Ratio, Proportion, Percentages, Profit-Loss
Overview
This topic forms the backbone of commercial mathematics at the primary level and appears consistently in MAHA TET Paper I and Paper II. Questions test your ability to apply these concepts to everyday situations—sharing sweets among children, calculating discounts at shops, or finding profit earned by a vendor. The examiner values conceptual clarity over rote memorisation.
For TET aspirants, mastery here serves two purposes: answering content questions correctly and understanding how to teach these interconnected concepts to young learners. Speed and accuracy come from understanding the underlying relationships rather than memorising isolated formulas.
The concepts build upon each other—ratio leads to proportion, proportion connects to percentages, and percentages underpin profit-loss calculations. Teach them in this sequence for both exam success and classroom effectiveness.
Key Concepts
- Ratio compares two quantities of the same kind using division. The ratio of a to b is written as a : b or a/b. Order matters: 3 : 5 is different from 5 : 3.
- Equivalent ratios are obtained by multiplying or dividing both terms by the same non-zero number. Example: 2 : 3 = 4 : 6 = 6 : 9.
- Proportion states that two ratios are equal. If a : b = c : d, then a, b, c, d are in proportion. The product of extremes equals the product of means: a × d = b × c.
- Percentage means "per hundred." To convert a fraction to percentage, multiply by 100. To convert percentage to fraction, divide by 100.
- Cost Price (CP) is what a seller pays to acquire goods. Selling Price (SP) is what the buyer pays to the seller.
- Profit occurs when SP > CP. Loss occurs when SP < CP. Both are always calculated on Cost Price.
- Discount is the reduction from Marked Price (MP). Selling Price = Marked Price − Discount.
- Unitary method finds the value of one unit first, then scales to the required quantity—essential for proportion problems.
Formulas / Key Facts
| Concept | Formula |
|---|---|
| Ratio simplification | Divide both terms by their HCF |
| Proportion test | a : b :: c : d ⟹ a × d = b × c |
| Fraction to percentage | (Fraction) × 100% |
| Percentage to fraction | (Percentage) ÷ 100 |
| Finding X% of Y | (X/100) × Y |
| Profit | SP − CP (when SP > CP) |
| Loss | CP − SP (when CP > SP) |
| Profit % | (Profit/CP) × 100 |
| Loss % | (Loss/CP) × 100 |
| SP when profit % given | SP = CP × (100 + Profit%)/100 |
| SP when loss % given | SP = CP × (100 − Loss%)/100 |
| Discount | MP − SP |
| Discount % | (Discount/MP) × 100 |
Key fact: Profit/Loss percentage is always calculated on Cost Price. Discount percentage is always calculated on Marked Price.
Worked Examples
Example 1: Ratio and Proportion
Problem: Divide ₹630 between Asha and Bina in the ratio 4 : 5.
Solution:
- Total parts = 4 + 5 = 9
- Value of one part = 630 ÷ 9 = ₹70
- Asha's share = 4 × 70 = ₹280
- Bina's share = 5 × 70 = ₹350
Answer: Asha gets ₹280, Bina gets ₹350.
Example 2: Percentage Application
Problem: In a class of 40 students, 15% are absent. How many students are present?
Solution:
- Number absent = 15% of 40 = (15/100) × 40 = 6
- Number present = 40 − 6 = 34
Answer: 34 students are present.
Example 3: Profit and Loss
Problem: A shopkeeper buys a toy for ₹240 and sells it for ₹270. Find the profit percentage.
Solution:
- CP = ₹240, SP = ₹270
- Profit = SP − CP = 270 − 240 = ₹30
- Profit % = (Profit/CP) × 100 = (30/240) × 100 = 12.5%
Answer: Profit percentage is 12.5%.
Example 4: Discount Calculation
Problem: A shirt has a marked price of ₹500. A discount of 20% is offered. Find the selling price.
Solution:
- Discount = 20% of 500 = (20/100) × 500 = ₹100
- SP = MP − Discount = 500 − 100 = ₹400
Answer: Selling price is ₹400.
Example 5: Combined Application
Problem: If 3 : 4 :: x : 20, find x.
Solution:
- Using proportion rule: 3 × 20 = 4 × x
- 60 = 4x
- x = 60 ÷ 4 = 15
Answer: x = 15.
Common Mistakes
| Wrong Thinking | Correct Approach |
|---|---|
| Calculating profit % on Selling Price | Profit % is always calculated on Cost Price |
| Writing ratio 5 : 3 when the question asks for smaller to larger | Read carefully—ratio order must match the question's order |
| Forgetting to simplify ratios to lowest terms | Always divide both terms by HCF: 12 : 18 = 2 : 3 |
| Confusing discount % calculation base | Discount % is on Marked Price, not on Cost Price or Selling Price |
| Adding percentages directly without finding actual values | 20% of 50 plus 30% of 50 ≠ 50% of 50; calculate each separately first |
| Mixing up proportion extremes and means | In a : b :: c : d, extremes are a and d, means are b and c |
Quick Reference
- Ratio = comparison by division; same units required; simplify using HCF.
- Proportion rule: Product of extremes = Product of means.
- Percentage = (Part/Whole) × 100.
- Profit/Loss % always on CP; Discount % always on MP.
- SP = CP + Profit = CP − Loss.
- Unitary method: Find value of 1 unit → multiply for required quantity.