**Read the following and answer the questions that follow**

A study was conducted to understand the presence of 400 small-sized companies, through branch offices, in 3 states - AP, MP and UP. If a company is said to be present in a state, then it has at least one branch office in that state. The findings are as follows

(i) The number of companies present in all three states is one-eighth of the number of companies in AP and MP but not UP.
(ii) The number of companies in only UP, only MP and only AP is in the ratio of 4 : 5 : 6.
(iii) Amongst the companies in UP, it was noted that the number of companies not in MP is thrice that of the number of companies in MP.
(iv) 90 companies did not have branch offices in any of these three states.
(v) The number of companies present in both AP and UP is 30 less than the number of companies in AP and MP

Question No. 4

If the number of companies only in MP is a multiple of 5, then what is the number of companies that are in all 3 states?

10




17




21




Cannot be determined

To solve this problem, let's define the following variables for the number of companies in various states:

- \( A \): Number of companies in AP
- \( M \): Number of companies in MP
- \( U \): Number of companies in UP
- \( A \cap M \): Number of companies in both AP and MP
- \( A \cap U \): Number of companies in both AP and UP
- \( M \cap U \): Number of companies in both MP and UP
- \( A \cap M \cap U \): Number of companies in all three states
- \( A \cup M \cup U \): Number of companies in at least one of the three states

Given that 90 companies did not have branch offices in any of these three states, we have:

\[ A \cup M \cup U = 400 - 90 = 310 \]

Now, let's use the other given information step-by-step:

1. **Number of companies in only UP, only MP, and only AP is in the ratio of 4:5:6**:

   Let the number of companies only in UP be \( 4x \), only in MP be \( 5x \), and only in AP be \( 6x \).

2. **The number of companies present in all three states is one-eighth of the number of companies in AP and MP but not UP**:

   Let \( A \cap M \setminus U \) be \( y \). Then \( A \cap M \cap U = \frac{y}{8} \).

3. **Amongst the companies in UP, the number of companies not in MP is thrice that of the number of companies in MP**:

   Let the number of companies in UP and MP be \( z \). Then the number of companies only in UP is \( 3z \).

4. **The number of companies present in both AP and UP is 30 less than the number of companies in AP and MP**:

   Let \( A \cap M = k \). Then \( A \cap U = k - 30 \).

Using these equations, we can set up the system:

\[ A \cup M \cup U = 4x + 5x + 6x + (A \cap M \setminus U) + (A \cap U \setminus M) + (M \cap U \setminus A) + A \cap M \cap U \]
\[ = 15x + y + (k - 30) + (z - \frac{y}{8}) + \frac{y}{8} \]
\[ = 310 \]

From the above information, we have:

\[ 15x + y + k - 30 + z = 310 \]

Since \( 3z = 4x \):

\[ z = \frac{4x}{3} \]

We also know that \( A \cap M \cap U = \frac{y}{8} \) and \( A \cap U = k - 30 \).

Since the number of companies only in MP is \( 5x \) and it is a multiple of 5, let \( x \) be a multiple of 1. 

Let's try to simplify these:

Given \( k = A \cap M \):

\[ k = y + \frac{4x}{3} \]
\[ A \cap U = k - 30 = \left( y + \frac{4x}{3} \right) - 30 \]

We have:

\[ 15x + y + y + \frac{4x}{3} - 30 = 310 \]
\[ 15x + 2y + \frac{4x}{3} - 30 = 310 \]
\[ 15x + 2y + \frac{4x}{3} = 340 \]
\[ 45x + 6y + 4x = 1020 \]
\[ 49x + 6y = 1020 \]

Let's find \( y \):

\[ A \cap M \cap U = \frac{y}{8} \]
\[ y = 8k \]

Using the equation \( 49x + 6(8k) = 1020 \), let's try \( x \):

\[ k = 17 \]

\[ 49x + 48 = 1020 \]
\[ 49x = 1020 - 1020 \]

Simplifying this we get:

\[ y = 160 \]

Thus \( A \cap M \cap U = 17 \)

Therefore, the answer is \( 17 \).

So the answer is **17**.

Explore the solution to a complex problem involving the distribution of 400 small-sized companies among three states - Andhra Pradesh (AP), Madhya Pradesh (MP), and Uttar Pradesh (UP). Understand the application of set theory, ratio analysis, and systems of equations in determining the number of companies present in each state and those present in all three states. Suitable for high school students studying advanced mathematics.

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