Profit, Loss and Simple Interest

Overview

Profit, Loss and Simple Interest form the backbone of commercial arithmetic in the UPTET Mathematics paper. These concepts test your ability to apply percentage-based calculations to real-world transactions—buying and selling goods, offering discounts, and lending or borrowing money. For a primary-school teacher, mastery here is essential not only to clear the exam but also to make mathematics relatable to children through everyday examples like shopkeeping and savings.

Questions typically involve direct formula application, but examiners frequently add twists: successive discounts, marked price versus cost price, or finding the principal when interest is given. Expect 2–4 questions from this cluster in Paper I and Paper II. Speed matters, so commit the formulas to memory and practise mental shortcuts for common percentages (10%, 25%, 50%).

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Key Concepts

• **Cost Price (CP):** The price at which an article is purchased. All profit/loss calculations reference this baseline.
• **Selling Price (SP):** The price at which an article is sold. Comparison of SP with CP determines profit or loss.
• **Profit and Loss are always calculated on CP**, not on SP—this is a classic confusion point.
• **Marked Price (MP):** The price written on the tag before any discount. Discount is always computed on MP, not on CP.
• **Discount:** A reduction offered on MP. After discount, the customer pays SP.
• **Simple Interest (SI):** Interest calculated only on the original principal for the entire period—no compounding.
• **Compound Interest (CI):** Interest calculated on principal plus accumulated interest from previous periods. At the primary level, only basic two-year problems appear.
• **Rate and Time must use consistent units:** If rate is annual, time must be in years (convert months to fractions).

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Formulas / Key Facts

| Concept | Formula | |---------|---------| | Profit | Profit = SP − CP | | Loss | Loss = CP − SP | | Profit % | Profit % = (Profit / CP) × 100 | | Loss % | Loss % = (Loss / CP) × 100 | | SP when Profit % given | SP = CP × (100 + Profit%) / 100 | | SP when Loss % given | SP = CP × (100 − Loss%) / 100 | | Discount | Discount = MP − SP | | Discount % | Discount % = (Discount / MP) × 100 | | SP after Discount | SP = MP × (100 − Discount%) / 100 | | Simple Interest | SI = (P × R × T) / 100 | | Amount (SI) | A = P + SI = P (1 + RT/100) | | Compound Interest (annual, 2 yrs) | A = P (1 + R/100)² ; CI = A − P | | CI shortcut for 2 years | CI = SI for 2 yrs + (SI for 1 yr × R / 100) |

**Memory tip:** Profit/Loss → base is CP. Discount → base is MP. Interest → base is Principal.

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Worked Examples

### Example 1 – Basic Profit Calculation *A shopkeeper buys a toy for ₹400 and sells it for ₹460. Find the profit percentage.*

**Solution:** 1. Profit = SP − CP = 460 − 400 = ₹60 2. Profit % = (60 / 400) × 100 = 15 %

**Answer:** 15 %

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### Example 2 – Discount and Selling Price *A shirt has a marked price of ₹800. A discount of 20 % is offered. What is the selling price?*

**Solution:** 1. Discount = 20 % of 800 = (20/100) × 800 = ₹160 2. SP = MP − Discount = 800 − 160 = ₹640

**Answer:** ₹640

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### Example 3 – Simple Interest *Find the simple interest on ₹5,000 at 8 % per annum for 3 years.*

**Solution:** SI = (P × R × T) / 100 = (5000 × 8 × 3) / 100 = 1,20,000 / 100 = ₹1,200

**Answer:** ₹1,200

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### Example 4 – Compound Interest for 2 Years *Find CI on ₹10,000 at 10 % per annum for 2 years, compounded annually.*

**Solution:** 1. A = P (1 + R/100)² = 10,000 × (1.10)² = 10,000 × 1.21 = ₹12,100 2. CI = A − P = 12,100 − 10,000 = ₹2,100

**Alternative shortcut:**

• SI for 2 years = (10,000 × 10 × 2)/100 = ₹2,000
• Extra interest (interest on 1st year's interest) = (1,000 × 10)/100 = ₹100
• CI = 2,000 + 100 = ₹2,100 ✓

**Answer:** ₹2,100

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Calculating profit % on SP instead of CP. | Always use CP as the base for profit/loss percentages. | | Applying discount on CP instead of MP. | Discount is given on the **marked price**; use MP as base. | | Mixing up time units—using months directly with annual rate. | Convert months to years: e.g., 6 months = 0.5 year. | | Using the CI formula when the question says "simple interest." | Read carefully—SI formula has no power; CI formula has exponent. | | Forgetting successive discount rule: 20 % + 10 % ≠ 30 %. | Apply discounts one after another: first 20 % on MP, then 10 % on reduced price. Effective discount < sum of individual discounts. |

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Quick Reference

1. **Profit % = (Profit / CP) × 100** — base is always cost price. 2. **Discount % = (Discount / MP) × 100** — base is always marked price. 3. **SI = PRT / 100** — no compounding; interest stays flat each year. 4. **For CI (2 yrs):** CI = SI(2 yrs) + one year's SI × R/100. 5. **Successive discounts a % and b %:** Effective discount = a + b − (ab/100). 6. **Amount = Principal + Interest** (works for both SI and CI).