Commercial Mathematics

🏪 The Shopkeeper’s Secret: Mastering Commercial Mathematics

Imagine you own a little candy shop. Every day, people come in to buy sweets, and you need to know if you’re making money or losing it. That’s what Commercial Mathematics is all about—the math of buying, selling, and money growing over time!

🍬 Part 1: Profit and Loss Calculations

The Story of Your Candy Shop

You buy a bag of 100 candies for ₹100 (that’s ₹1 per candy). This is called the Cost Price (CP)—what YOU paid.

You sell each candy for ₹1.50. This is the Selling Price (SP)—what the CUSTOMER pays.

Did you make money or lose it?

• You spent: ₹100
• You got back: 100 × ₹1.50 = ₹150
• Profit = SP - CP = ₹150 - ₹100 = ₹50 🎉

The Magic Formulas

PROFIT = SP - CP (when SP > CP)
LOSS = CP - SP (when CP > SP)

Profit and Loss Percentage

The shopkeeper next door also made ₹50 profit, but he spent ₹500 to make it. Who’s the better businessperson?

You need to calculate the percentage!

Profit % = (Profit / CP) × 100
Loss % = (Loss / CP) × 100

Your profit % = (50 / 100) × 100 = 50% 🚀

Neighbor’s profit % = (50 / 500) × 100 = 10%

You’re 5 times better at making profit!

Quick Example

A toy costs ₹80. You sell it for ₹100.

• Profit = ₹100 - ₹80 = ₹20
• Profit % = (20/80) × 100 = 25%

🏷️ Part 2: Discount and Marked Price

The Price Tag Trick

Walk into any shop. See that big price on the tag? That’s the Marked Price (MP)—the price the shop WANTS you to see.

But then comes the magic word: “SALE! 20% OFF!”

That reduction is the Discount.

The Real Formula

Discount = MP × (Discount% / 100)
Selling Price = MP - Discount

Example: The T-Shirt

A t-shirt has a tag showing ₹500 (Marked Price).

The shop offers 20% discount.

• Discount = 500 × (20/100) = ₹100
• You pay = 500 - 100 = ₹400

The Shopkeeper’s Secret 🤫

The shopkeeper bought that t-shirt for ₹300 (Cost Price).

Even after giving you a discount:

• SP = ₹400
• CP = ₹300
• Profit = ₹100 (that’s 33% profit!)

Lesson: The marked price is usually set HIGH so shopkeepers can give discounts and STILL make profit!

🎯 Part 3: Successive Discounts

Discount on Discount? Yes!

Sometimes shops say: “30% + 20% OFF!”

This does NOT mean 50% off! Let’s see why.

The Step-by-Step Method

Product Price: ₹1000

First discount (30%):

• Discount = 1000 × 0.30 = ₹300
• New price = 1000 - 300 = ₹700

Second discount (20%) on the NEW price:

• Discount = 700 × 0.20 = ₹140
• Final price = 700 - 140 = ₹560

The Shortcut Formula

Final Price = MP × (1 - d₁/100) × (1 - d₂/100)

For 30% + 20%:

• Final = 1000 × 0.70 × 0.80 = ₹560

The Single Equivalent Discount

What ONE discount equals 30% + 20%?

• You paid ₹560 for a ₹1000 item
• Total discount = ₹440
• Single equivalent = 44% (not 50%!)

Formula: Single Discount % = a + b - (a×b)/100 = 30 + 20 - (30×20)/100 = 50 - 6 = 44%

⚖️ Part 4: False Weights Problems

The Cheating Grocer

Some dishonest shopkeepers use weights that are lighter than they should be!

If a shopkeeper uses an 800g weight instead of 1kg (1000g):

• You pay for 1000g
• You get only 800g
• The shopkeeper keeps 200g worth for free!

The Formula

Gain % = (Error / True Weight - Error) × 100

Or simply:

Gain % = (Error / Actual Weight Given) × 100

Example

A shopkeeper uses 950g instead of 1kg.

• Error = 1000 - 950 = 50g
• Gain % = (50 / 950) × 100 = 5.26%

Double Cheating 😱

Some shopkeepers cheat TWICE:

1. Buy with heavier weights (get MORE than they pay for)
2. Sell with lighter weights (give LESS than they charge for)

Total Gain % = ((True Weight)² - (False Weight)²)
               / (False Weight)² × 100

💰 Part 5: Simple Interest (SI)

The Story of Growing Money

You put ₹1000 in a bank. The bank says: “I’ll give you 10% interest every year.”

After 1 year, you get:

• Interest = 1000 × 10/100 = ₹100
• Total = ₹1100

After 2 years:

• Interest = 1000 × 10/100 × 2 = ₹200
• Total = ₹1200

The interest stays the same every year! (That’s why it’s called “simple”)

The Golden Formula

SI = (P × R × T) / 100

Where:
P = Principal (starting money)
R = Rate (% per year)
T = Time (in years)

Amount Formula

Amount (A) = P + SI = P(1 + RT/100)

Example

You deposit ₹5000 at 8% for 3 years.

• SI = (5000 × 8 × 3) / 100 = ₹1200
• Amount = 5000 + 1200 = ₹6200

📈 Part 6: Compound Interest (CI)

Money That Grows Like a Snowball

Unlike Simple Interest, Compound Interest grows on the previous interest too!

Start with ₹1000 at 10% per year:

Year 1:

• Interest on ₹1000 = ₹100
• Total = ₹1100

Year 2:

• Interest on ₹1100 = ₹110 (not ₹100!)
• Total = ₹1210

Year 3:

• Interest on ₹1210 = ₹121
• Total = ₹1331

The Power Formula

A = P(1 + R/100)ⁿ

Where:
A = Amount after n years
P = Principal
R = Rate per period
n = Number of periods

CI Formula

CI = A - P = P[(1 + R/100)ⁿ - 1]

Example

₹10,000 at 5% compound interest for 2 years:

• A = 10000 × (1.05)² = 10000 × 1.1025 = ₹11,025
• CI = 11,025 - 10,000 = ₹1,025

Compounding Frequency

Money can compound:

• Yearly: Use R and n as years
• Half-yearly: Use R/2 and 2n
• Quarterly: Use R/4 and 4n
• Monthly: Use R/12 and 12n

⚔️ Part 7: SI vs CI Comparison

The Great Battle

For the same P, R, and T:

The 2-Year Shortcut

For 2 years, the difference between CI and SI:

CI - SI = P × (R/100)²

Example

₹10,000 at 10% for 2 years:

• SI = 10000 × 10 × 2 / 100 = ₹2000
• CI - SI = 10000 × (10/100)² = 10000 × 0.01 = ₹100
• CI = 2000 + 100 = ₹2100

The 3-Year Formula

CI - SI = P × (R/100)² × (3 + R/100)

🏠 Part 8: Installments and EMI

Buy Now, Pay Later

You want a phone worth ₹12,000 but don’t have all the money. The shop offers:

“Pay ₹1,100 every month for 12 months!”

That’s an EMI (Equated Monthly Installment).

Understanding EMI

EMI includes:

1. Part of the actual price
2. Interest on the remaining amount

The Simple Installment Formula

For n equal installments at rate R% per period:

If Cash Price = P and Installment = I:

P = I/(1+R/100) + I/(1+R/100)² + ... + I/(1+R/100)ⁿ

Simpler Case: Equal Installments (SI)

For Simple Interest:

Total Amount = Installment × Number of Installments
Interest = Total Amount - Cash Price

Example: TV on Installments

A TV costs ₹20,000 cash. Or pay ₹5,500 for 4 months.

• Total paid = 5500 × 4 = ₹22,000
• Extra paid (interest) = 22,000 - 20,000 = ₹2,000

EMI Formula (Compound Interest)

EMI = P × R × (1+R)ⁿ / [(1+R)ⁿ - 1]

Where:
P = Principal (loan amount)
R = Monthly interest rate (decimal)
n = Number of months

Quick EMI Example

Loan: ₹100,000 at 12% yearly (1% monthly) for 12 months:

EMI = 100000 × 0.01 × (1.01)¹² / [(1.01)¹² - 1]
    = 1000 × 1.1268 / 0.1268
    = ₹8,884 per month (approx)

🎯 Quick Reference Summary

graph TD
    A["Commercial Math"] --> B["Buying & Selling"]
    A --> C["Interest"]
    B --> D["Profit/Loss"]
    B --> E["Discounts"]
    B --> F["False Weights"]
    C --> G["Simple Interest"]
    C --> H["Compound Interest"]
    C --> I["Installments/EMI"]

The Essential Formulas

🌟 The Shopkeeper’s Wisdom

Remember: Commercial Mathematics is just the language of money. Whether you’re buying candies, getting a discount, putting money in a bank, or buying a phone on EMI—the same simple formulas work everywhere!

Master these, and you’ll never be tricked by a shopkeeper, always know the real cost of a loan, and understand why compound interest is called the “eighth wonder of the world”!

Now go forth and calculate with confidence! 🚀