Comparing Quantities — Ratios, Percentages, and Interest
"Every day we compare quantities — 'Is this smartphone a better deal than that one?' 'How much interest will my savings earn?' Ratios and percentages help us answer these questions."
1. Ratio — Comparing Two Quantities
Ratio compares two quantities of the SAME unit. Written as a:b or a/b. 'Pronounced "a is to b."'
Example: 'If there are 3 boys and 5 girls in a class, the ratio of boys to girls is 3:5. Ratio of girls to total students is 5:8.'
Rules for Ratios
- Quantities must be in the SAME unit. Convert if needed. 'Compare 1 m to 50 cm → 100 cm : 50 cm = 2:1.'
- A ratio is in SIMPLEST form when the two numbers have no common factor other than 1.
- Ratio is a PURE number — no units in the final form.
Equivalent Ratios
'Multiply or divide both terms by the SAME number. 2:3 = 4:6 = 6:9 = 8:12.'
2. Proportion — Equality of Two Ratios
Proportion: Two ratios are EQUAL. Written as a:b :: c:d or a/b = c/d.
'EXTREME terms: a and d. MIDDLE terms: b and c.'
Cross-Product Rule: a : b = c : d means a × d = b × c (product of extremes = product of means).
Worked Example: 'Check if 4:7 and 12:21 are in proportion.' 4 × 21 = 84. 7 × 12 = 84. Product are equal → 4:7 :: 12:21. YES, they are in proportion.
3. Unitary Method — Find the Value of One, Then Many
Steps: 1. Find the value of ONE item (unit). 2. Multiply to find the value of the REQUIRED number of items.
Example 1: '8 kg of AP rice costs Rs 560. Find the cost of 15 kg.' 1 kg cost = 560 ÷ 8 = Rs 70. 15 kg cost = 70 × 15 = Rs 1050.
Example 2: 'A car travels 180 km in 3 hours. How far will it travel in 5 hours?' Distance in 1 hour = 180 ÷ 3 = 60 km. Distance in 5 hours = 60 × 5 = 300 km.
4. Percentage — 'Per Hundred'
Percentage means 'out of 100.' Symbol: %. '55% means 55 out of every 100.'
Converting Fractions and Decimals to Percentages
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 3/8 | 0.375 | 37.5% |
| 1 | 1.0 | 100% |
| 7/5 | 1.4 | 140% |
Conversion Rules:
- Fraction → Percentage: Multiply by 100%. (3/4) × 100% = 75%.
- Decimal → Percentage: Multiply by 100% (shift decimal 2 places right). 0.625 × 100% = 62.5%.
- Percentage → Fraction: Divide by 100 and simplify. 40% = 40/100 = 2/5.
- Percentage → Decimal: Divide by 100 (shift decimal 2 places left). 37.5% = 0.375.
Finding Percentage of a Quantity
'30% of 200 = (30/100) × 200 = 60.'
AP Context: 'If 65% of AP's farmers use modern irrigation, find the number of farmers out of 50,000 who use it.' 65% of 50000 = (65/100) × 50000 = 32,500 farmers.
Expressing One Quantity as Percentage of Another
Formula: (Quantity A / Quantity B) × 100%.
Example: 'In a class of 40 students, 18 are girls. What percentage are girls?' (18/40) × 100% = 45%.
5. Profit and Loss
Cost Price (CP): The price at which an item is BOUGHT. Selling Price (SP): The price at which an item is SOLD.
Formulas
Profit: SP > CP → Profit = SP − CP. Loss: SP < CP → Loss = CP − SP. Profit Percentage: (Profit / CP) × 100%. Loss Percentage: (Loss / CP) × 100%.
Worked Examples
Example 1: 'An AP farmer buys seeds for Rs 2000 and sells the crop for Rs 3500. Find profit percentage.' Profit = 3500 − 2000 = Rs 1500. Profit% = (1500/2000) × 100% = 75%.
Example 2: 'A shopkeeper buys 50 kg of rice at Rs 40/kg and sells at Rs 45/kg. Find overall profit percentage.' Total CP = 50 × 40 = Rs 2000. Total SP = 50 × 45 = Rs 2250. Profit = Rs 250. Profit% = (250/2000) × 100% = 12.5%.
Example 3: 'A book is bought for Rs 250 and sold for Rs 200. Find loss percentage.' Loss = 250 − 200 = Rs 50. Loss% = (50/250) × 100% = 20%.
6. Simple Interest
Interest: Extra money paid when borrowing or earned when saving. Principal (P): The original amount borrowed/invested. Rate (R): Percentage per year. Time (T): Duration in YEARS. Amount (A): Total money after interest = P + SI.
Simple Interest Formula
SI = (P × R × T) / 100
Worked Examples
Example 1: 'Find SI on Rs 5000 at 8% per annum for 3 years.' SI = (5000 × 8 × 3) / 100 = 120000/100 = Rs 1200. Amount = 5000 + 1200 = Rs 6200.
Example 2: 'AP farmer takes a loan of Rs 50,000 at 6% per annum for 2 years for buying fertilisers. Find total interest paid.' SI = (50000 × 6 × 2) / 100 = 600000/100 = Rs 6000.
Example 3: 'Find the rate if Rs 4000 yields Rs 800 as simple interest in 4 years.' 800 = (4000 × R × 4) / 100 → 800 = 160R → R = 5% per annum.
Example 4: 'Find the time for Rs 6000 to become Rs 7200 at 5% per annum simple interest.' SI = 7200 − 6000 = Rs 1200. 1200 = (6000 × 5 × T) / 100 → 1200 = 300T → T = 4 years.
7. Common Mistakes and Fixes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| 50% = 0.5 and 50% of 100 = 50, so 0.5 = 50 | Confusing decimal with percentage | 0.5 = 50% = 0.5 (they are equal, not 50) |
| Profit% = (Profit/SP) × 100 | Profit% is ALWAYS on CP | Profit% = (Profit/CP) × 100 |
| Simple Interest = P × R × T | Missing division by 100 | SI = (P × R × T)/100 |
| Ratio 30 cm to 1 m = 30:1 | Units must match | 30 cm : 100 cm = 3:10 = 0.3 |
8. AP SSC Exam Focus
| Topic | Marks | Question Type |
|---|---|---|
| Ratio and proportion | 2-3 | Simplify, find missing term |
| Percentage | 2-3 | Convert, find value |
| Profit and loss | 3-4 | Word problems |
| Simple interest | 3-4 | Application problems |
| Unitary method | 2-3 | Word problems |
Quick Self-Test
Q1. Simplify the ratio 45:60. A1. Divide by 15 → 3:4.
Q2. Find the value of x: 8:12 :: x:18. A2. 8 × 18 = 12x → 144 = 12x → x = 12.
Q3. Convert 7/16 to percentage. A3. (7/16) × 100% = 43.75%.
Q4. If a retailer buys goods for Rs 400 and sells them for Rs 460, what is the profit percentage? A4. Profit = Rs 60. Profit% = (60/400) × 100% = 15%.
Q5. Find simple interest on Rs 8000 at 6% per annum for 5 years. A5. SI = (8000 × 6 × 5)/100 = Rs 2400.
Q6. What principal earns Rs 900 as simple interest at 5% per annum in 6 years? A6. 900 = (P × 5 × 6)/100 → 900 = 30P/100 → P = Rs 3000.
Q7. If 30% of a number is 150, find the number. A7. (30/100) × N = 150 → N = 150 × 100/30 = 500.
