There are two companies X and Y. Each company has three groups: HR, Marketing, and Finance. Consider the following information:

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

**Company Y:**
- Existing members in HR = 6 (no new members added here).
- Existing members in Finance = 10.

Additional information:

(i) Total members (existing + new) in Finance group of company Y is half the total members (existing + new) in Marketing group of company X.

(ii) New members added in Finance group of company Y is half the new members added in Marketing group of company Y.

(iii) Total members (existing + new) in Marketing group of company X is 6 more than total members (existing + new) in Marketing group of company Y.

(iv) Total members (existing + new) in HR group of company X and Finance group of company Y together equal 28.

**Question:** Find the total number of existing members in company Y.

Question

There are two companies X and Y. Each company has three groups: HR, Marketing, and Finance. Consider the following information:

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

**Company Y:**
- Existing members in HR = 6 (no new members added here).
- Existing members in Finance = 10.

Additional information:

(i) Total members (existing + new) in Finance group of company Y is half the total members (existing + new) in Marketing group of company X.

(ii) New members added in Finance group of company Y is half the new members added in Marketing group of company Y.

(iii) Total members (existing + new) in Marketing group of company X is 6 more than total members (existing + new) in Marketing group of company Y.

(iv) Total members (existing + new) in HR group of company X and Finance group of company Y together equal 28.

**Question:** Find the total number of existing members in company Y.

Accepted Answer

**Solution:**

1. Let the existing members in HR, Marketing, and Finance groups of company X be $H_x$, $M_x$, and $F_x$ respectively.
   - Given total existing members in company X: $H_x + M_x + F_x = 64$.

2. When 2 new members are added to each group in company X, the new totals become:
   - HR: $H_x + 2$
   - Marketing: $M_x + 2$
   - Finance: $F_x + 2$

3. According to the problem, the ratio of total members in HR and Marketing combined to Finance is 4:3:
   $$\frac{(H_x + 2) + (M_x + 2)}{F_x + 2} = \frac{4}{3}$$

Simplify numerator:
   $$(H_x + M_x + 4) = \frac{4}{3}(F_x + 2)$$

4. From step 1, $H_x + M_x = 64 - F_x$, so:
   $$64 - F_x + 4 = \frac{4}{3}(F_x + 2)$$
   $$68 - F_x = \frac{4}{3}F_x + \frac{8}{3}$$

5. Multiply both sides by 3 to clear denominators:
   $$3(68 - F_x) = 4F_x + 8$$
   $$204 - 3F_x = 4F_x + 8$$

6. Rearranging:
   $$204 - 8 = 4F_x + 3F_x$$
   $$196 = 7F_x$$
   $$F_x = 28$$

7. Now find $H_x + M_x$:
   $$H_x + M_x = 64 - 28 = 36$$

8. New totals in company X groups:
   - HR: $H_x + 2 = H_x + 2$
   - Marketing: $M_x + 2 = M_x + 2$
   - Finance: $28 + 2 = 30$

9. Let the existing members in company Y be $H_y = 6$, $M_y$, and $F_y = 10$.
   Let new members added in Marketing and Finance groups of company Y be $m$ and $f$ respectively.
   Given no new members in HR group of Y.

10. From (ii): New members in Finance group of Y is half of new members in Marketing group:
    $$f = \frac{m}{2}$$

11. From (i): Total members in Finance group of Y (existing + new) is half of total members in Marketing group of X (existing + new):
    $$F_y + f = \frac{1}{2}(M_x + 2)$$
    $$10 + f = \frac{1}{2}(M_x + 2)$$

12. From (iii): Total members in Marketing group of X is 6 more than total members in Marketing group of Y:
    $$(M_x + 2) = (M_y + m) + 6$$

13. From (iv): Total members in HR group of X and Finance group of Y equals 28:
    $$(H_x + 2) + (F_y + f) = 28$$
    Substitute known values:
    $$(H_x + 2) + (10 + f) = 28$$
    $$H_x + f = 16$$

14. Recall from step 7: $H_x + M_x = 36$, so $M_x = 36 - H_x$.

15. Substitute $M_x$ in equation from step 11:
    $$10 + f = \frac{1}{2}((36 - H_x) + 2) = \frac{1}{2}(38 - H_x) = 19 - \frac{H_x}{2}$$
    $$f = 9 - \frac{H_x}{2}$$

16. From step 13:
    $$H_x + f = 16$$
    Substitute $f$:
    $$H_x + 9 - \frac{H_x}{2} = 16$$
    $$\frac{H_x}{2} + 9 = 16$$
    $$\frac{H_x}{2} = 7$$
    $$H_x = 14$$

17. Now find $f$:
    $$f = 9 - \frac{14}{2} = 9 - 7 = 2$$

18. Find $M_x$:
    $$M_x = 36 - 14 = 22$$

19. From step 12:
    $$(M_x + 2) = (M_y + m) + 6$$
    Substitute $M_x$:
    $$22 + 2 = M_y + m + 6$$
    $$24 = M_y + m + 6$$
    $$M_y + m = 18$$

20. From step 10:
    $$f = \frac{m}{2}$$
    We know $f = 2$, so:
    $$2 = \frac{m}{2} \Rightarrow m = 4$$

21. Calculate $M_y$:
    $$M_y + 4 = 18 \Rightarrow M_y = 14$$

22. Total existing members in company Y:
    $$H_y + M_y + F_y = 6 + 14 + 10 = 30$$

**Answer:** 30

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