Numerical Applications

Alligations and Mixtures

When two or more persons start a business enterprise jointly, they are called partners and the deal is known as partnership. The partners start a business by investing money. They share the profit earned or the loss incurred in the business in proportion to the money invested by each partner.

**Distribution of Profit:**

i) When investments are for the same time:

The profit or loss is distributed among the partners in the ratio of their investments. Suppose A and B invest ₹*x* and ₹*y* respectively for one year in a business, then at the end of the year:

A’s share of profit : B's share of profit = *x* : *y*

ii) When investments are for different time periods:

In this case, equivalent capitals are calculated for a unit of time by taking the product of the capital and the number of units of time. Now, profit or loss is divided in the ratio of these equivalent capitals. Suppose A invests ₹*x* for *p* months and B invests ₹*y* for *q* months, then

A’s share of profit : B's share of profit = *xp* : *yq*

Let us consider an example to understand this concept.

Two persons A and B invested ₹12,000 and ₹15,000 respectively and started a business. We need to find the share of each, out of an annual profit of ₹1,800.

First, we find the ratio of profit.

A’s share of profit: B's share of profit

= A’s investment: B's investment

= ₹12,000 : ₹15,000

= 4 : 5

Now, we can easily find the share of each A and B.

$\mathrm{A}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b91,800\times \frac{4}{4+5}=\u20b9800\phantom{\rule{0ex}{0ex}}\mathrm{B}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b91,800\times \frac{5}{4+5}=\u20b91,000$

Now, if A invested ₹12,000 for 10 months and B invested ₹15,000 for one year, then let us find the share of each, out of an annual profit of ₹1,800.

In that case, A and B share the profit in the ratio of their equivalent capitals.

A’s share of profit …

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