Partnership
Meaning of Partnership
Partnership is a business relationship between two or more individuals who agree to share in the profits or losses of a business. Each partner contributes money, property, labor, or skill to the business and expects to share in the profits and losses of the business. The partners can be individuals, corporations, or other entities.
Types of Partnership
Partnerships can be classified into several types based on different criteria:
Based on Liability:
- General Partnership: In this type, all partners have unlimited liability, meaning they are personally responsible for the debts of the business.
- Limited Partnership: This partnership has at least one general partner with unlimited liability and at least one limited partner whose liability is limited to the amount of their investment in the business.
- Limited Liability Partnership (LLP): In an LLP, all partners have limited liability, which means their personal assets are protected from business debts.
Based on Duration:
- Partnership at Will: This partnership continues indefinitely until any partner decides to dissolve it.
- Particular Partnership: This partnership is formed for a specific purpose or a limited duration. It dissolves once the purpose is achieved or the time period expires.
Key Concepts in Partnership Problems
Partnership problems in competitive exams typically revolve around the distribution of profits and losses among partners based on their investments and the duration of their investments.
Capital Contribution:
This is the amount of money or assets each partner invests in the business.
Profit/Loss Ratio:
This is the ratio in which the partners agree to share the profits or losses of the business. If not specified, it is assumed to be equal.
Duration of Investment:
The period for which each partner's capital is invested in the business.
Calculating Profit/Loss Distribution
The profit or loss is distributed among partners in proportion to the product of their capital and the time period for which it was invested.
The ratio of profit sharing is equal to the ratio of the product of investment and time.
Profit Sharing Ratio = (Capital 1 × Time 1) : (Capital 2 × Time 2) : ...
Scenario 1: Investments are for the same duration
If all partners invest their capital for the same period, the profit-sharing ratio is simply the ratio of their investments.
Profit Sharing Ratio = Capital 1 : Capital 2 : ...
Example: A and B enter into a partnership. A invests ₹50,000 and B invests ₹80,000. If the total profit is ₹13,000, find the share of each partner.
Since the duration of investment is the same, the profit-sharing ratio is A's Investment : B's Investment = 50,000 : 80,000 = 5 : 8.
Total parts = 5 + 8 = 13.
A's share = (5/13) * 13,000 = ₹5,000.
B's share = (8/13) * 13,000 = ₹8,000.
Scenario 2: Investments are for different durations
In this case, we multiply the investment by the duration of the investment for each partner to find the profit-sharing ratio.
Example: A, B, and C invest ₹40,000, ₹60,000, and ₹80,000 respectively in a business. A invests for 12 months, B for 8 months, and C for 6 months. If the total profit is ₹46,000, find the share of each partner.
A's ratio = 40,000 * 12 = 4,80,000
B's ratio = 60,000 * 8 = 4,80,000
C's ratio = 80,000 * 6 = 4,80,000
Profit Sharing Ratio = 4,80,000 : 4,80,000 : 4,80,000 = 1 : 1 : 1.
Total parts = 1 + 1 + 1 = 3.
A's share = (1/3) * 46,000 = ₹15,333.33 (approx)
B's share = (1/3) * 46,000 = ₹15,333.33 (approx)
C's share = (1/3) * 46,000 = ₹15,333.33 (approx)
Working Partners and Sleeping Partners
In some partnerships, one or more partners might actively manage the business (working partners), while others may only contribute capital (sleeping partners). Working partners are often entitled to a salary or commission in addition to their share of profits. This needs to be accounted for before distributing the remaining profit according to the capital/time ratio.
Steps to solve problems with working partners:
- Calculate the total amount of salary/commission given to working partners.
- Subtract this amount from the total profit.
- Distribute the remaining profit among all partners based on their capital and time ratio.
- Add the share of profit to the salary/commission of the working partners to get their total earnings.
Example: P and Q enter into a partnership investing ₹50,000 and ₹70,000 respectively. P is a working partner and is entitled to 10% commission on profit, and the remaining profit is divided in the ratio of their investment. If the total profit is ₹32,000, find the share of each.
P's commission = 10% of 32,000 = (10/100) * 32,000 = ₹3,200.
Profit remaining = 32,000 - 3,200 = ₹28,800.
Investment ratio = 50,000 : 70,000 = 5 : 7.
Total parts = 5 + 7 = 12.
P's share of remaining profit = (5/12) * 28,800 = 5 * 2,400 = ₹12,000.
Q's share of remaining profit = (7/12) * 28,800 = 7 * 2,400 = ₹16,800.
P's total earning = P's commission + P's share of profit = 3,200 + 12,000 = ₹15,200.
Q's total earning = Q's share of profit = ₹16,800.
Interest on Capital/Drawings
Sometimes, interest on capital invested by partners or interest on drawings (money withdrawn by partners) might be considered.
- Interest on Capital: If interest on capital is allowed, it is usually treated as an expense and deducted from the profit before distribution.
- Interest on Drawings: If partners withdraw money during the year, they might be charged interest on these drawings. This interest is added to the profit.
The order of operations is crucial:
- Add interest on drawings to the profit.
- Subtract interest on capital and salaries/commissions to working partners.
- Distribute the remaining profit based on the agreed ratio (usually capital ratio).
Average
Meaning of Average
The average, also known as the mean, is a single value that represents the central or typical value of a set of numbers. It is calculated by summing all the numbers in the set and then dividing by the count of numbers in the set.
Formula for Average
Average = (Sum of all observations) / (Number of observations)
Calculating Average
To find the average, you need two pieces of information: the sum of all the values and the total number of values.
Example: Find the average of the numbers 10, 20, 30, 40, and 50.
Sum of observations = 10 + 20 + 30 + 40 + 50 = 150.
Number of observations = 5.
Average = 150 / 5 = 30.
Relationship between Sum, Average, and Number of Observations
The formula can be rearranged to find the sum if the average and the number of observations are known.
Sum of all observations = Average × Number of observations
This relationship is very useful in solving average problems.
Types of Average Problems
1. Average of a Series of Numbers
This is the basic calculation as shown above. For consecutive numbers, there are shortcuts.
- Consecutive Integers: The average is the middle number if the count is odd, or the average of the two middle numbers if the count is even.
- Arithmetic Progression (AP): The average is the first term plus the last term, divided by 2. Average = (First Term + Last Term) / 2.
Example: Find the average of the first 10 even numbers.
The first 10 even numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
This is an AP with First Term = 2 and Last Term = 20.
Average = (2 + 20) / 2 = 22 / 2 = 11.
Alternatively, sum = 2 * (10 * 11 / 2) = 110. Average = 110 / 10 = 11.
2. Average when a New Item is Added
When a new item is added to a group, the average changes. We can calculate the new sum and divide by the new count.
Example: The average age of 5 boys is 15 years. If a new boy joins the group, the average age becomes 16 years. What is the age of the new boy?
Initial sum of ages = 5 boys * 15 years/boy = 75 years.
New number of boys = 5 + 1 = 6.
New sum of ages = 6 boys * 16 years/boy = 96 years.
Age of the new boy = New sum - Initial sum = 96 - 75 = 21 years.
3. Average when an Item is Removed
Similar to adding an item, removing an item also changes the average.
Example: The average weight of 10 cricketers is 55 kg. If one cricketer leaves the group, the average weight of the remaining 9 cricketers becomes 54 kg. What is the weight of the cricketer who left?
Initial sum of weights = 10 cricketers * 55 kg/cricketer = 550 kg.
New number of cricketers = 9.
New sum of weights = 9 cricketers * 54 kg/cricketer = 486 kg.
Weight of the cricketer who left = Initial sum - New sum = 550 - 486 = 64 kg.
4. Average when an Item is Replaced
When one item is replaced by another, the change in the total sum is the difference between the new item and the old item.
Example: The average weight of 3 men is increased by 2 kg when one of them whose weight is 50 kg is replaced by a new man. Find the weight of the new man.
Initial average weight = Let it be 'A'.
Initial sum of weights = 3 * A.
New average weight = A + 2 kg.
New sum of weights = 3 * (A + 2) = 3A + 6 kg.
The change in sum is due to the replacement: New Sum - Old Sum = (Weight of New Man) - (Weight of Old Man).
(3A + 6) - 3A = (Weight of New Man) - 50.
6 = (Weight of New Man) - 50.
Weight of New Man = 50 + 6 = 56 kg.
5. Average of Ages
These problems often involve calculating the average age of a group at different points in time (e.g., 5 years ago, 3 years hence). Remember to adjust the ages of all members of the group for the change in time.
Example: The average age of a family of 5 members is 24 years. If the youngest member is 8 years old, find the average age of the family at the time of his birth.
Sum of ages of 5 members = 24 * 5 = 120 years.
The youngest member is 8 years old. At the time of his birth, everyone in the family was 8 years younger.
So, the total reduction in age for the entire family = 5 members * 8 years/member = 40 years.
Sum of ages at the time of his birth = 120 - 40 = 80 years.
Average age at the time of his birth = 80 / 5 = 16 years.
6. Weighted Average
A weighted average is used when different items in a set have different levels of importance or frequency. For example, when calculating the average marks of students in different subjects where each subject might have a different weightage.
Formula: Weighted Average = (Σ (Weight_i × Value_i)) / (Σ Weight_i)
Where:
- Value_i is the value of the i-th item.
- Weight_i is the weight assigned to the i-th item.
Example: A student scores 60 marks in Physics (weight 3), 70 marks in Chemistry (weight 2), and 80 marks in Maths (weight 5). Find the weighted average score.
Sum of (Weight × Value) = (3 * 60) + (2 * 70) + (5 * 80) = 180 + 140 + 400 = 720.
Sum of Weights = 3 + 2 + 5 = 10.
Weighted Average = 720 / 10 = 72.
Time and Work
Basic Concepts
Time and Work problems deal with the relationship between the amount of work done, the rate at which work is done, and the time taken to complete the work. The fundamental principle is that the amount of work done is directly proportional to the rate of work and the time spent working.
Work = Rate × Time
From this, we can derive:
- Rate = Work / Time
- Time = Work / Rate
In most problems, the "Work" is considered as a single unit (i.e., completing the entire task). So, the formulas simplify to:
- 1 Unit of Work = Rate × Time
- Rate = 1 / Time (This means if someone takes 'T' days to complete a work, their rate of work is 1/T of the work per day).
- Time = 1 / Rate
Calculating Individual Rates
If a person 'A' can complete a work in 'x' days, then A's rate of work is 1/x of the work per day.
If a person 'B' can complete the same work in 'y' days, then B's rate of work is 1/y of the work per day.
Calculating Combined Rates
When multiple people work together, their rates of work add up.
If A takes 'x' days and B takes 'y' days to complete a work individually, then working together, their combined rate is (1/x + 1/y) of the work per day.
Combined Rate = Rate of A + Rate of B = (1/x) + (1/y)
To find the time taken by them to complete the work together, we take the reciprocal of the combined rate.
Time Taken Together = 1 / Combined Rate = 1 / ((1/x) + (1/y))
This can be simplified using LCM or cross-multiplication:
Time Taken Together = (x * y) / (x + y)
Example: A can complete a work in 10 days and B can complete the same work in 15 days. In how many days will they complete the work together?
A's rate = 1/10 work/day.
B's rate = 1/15 work/day.
Combined rate = (1/10) + (1/15).
LCM of 10 and 15 is 30.
Combined rate = (3/30) + (2/30) = 5/30 = 1/6 work/day.
Time taken together = 1 / (1/6) = 6 days.
Using the shortcut: Time = (10 * 15) / (10 + 15) = 150 / 25 = 6 days.
Working with Three or More People
The same principle applies. If A takes 'x' days, B takes 'y' days, and C takes 'z' days, their combined rate is (1/x + 1/y + 1/z).
Time Taken Together = 1 / ((1/x) + (1/y) + (1/z))
Example: A, B, and C can complete a work in 12, 15, and 20 days respectively. How many days will they take to complete the work together?
A's rate = 1/12.
B's rate = 1/15.
C's rate = 1/20.
Combined rate = (1/12) + (1/15) + (1/20).
LCM of 12, 15, and 20 is 60.
Combined rate = (5/60) + (4/60) + (3/60) = 12/60 = 1/5 work/day.
Time taken together = 1 / (1/5) = 5 days.
Working in Alternating Days
In these problems, individuals work on alternate days. We calculate the work done in one full cycle (e.g., 2 days if two people alternate).
Example: A can do a work in 10 days and B can do it in 20 days. They start working together, but B works only on every third day. In how many days will the work be completed?
A's rate = 1/10.
B's rate = 1/20.
Consider a cycle of 3 days:
- Day 1: A works (1/10 work done)
- Day 2: A works (1/10 work done)
- Day 3: B works (1/20 work done)
Work done in 3 days = (1/10) + (1/10) + (1/20) = (2/20) + (2/20) + (1/20) = 5/20 = 1/4 of the work.
Since 1/4 work is done in 3 days, the full work will be done in 3 * 4 = 12 days.
Let's verify: Work done in 9 days (3 cycles of 3 days) = 3 * (1/4) = 3/4 work. Day 10: A works (1/10). Total work = 3/4 + 1/10 = 15/20 + 2/20 = 17/20. Day 11: A works (1/10). Total work = 17/20 + 1/10 = 17/20 + 2/20 = 19/20. Day 12: B works (1/20). Total work = 19/20 + 1/20 = 20/20 = 1 (Full work). So, it takes 12 days.
Work Done by Men, Women, and Children
These problems often involve different efficiencies for men, women, and children. We first establish the relationship between their working rates.
Example: 3 men can do a work in 10 days. 5 women can do the same work in 12 days. 8 children can do the same work in 15 days. How many days will 4 men, 7 women, and 10 children take to complete the work?
From the given information:
- 3 Men * 10 days = 30 Man-days (to complete the work)
- 5 Women * 12 days = 60 Woman-days (to complete the work)
- 8 Children * 15 days = 120 Child-days (to complete the work)
This means the work requires 30 Man-days, 60 Woman-days, or 120 Child-days.
We can establish equivalence: 30 M = 60 W => 1 M = 2 W 60 W = 120 C => 1 W = 2 C Therefore, 1 M = 2 W = 4 C.
Now, convert the requirement (4 men, 7 women, 10 children) into a single unit, say 'Men':
- 4 Men = 4 M
- 7 Women = 7 * (1/2) M = 3.5 M
- 10 Children = 10 * (1/4) M = 2.5 M
Total equivalent men = 4 + 3.5 + 2.5 = 10 Men.
We know that 30 Man-days are required for the work.
So, if we have 10 Men, the time taken will be:
Time = Total Man-days / Number of Men = 30 / 10 = 3 days.
Work Done by Pipes and Cisterns
This is a variation of Time and Work problems, where pipes fill a cistern and others empty it.
- A pipe that fills a cistern is considered positive work (+ve rate).
- A pipe that empties a cistern is considered negative work (-ve rate).
The logic is the same: calculate the rates and combine them.
Example: Pipe A can fill a cistern in 10 hours. Pipe B can fill it in 15 hours. Pipe C can empty it in 12 hours. If all three pipes are opened simultaneously, in how many hours will the cistern be filled?
A's rate (filling) = +1/10 cistern/hour.
B's rate (filling) = +1/15 cistern/hour.
C's rate (emptying) = -1/12 cistern/hour.
Combined rate = (1/10) + (1/15) - (1/12).
LCM of 10, 15, and 12 is 60.
Combined rate = (6/60) + (4/60) - (5/60) = (6 + 4 - 5) / 60 = 5/60 = 1/12 cistern/hour.
Time taken to fill = 1 / (1/12) = 12 hours.
Efficiency-Based Problems
Sometimes, problems state efficiency relative to others.
Example: A is twice as efficient as B. B is thrice as efficient as C. If A can complete a work in 30 days, how many days will B and C take to complete the same work together?
Let C's efficiency = 1 unit.
B's efficiency = 3 * C's efficiency = 3 units.
A's efficiency = 2 * B's efficiency = 2 * 3 = 6 units.
A's rate = 1/30 work/day.
Since A's efficiency is 6 units, and A completes work in 30 days, the total work units = 6 units * 30 days = 180 unit-days.
B's rate = 3 units. Time for B alone = Total Work Units / B's Efficiency = 180 / 3 = 60 days.
C's rate = 1 unit. Time for C alone = Total Work Units / C's Efficiency = 180 / 1 = 180 days.
Now, find the time for B and C together:
B's rate = 1/60.
C's rate = 1/180.
Combined rate = (1/60) + (1/180) = (3/180) + (1/180) = 4/180 = 1/45 work/day.
Time taken together = 45 days.