SATHEE: Partnership - All Formulas & Shortcuts

Partnership - All Formulas & Shortcuts

Quick reference guide for all Partnership formulas with instant profit/loss sharing techniques

📘 Basic Definitions

1. Types of Partnership

Simple Partnership:

All partners invest for the SAME time period Profit ratio = Investment ratio

Compound Partnership:

Partners invest for DIFFERENT time periods Profit ratio = Capital × Time ratio

Active Partner:

Manages the business
May receive salary/commission before profit distribution

Sleeping/Dormant Partner:

Only invests capital
Does not manage business

2. Fundamental Formula

For Simple Partnership:

Profit Ratio = Investment Ratio A:B = Investment_A : Investment_B

For Compound Partnership:

Profit Ratio = (Capital_A × Time_A) : (Capital_B × Time_B)

Distribution:

A’s share = Total Profit × A’s ratio / Sum of ratios

⚡ Simple Partnership

3. Equal Time, Different Investments

If A invests ₹a, B invests ₹b for same time t:

Profit Ratio:

A:B = a:b

Individual Shares:

A’s share = Total × a/(a+b) B’s share = Total × b/(a+b)

Example: A invests ₹30,000, B invests ₹50,000 for 1 year. Profit = ₹16,000

Ratio = 30000:50000 = 3:5 A’s share = 16000 × 3/8 = ₹6,000 B’s share = 16000 × 5/8 = ₹10,000

4. Three or More Partners (Equal Time)

A invests ₹a, B invests ₹b, C invests ₹c for time t:

Profit Ratio:

A:B:C = a:b:c

Shares:

A’s share = Total × a/(a+b+c) B’s share = Total × b/(a+b+c) C’s share = Total × c/(a+b+c)

🔥 Compound Partnership

5. Different Capital, Different Time

A invests ₹a for t₁ months, B invests ₹b for t₂ months:

Profit Ratio:

A:B = (a × t₁):(b × t₂)

Example: A invests ₹40,000 for 6 months, B invests ₹30,000 for 8 months

A:B = (40000 × 6):(30000 × 8) = 240000:240000 = 1:1 (Equal profit share!)

6. Variable Capital Method

When capital changes during the year:

Steps:

Calculate capital × time for each period
Sum all (capital × time) for each partner
Find ratio of totals

Example: A invests ₹20,000 for 3 months, then adds ₹10,000 for 9 more months

A’s total = (20000 × 3) + (30000 × 9) = 60000 + 270000 = 330000

If B invests ₹40000 for 12 months: B’s total = 40000 × 12 = 480000

Ratio A:B = 330000:480000 = 11:16

💡 Advanced Formulas

7. Working vs Sleeping Partners

A is working partner, B is sleeping partner:

If A gets salary s before profit distribution:

Remaining Profit = Total Profit - Salary Divide remaining profit in ratio

A’s total = Salary + A’s share from remaining B’s total = B’s share from remaining

Example: Total profit ₹50,000. A (working) gets ₹5,000 salary. Invest ratio A:B = 2:3

Remaining = 50000 - 5000 = 45000 A’s share from 45000 = 45000 × 2/5 = 18000 B’s share from 45000 = 45000 × 3/5 = 27000

A’s total = 5000 + 18000 = ₹23,000 B’s total = ₹27,000

8. Partnership with Different Profit Terms

A and B agree to share profit in ratio x:y, but investments are in ratio a:b:

This is a contract-based partnership

Profit is distributed as per agreement (x:y) Not based on investment ratio (a:b)

Common in:

Joint ventures
Special partnership agreements
Skill-based partnerships

If profit ratio is known and one investment is given:

Given: A invests ₹a, time is same for all, profit ratio A:B = x:y

Find B’s investment:

Investment_B = Investment_A × (y/x)

Example: A invests ₹60,000. Profit ratio A:B = 3:4. Find B’s investment.

B’s investment = 60000 × 4/3 = ₹80,000

🎯 Special Cases

10. One Partner Joins Later

A starts business, B joins after t months:

If A invests ₹a for 12 months, B invests ₹b for (12-t) months:

A:B = (a × 12) : [b × (12-t)]

Example: A invests ₹50,000 for 12 months. B joins after 4 months with ₹60,000.

A:B = (50000 × 12) : (60000 × 8) = 600000 : 480000 = 5:4

11. Partner Leaves Early

A invests for full year, B leaves after t months:

A invests ₹a for 12 months, B invests ₹b for t months:

A:B = (a × 12):(b × t)

Example: A: ₹40,000 for 12 months. B: ₹80,000 but leaves after 6 months.

A:B = (40000 × 12):(80000 × 6) = 480000:480000 = 1:1

12. Partnership with Loss

Same formulas apply for loss distribution!

Loss Ratio = Investment × Time Ratio (Same as profit ratio)

Example: A:B invest ratio 3:2, Loss = ₹25,000

A’s loss = 25000 × 3/5 = ₹15,000 B’s loss = 25000 × 2/5 = ₹10,000

📊 Quick Shortcuts

Shortcut 1: Same Capital, Different Time

If investments are equal but time differs:

Profit Ratio = Time Ratio

Example: Both invest ₹50,000. A for 8 months, B for 6 months.

A:B = 8:6 = 4:3

Shortcut 2: Investments in Arithmetic Progression

If investments are: a, a+d, a+2d, …

Sum = n/2 × [2a + (n-1)d]

Example: 3 partners invest ₹10,000, ₹15,000, ₹20,000 (AP with d=5000)

Ratio = 10:15:20 = 2:3:4

Shortcut 3: Finding Total Capital

If profit ratio and one partner’s profit are known:

Given: A’s profit = P, ratio A:B:C = x:y:z

Total Profit = P × (x+y+z)/x B’s Profit = P × y/x C’s Profit = P × z/x

Example: A’s profit = ₹12,000, ratio 3:4:5

Total = 12000 × 12/3 = ₹48,000 B’s profit = 12000 × 4/3 = ₹16,000 C’s profit = 12000 × 5/3 = ₹20,000

Shortcut 4: Replacement of Partner

C replaces B after t months in A and B partnership:

Original: A and B for t months at ratio a:b New: A continues, C replaces B for (12-t) months

A’s ratio = a × 12 B’s ratio = b × t C’s ratio = c × (12-t)

Final ratio = (a × 12) : (b × t) : [c × (12-t)]

🔥 Exam Patterns

Pattern 1: Partnership with Removal

Q: A starts with ₹30,000. After 4 months, B joins with ₹40,000. After 8 months, A withdraws ₹10,000. Find profit ratio at year end.

Solution:

A: (30000 × 8) + (20000 × 4) = 240000 + 80000 = 320000 B: 40000 × 8 = 320000

A:B = 320000:320000 = 1:1

Pattern 2: Active Partner Commission

Q: A and B invest ₹20,000 and ₹30,000. A manages and gets 10% of profit as commission. Total profit ₹25,000. Find shares.

Solution:

A’s commission = 25000 × 10/100 = ₹2,500 Remaining = 25000 - 2500 = ₹22,500

Ratio = 20000:30000 = 2:3 A’s share from 22500 = 22500 × 2/5 = ₹9,000 B’s share from 22500 = 22500 × 3/5 = ₹13,500

A’s total = 2500 + 9000 = ₹11,500 B’s total = ₹13,500

Pattern 3: Finding Investment Period

Q: A invests ₹50,000 for 12 months. B invests ₹75,000 for x months. Profit ratio 2:3. Find x.

Solution:

(50000 × 12) : (75000 × x) = 2:3 600000 : 75000x = 2:3

Cross multiply: 600000 × 3 = 75000x × 2 1800000 = 150000x x = 12 months

💎 Mental Math Tricks

Trick 1: Simplifying Ratios

When given large numbers:

₹60,000 : ₹80,000 : ₹1,00,000 = 60:80:100 = 6:8:10 = 3:4:5

Always divide by HCF/GCD

Trick 2: Quick Profit Calculation

For 2 partners with ratio a:b and total profit P:

First partner = P × a/(a+b) Second partner = P × b/(a+b)

Mental shortcut for common ratios:

1:1 → 50%-50%
2:3 → 40%-60%
3:4 → 43%-57% (approx)
1:2 → 33%-67%

Trick 3: Time Period Conversion

Remember monthly to yearly:

3 months = 1/4 year 4 months = 1/3 year 6 months = 1/2 year 8 months = 2/3 year 9 months = 3/4 year

🎓 Golden Rules

Investment × Time = Profit Ratio (for compound partnership)
Working partner’s salary is deducted BEFORE profit distribution
Loss is distributed same as profit (in investment ratio)
Time must be in same units (all months or all years)
Partnership profit ≠ Individual profit from business operations

🔍 Common Mistakes

❌ Forgetting to multiply capital by time in compound partnership ❌ Not deducting salary before distributing profit ❌ Mixing months and years in time calculation ❌ Adding ratios instead of using proportion ✅ Always check if partnership is simple or compound ✅ Calculate capital × time for each partner separately ✅ Simplify ratio by dividing by HCF ✅ Verify: Sum of all shares = Total profit

📝 Summary Table

Scenario Formula

Simple Partnership Profit Ratio = Investment Ratio

Compound Partnership Profit Ratio = (Capital × Time) Ratio

A’s Share Total × A’s ratio / Sum of ratios

With Salary Deduct salary first, then distribute

Loss Sharing Same as profit ratio

Partner Joins Later Multiply by months invested

Equal Investment Ratio = Time ratio

Equal Time Ratio = Investment ratio

Scenario	Formula
Simple Partnership	Profit Ratio = Investment Ratio
Compound Partnership	Profit Ratio = (Capital × Time) Ratio
A’s Share	Total × A’s ratio / Sum of ratios
With Salary	Deduct salary first, then distribute
Loss Sharing	Same as profit ratio
Partner Joins Later	Multiply by months invested
Equal Investment	Ratio = Time ratio
Equal Time	Ratio = Investment ratio