Partnership: Special Cases

The working partner is paid a salary or commission out of the profit first; the rest is shared by all in the ratio of capital × time.

The idea

A sleeping partner only puts in money. A working partner puts in money and also does the work of running the business, so he is paid for his effort separately — either a percentage of the profit (commission or management fee) or a fixed salary per month. This sum comes out of the profit first. What is left is shared among all (the working partner included) in the ratio of capital × time.

A trick to keep numbers whole: take the total profit as a convenient number of “units” for which the percent and all shares come out whole (for example 40 units for 10% and 12 parts). Do all the work in units and at the end multiply by the rupees in one unit.

Steps

1. Take the working partner’s fee or salary out of the total profit (for a percent, use convenient units).
2. Balance = total − fee.
3. Share the balance among all the partners by capital × time.
4. The working partner’s total = fee + his share.

Part of the profit split equally, or some money set aside first (reserve fund or charity).

The idea

The partners may also agree that “40% of the profit is split equally, the other 60% by capital.” Then each partner gets two pieces: an equal piece (the same for everyone) and a ratio piece (by capital × time). Another form: some percent of the profit first goes to a reserve fund or charity and only the rest is shared among the partners.

When the difference between two partners is asked, the equal piece is the same for both and cancels in the difference; the difference comes only from the ratio piece.

Steps

1. Take the total profit as convenient units.
2. Find the units of the part split equally (or set aside); divide the equal part by the number of partners.
3. Share the remaining units by capital × time.
4. Add each partner’s units and, as the question needs, find the value of 1 unit.

When a new partner comes in, the old partners give up part of their shares; the ratio of that sacrifice and its compensation.

The idea

Two partners A and B used to share the profit $3:2$ — A $\frac{3}{5}$ and B $\frac{2}{5}$. Now C comes in and wants a share too. It has to come from somewhere: A and B each give up part of their own share. What each gave up is his sacrifice: sacrifice $=$ old share $-$ new share. The new partner usually pays the old partners some money (goodwill or premium) for this sacrifice, and that money is divided in the ratio of the sacrifice.

Two main forms: (1) the new partner’s share is given and the rest goes in the old ratio — then the ratio of sacrifice is simply the old ratio (both give up the same fraction of their shares). (2) the new ratio is given directly — then subtract new from old shares to get the sacrifices. If an old partner’s share has increased, his “sacrifice” is negative, i.e. he has gained.

Steps

1. Write the old shares as fractions (in the ratio of capitals).
2. Write the new shares as fractions (from the ratio given, or from the fractions given).
3. Each old partner’s sacrifice = old − new; then the ratio of sacrifices.
4. Divide the compensation in the ratio of the sacrifices.

The more capital-months, the more of the loss to bear; subtracting the loss from the capital gives the money coming back.

The idea

If at the end there is a loss instead of a profit, the rule of sharing stays the same: the partner whose money was bigger or stayed longer bears more of the loss. Ratio of loss = the profit ratio = capital × time. Just read “loss” where you read “profit”.

Often the next question is how much of his capital a partner gets back after the loss: money coming back = capital put in − his share of the loss. In some questions the loss share could exceed the capital — then read what is asked carefully.

Steps

1. Find each partner’s capital × time and make the simple ratio.
2. Divide the total loss into parts to find 1 part.
3. Each partner’s loss = his parts × 1 part.
4. If the money back is asked, capital − loss.

When each year’s profit (or loss) is given separately and the capitals also change, share each year on its own.

The idea

If one total profit for the whole time is given, we may add up all the capital × time together and form one ratio. But when each year’s profit is given separately (₹36,000 in year 1, a loss of ₹12,000 in year 2 …), the capitals and the profits differ from year to year, so they cannot be merged into one ratio. Share each year’s profit in the ratio of that year’s capitals, then add the partner’s yearly shares (subtract the loss).

Steps

1. Make the simple ratio of each year’s capitals.
2. From that year’s profit or loss find the partner’s share.
3. Add the yearly shares; subtract the share of a loss year.

The deed says: pay interest on capital first, then share the balance equally or in some ratio.

The idea

In some partnerships, before the profit is shared every partner is first given simple interest on his capital at a fixed rate, and then the balance of the profit is shared equally or in some ratio. A partner gets in all = interest + share of the balance.

After the interest is paid, the balance of the profit = total profit − the total interest of all partners. This balance is shared by the rule given.

Steps

1. Find each partner’s interest ($I=\frac{CrT}{100}$).
2. Total profit − total interest = the balance.
3. Share the balance by the given rule (equally or in a ratio).
4. The partner asked gets in all = his interest + his share of the balance.

One page to pick the method as soon as you read a question.

Here: $C$ = capital, $r$ = rate % per year, $t$ = years, $P$ = total profit, $I$ = interest; “sacrifice” = old fractional share − new fractional share. The capital × time rule and the methods for joining and leaving are in the first chapter (Partnership) and apply here too.

All kinds mixed together; spot the method first, then solve.

These questions are not in method order — just like the exam. For each, first decide which condition applies.

Mixed questions from the hard end of the exam.

Long exam questions: a special condition together with a partner joining late, leaving early or changing capital.

Each question here combines two methods. First find each partner’s capital × time (the method of the first chapter), then apply the condition (salary, equal split, loss, interest, new partner).

Read the front, think, then flip.