. This topic—covering percentage, ratio-proportion, profit-loss and interest—appears consistently because these concepts underpin everyday numerical reasoning that primary teachers must master and teach effectively.
For TET-1 (classes 1-5) and TET-2 (classes 6-8), expect 3-5 questions directly testing these areas. The questions are typically word problems requiring quick calculation and conceptual clarity. Mastery here also helps in data interpretation and mensuration problems where percentage calculations frequently appear.
Students must build fluency in converting between fractions, decimals and percentages, and develop the habit of identifying base values correctly. Most errors stem from confusion about "of what" a percentage is taken or mixing up cost price with selling price.
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Key Concepts
- **Percentage means "per hundred"**: 25% = 25/100 = 0.25. Always convert percentages to fractions or decimals for calculation.
- **Ratio compares two quantities of the same unit**: The ratio a:b means for every 'a' units of the first, there are 'b' units of the second. Ratios have no units.
- **Proportion states two ratios are equal**: If a:b = c:d, then ad = bc (cross-multiplication rule).
- **Cost Price (CP) is what you pay; Selling Price (SP) is what you receive**: Profit or loss is always calculated on CP unless stated otherwise.
- **Simple Interest (SI) remains constant each year**: Interest is calculated only on the original principal.
- **Compound Interest (CI) grows each period**: Interest is calculated on principal plus accumulated interest.
- **Successive percentage changes cannot simply be added**: A 10% increase followed by 10% decrease does not return to the original value.
- **"Of" in percentage problems means multiplication**: 30% of 50 = (30/100) × 50.
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Formulas / Key Facts
**Percentage**
- Percentage = (Part / Whole) × 100
- Value after x% increase = Original × (1 + x/100)
- Value after x% decrease = Original × (1 − x/100)
- Net effect of successive changes of a% and b% = (a + b + ab/100)%
**Ratio and Proportion**
- If a:b = c:d, then a×d = b×c
- To divide amount M in ratio a:b, shares are Ma/(a+b) and Mb/(a+b)
- Compounded ratio of a:b and c:d = ac:bd
**Profit and Loss**
- Profit = SP − CP (when SP > CP)
- Loss = CP − SP (when CP > SP)
- Profit% = (Profit/CP) × 100
- Loss% = (Loss/CP) × 100
- SP = CP × (1 + Profit%/100) or CP × (1 − Loss%/100)
**Simple Interest**
- SI = (P × R × T) / 100, where P = Principal, R = Rate% per annum, T = Time in years
- Amount = P + SI
**Compound Interest**
- Amount = P × (1 + R/100)^T
- CI = Amount − P
- For half-yearly compounding: Rate = R/2, Time = 2T
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Worked Examples
### Example 1: Percentage Increase and Decrease **Problem**: A shopkeeper increases the price of an item by 20% and then offers a 20% discount. What is the net percentage change?
**Solution**:
- Let original price = 100
- After 20% increase = 100 × 1.20 = 120
- After 20% discount on 120 = 120 × 0.80 = 96
- Net change = 96 − 100 = −4
- Net percentage change = −4%
**Alternative using formula**: Net effect = 20 + (−20) + (20 × −20)/100 = 0 − 400/100 = −4%
**Answer**: 4% decrease
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### Example 2: Ratio and Proportion **Problem**: Divide Rs 1,800 among A, B and C in the ratio 2:3:4. Find each person's share.
**Solution**:
- Total parts = 2 + 3 + 4 = 9
- Value of 1 part = 1800/9 = 200
- A's share = 2 × 200 = Rs 400
- B's share = 3 × 200 = Rs 600
- C's share = 4 × 200 = Rs 800
**Verification**: 400 + 600 + 800 = 1800 ✓
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### Example 3: Profit and Loss **Problem**: A trader sells an article for Rs 540, making a profit of 8%. Find the cost price.
**Solution**:
- SP = CP × (1 + Profit%/100)
- 540 = CP × (1 + 8/100)
- 540 = CP × 1.08
- CP = 540/1.08 = 500
**Answer**: Rs 500
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### Example 4: Simple vs Compound Interest **Problem**: Find SI and CI on Rs 10,000 at 10% per annum for 2 years.
**Solution for SI**:
- SI = (10000 × 10 × 2)/100 = Rs 2,000
**Solution for CI**:
- Amount = 10000 × (1 + 10/100)² = 10000 × 1.21 = Rs 12,100
- CI = 12100 − 10000 = Rs 2,100
**Difference** = CI − SI = Rs 100
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Common Mistakes
- **Taking percentage of wrong base**: When price increases by 20% then decreases by 20%, students assume no change. The decrease is on the increased price, not the original. → Always identify the current base before applying percentage.
- **Calculating profit/loss percentage on SP instead of CP**: Profit% formula uses CP in denominator. → Remember: Profit/Loss percentages are always on Cost Price unless explicitly stated.
- **Adding successive percentages directly**: A 10% increase followed by 20% increase is not 30% increase. → Use the formula: (a + b + ab/100)% or calculate step by step.
- **Confusing ratio with actual values**: A ratio of 3:5 doesn't mean amounts are 3 and 5. → Ratios only show relative proportion; multiply by a common factor to get actual values.
- **Forgetting to convert time units in interest problems**: If rate is per annum but time is in months, students forget to convert. → Always express time in years when rate is annual: 6 months = 0.5 years.
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Quick Reference
- **Percentage to fraction**: x% = x/100
- **Profit% = (Profit/CP) × 100** — always on Cost Price
- **SI formula**: SI = PTR/100
- **CI formula**: A = P(1 + R/100)^T
- **Dividing in ratio a:b**: First share = Total × a/(a+b)
- **Successive % change**: Net = a + b + ab/100