There are two companies X and Y, each having three groups: HR, Marketing, and Finance. The following information is given:

- Company X has 64 existing members in total. If 2 new members are added to each group, the ratio of total members (existing + new) in HR and Marketing combined to the total members in Finance becomes $4 : 3$.

- Company Y has 6 existing members in HR and 10 existing members in Finance. No new members are added to HR in company Y.

- Additional information:
1. Total members (new + existing) in Finance group of company Y is half the total members (new + existing) in Marketing group of company X.
2. New members added in Finance group of company Y is half the new members added in Marketing group of company Y.
3. Total members (new + existing) in Marketing group of company X is 6 more than total members (new + existing) in Marketing group of company Y.
4. Total members (new + existing) in HR group of company X and Finance group of company Y together equal 28.

Find the ratio of total members (new + existing) of company X to company Y.

Question

There are two companies X and Y, each having three groups: HR, Marketing, and Finance. The following information is given:

- Company X has 64 existing members in total. If 2 new members are added to each group, the ratio of total members (existing + new) in HR and Marketing combined to the total members in Finance becomes $4 : 3$.

- Company Y has 6 existing members in HR and 10 existing members in Finance. No new members are added to HR in company Y.

- Additional information:
  1. Total members (new + existing) in Finance group of company Y is half the total members (new + existing) in Marketing group of company X.
  2. New members added in Finance group of company Y is half the new members added in Marketing group of company Y.
  3. Total members (new + existing) in Marketing group of company X is 6 more than total members (new + existing) in Marketing group of company Y.
  4. Total members (new + existing) in HR group of company X and Finance group of company Y together equal 28.

Find the ratio of total members (new + existing) of company X to company Y.

Accepted Answer

**Solution:**

1. Let the existing members in company X be:
   - HR = $x$
   - Marketing = $y$
   - Finance = $z$

Given total existing members in X: $x + y + z = 64$.

2. When 2 new members are added to each group in X, the new totals become:
   - HR = $x + 2$
   - Marketing = $y + 2$
   - Finance = $z + 2$

The ratio of (HR + Marketing) to Finance is given as:
   $$\frac{(x + 2) + (y + 2)}{z + 2} = \frac{4}{3}$$

Simplify numerator:
   $$\frac{x + y + 4}{z + 2} = \frac{4}{3}$$

Cross-multiplied:
   $$3(x + y + 4) = 4(z + 2)$$

$$3x + 3y + 12 = 4z + 8$$

Rearranged:
   $$3x + 3y - 4z = -4 \quad (1)$$

3. From total existing members:
   $$x + y + z = 64 \quad (2)$$

4. From (1) and (2), multiply (2) by 3:
   $$3x + 3y + 3z = 192$$

Subtract (1) from this:
   $$(3x + 3y + 3z) - (3x + 3y - 4z) = 192 - (-4)$$

$$3z + 4z = 196$$

$$7z = 196 \Rightarrow z = 28$$

5. Substitute $z=28$ into (2):
   $$x + y + 28 = 64 \Rightarrow x + y = 36$$

6. Company Y existing members:
   - HR = 6 (no new members added)
   - Finance = 10
   - Marketing = unknown, say $m$

7. Let new members added in Marketing and Finance of company Y be $m_n$ and $f_n$ respectively.

Given:
   - New members in Finance group of Y is half of new members in Marketing group of Y:
     $$f_n = \frac{m_n}{2} \quad (3)$$

- Total members in Finance group of Y (new + existing):
     $$10 + f_n$$

- Total members in Marketing group of X (new + existing):
     $$(y + 2)$$

Given:
   $$10 + f_n = \frac{y + 2}{2} \quad (4)$$

8. Total members in Marketing group of Y (new + existing):
   $$m + m_n$$

Given:
   $$(y + 2) = (m + m_n) + 6 \quad (5)$$

9. Total members in HR group of X (new + existing):
   $$x + 2$$

Total members in Finance group of Y (new + existing):
   $$10 + f_n$$

Given sum:
   $$(x + 2) + (10 + f_n) = 28 \Rightarrow x + f_n = 16 \quad (6)$$

10. From (3), (4), and (6), express $f_n$ and $m_n$ in terms of $x$ and $y$:

From (4):
    $$10 + f_n = \frac{y + 2}{2} \Rightarrow f_n = \frac{y + 2}{2} - 10$$

From (6):
    $$x + f_n = 16 \Rightarrow f_n = 16 - x$$

Equate both expressions for $f_n$:
    $$16 - x = \frac{y + 2}{2} - 10$$

Multiply both sides by 2:
    $$32 - 2x = y + 2 - 20$$

$$32 - 2x = y - 18$$

Rearranged:
    $$y = 50 - 2x \quad (7)$$

11. Recall from step 5:
    $$x + y = 36$$

Substitute $y$ from (7):
    $$x + (50 - 2x) = 36$$

$$50 - x = 36 \Rightarrow x = 14$$

12. Substitute $x=14$ into (7):
    $$y = 50 - 2(14) = 50 - 28 = 22$$

13. Now, $z = 28$, $x = 14$, $y = 22$.

14. Calculate new members in Finance group of Y:
    $$f_n = 16 - x = 16 - 14 = 2$$

15. New members in Marketing group of Y:
    $$m_n = 2 f_n = 2 	imes 2 = 4$$

16. Total members in Marketing group of Y:
    $$m + m_n = ?$$

From (5):
    $$(y + 2) = (m + m_n) + 6 \Rightarrow 22 + 2 = (m + 4) + 6$$

$$24 = m + 10 \Rightarrow m = 14$$

17. Total members in company X (new + existing):
    $$ (x + 2) + (y + 2) + (z + 2) = 16 + 24 + 30 = 70$$

18. Total members in company Y (new + existing):
    $$ (6 + 0) + (14 + 4) + (10 + 2) = 6 + 18 + 12 = 36$$

19. Required ratio:
    $$\frac{70}{36} = \frac{35}{18}$$

**Answer:** $35 : 18$

Answer

Answer

Answer

Answer