Percentage and Ratio

Overview

Percentage and Ratio form the quantitative backbone of the TN TET Mathematics section. These concepts appear directly in exam questions and also underpin topics like profit-loss, simple interest, data interpretation and time-work problems. A strong grasp here means faster problem-solving across multiple question types.

For TN TET, expect questions that test your ability to convert between fractions, decimals and percentages, find percentage increase/decrease, simplify ratios, apply proportion rules and solve word problems using the unitary method. These are not just calculation questions—they require conceptual clarity about what "per hundred" means and how quantities relate to each other. Mastering this topic builds the foundation for teaching these concepts effectively to primary and upper-primary students.

Key Concepts

• **Percentage means "per hundred"**: 25% literally means 25 out of 100, written as 25/100 or 0.25. Always think of percentage as a fraction with denominator 100.

• **Ratio compares two quantities of the same kind**: A ratio 3:5 means for every 3 units of the first quantity, there are 5 units of the second. 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 rule). This is the foundation for solving proportion problems.

• **Unitary method finds the value of one unit first**: If 5 pens cost ₹40, one pen costs ₹40÷5 = ₹8. From this unit value, you can find the cost of any number of pens.

• **Percentage change = (Change ÷ Original) × 100**: Whether increase or decrease, always divide by the original (starting) value, not the new value.

• **Successive percentages don't add directly**: A 10% increase followed by 10% decrease does NOT bring you back to the original. The net effect must be calculated step by step.

• **Part-to-whole vs part-to-part**: Ratios can express parts to whole (like 2 out of 5) or parts to parts (like 2:3). Converting between these is essential.

Formulas / Key Facts

| Concept | Formula/Fact | |---------|--------------| | Percentage to fraction | x% = x/100 | | Fraction to percentage | (a/b) × 100% | | Percentage of a number | x% of N = (x/100) × N | | Percentage increase | [(New − Original) ÷ Original] × 100 | | Percentage decrease | [(Original − New) ÷ Original] × 100 | | If A is x% more than B | A = B × (1 + x/100) | | If A is x% less than B | A = B × (1 − x/100) | | Ratio a:b in fraction form | a/(a+b) and b/(a+b) of total | | Proportion rule | If a:b = c:d, then ad = bc | | Successive % change | Final = Original × (1 ± p/100) × (1 ± q/100) | | Common fraction-percent pairs | 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/3 ≈ 33.33% |

Worked Examples

**Example 1: Basic Percentage Calculation**

*A school has 800 students. If 35% are girls, how many boys are in the school?*

Step 1: Find number of girls = 35% of 800 = (35/100) × 800 = 280

Step 2: Number of boys = Total − Girls = 800 − 280 = **520 boys**

Alternative method: Boys = 65% of 800 = (65/100) × 800 = 520

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**Example 2: Percentage Increase/Decrease**

*A shopkeeper increases the price of an item from ₹250 to ₹300. Find the percentage increase.*

Step 1: Change in price = 300 − 250 = ₹50

Step 2: Percentage increase = (Change ÷ Original) × 100 = (50 ÷ 250) × 100 = **20%**

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**Example 3: Ratio and Proportion**

*The ratio of Ravi's age to Suresh's age is 3:5. If Suresh is 25 years old, find Ravi's age.*

Method 1 (Proportion): 3:5 = Ravi's age : 25

Cross-multiply: Ravi's age = (3 × 25) ÷ 5 = 75 ÷ 5 = **15 years**

Method 2 (Unitary): 5 parts = 25 years, so 1 part = 5 years. Ravi = 3 parts = 15 years.

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**Example 4: Unitary Method**

*If 12 workers can complete a task in 8 days, how many days will 16 workers take to complete the same task?*

Step 1: Total work = 12 workers × 8 days = 96 worker-days

Step 2: Days for 16 workers = 96 ÷ 16 = **6 days**

(Note: More workers means fewer days—inverse proportion)

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**Example 5: Dividing in a Given Ratio**

*Divide ₹1200 between A and B in the ratio 3:5.*

Step 1: Total parts = 3 + 5 = 8 parts

Step 2: Value of 1 part = 1200 ÷ 8 = ₹150

Step 3: A's share = 3 × 150 = ₹450; B's share = 5 × 150 = **₹750**

Common Mistakes

• **Using new value instead of original for percentage change** → Always use the ORIGINAL value as the base. "Increased from 200 to 250" means base is 200, not 250.

• **Adding successive percentages directly** → A 20% increase then 20% decrease is NOT zero change. Calculate: 100 → 120 → 96. Net effect is 4% decrease.

• **Confusing ratio parts with actual values** → In ratio 2:3, the values are NOT 2 and 3. They represent 2 parts and 3 parts of some total. Always find the value of one part first.

• **Forgetting to check if proportion is direct or inverse** → More workers = less time (inverse). More speed = less time (inverse). More items = more cost (direct). Identify the relationship before calculating.

• **Converting percentages incorrectly** → 5% is 5/100 = 0.05, NOT 0.5. A common error is misplacing the decimal point.

• **Mixing up "of" and "more than"** → "A is 20% of B" means A = 0.2B. "A is 20% more than B" means A = 1.2B. Read carefully.

Quick Reference

• x% = x/100; to find x% of N, multiply N by x/100

• Percentage change = (Difference ÷ Original) × 100

• To divide amount in ratio a:b → A gets a/(a+b) of total, B gets b/(a+b)

• Unitary method: Find value of 1 unit first, then scale up or down

• Direct proportion: both increase or both decrease together

• Inverse proportion: one increases while the other decreases

• Memorise: 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/3 ≈ 33.33%