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 : B's share of profit

= A’s investment × time period : B's investment × time period

= 12000 × 10 : 15000 × 12

= 2 : 3

Now,

$\mathrm{A}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b91,800\times \frac{2}{2+3}=\u20b9720\phantom{\rule{0ex}{0ex}}\mathrm{B}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b91,800\times \frac{3}{2+3}=\u20b91,080$

Let us go through the following examples to understand the concept better.

**Example 1:**

Suresh, Mukesh and Ramesh started a business by investing ₹1,20,000, ₹1,35,000 and ₹1,50,000 respectively. They made a profit of ₹54,000. How much profit did each of them get?

**Solution:**

Let us first find the profit-sharing ratio.

Suresh’s share of profit : Mukesh's share of profit : Ramesh's share of profit

= Suresh’s investment : Mukesh's investment : Ramesh's investment

= ₹1,20,000 : ₹1,35,000 : ₹1,50,000

= 8 : 9 : 10

$\mathrm{Suresh}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b954,000\times \frac{8}{8+9+10}=\u20b916,000\phantom{\rule{0ex}{0ex}}\mathrm{Mukesh}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b954,000\times \frac{9}{8+9+10}=\u20b918,000\phantom{\rule{0ex}{0ex}}\mathrm{Ramesh}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b954,000\times \frac{10}{8+9+10}=\u20b920,000$

**Example 2:**

John started a business investing ₹50,000. After 4 months, Peter joined him with a capital of ₹45,000. After another 6 months, Steve joined them with a capital of ₹60,000. At the end of the year, they made a profit of ₹12,100. Find the share of each.

**Solution:**

Let us find the equivalent capitals,

John’s share of profit : Peter's share of profit : Steve's share of profit

= John’s investment : Peter's investment : Steve's investment

= ₹50,000 × 12 : ₹45,000 × 8 : ₹60,000 × 6

= 5 : 3 : 3

$\mathrm{John}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b912,100\times \frac{5}{5+3+3}=\u20b95,500\phantom{\rule{0ex}{0ex}}\mathrm{Peter}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b912,100\times \frac{3}{5+3+3}=\u20b93,300\phantom{\rule{0ex}{0ex}}\mathrm{Steve}\text{'}\mathrm{s}\mathrm{share}\mathrm{of}\mathrm{profit}=\u20b912,100\times \frac{3}{5+3+3}=\u20b93,300$

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 shar…

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