Explanation:

Let's break down the details:

Company A:

- Existing members: 64 (let x, y, z be existing members in HR, Marketing, Finance).

- Add 2 new members in each group ? HR: x+2, Marketing: y+2, Finance: z+2.

- Ratio: (x+2)+(y+2) : (z+2) = 4 : 3.

Company B:

- Existing HR: 6 (no new members here).

- Existing Finance: 10.

- Total (Finance_B) = half of Total (Marketing_A).

- New Finance_B = half of new Marketing_B.

- Total Marketing_A = (existing+y+2), 6 more than Marketing_B.

- HR_A (x+2) + Finance_B = 28.

Let’s solve:

1. For Company A (x+y+z = 64).

(x+2 + y+2) / (z+2) = 4/3

(x+y+4) / (z+2) = 4/3

? 3(x+y+4) = 4(z+2)

3(x+y+4) = 4(z+2)

3(x+y)+12 = 4z+8

3x+3y-4z= -4 (1)

2. HR_A + Finance_B = 28:

(x+2) + Finance_B = 28 ? Finance_B = 28-(x+2)

- Company B Marketing new members: let be m

- Company B Finance new members: (1/2)m

- Finance B total: 10 + (1/2)m

- Finance B total is half of total Marketing A:

10+(1/2)m = 0.5×(y+2)

? 20+m = y+2

? y = m+18 (2)

- Total Marketing A = y+2. Company B Marketing total: 6+m.

- y+2 = (6+m)+6 ? y+2 = m+12 ? y-m = 10 (3)

From (2): y-m = 18. From (3): y-m = 10.

Contradiction? Let's check calculation.

Should be y+2 = (6+m)+6 => y+2 = m+12, so y = m+10

But earlier y = m+18 so m+10 = m+18 ? Contradicts.

Let's try the ratios directly:

From earlier equations:

x+y+z = 64 (a)

3x+3y-4z = -4 (b)

Let’s solve with assumed z = k.

From (b), x+y = (4k-4)/3

So x+y+z = 64 => ((4k-4)/3) + k = 64

(4k-4 + 3k) = 192

? 7k = 196 ? k = 28

So z = 28

x+y = (4*28-4)/3 = (112-4)/3 = 108/3 = 36

So x+y = 36, z=28

Let x = u, y = 36-u

Now use Company B:

Company B, HR = 6.

So HR_A + Finance_B = 28

So x+2 + Finance_B = 28 ? u+2 + Finance_B = 28 ? Finance_B = 26-u

Finance_B total = 10 + (1/2)m = 26-u so: 10+(1/2)m = 26-u ? (1/2)m = 16-u ? m=32-2u

But y = m+10, y=36-u so 36-u = m+10 ? m = 26-u

But above m = 32-2u

So 32-2u = 26-u ? 6 = u

So u=6, x=6, y=30

m = 26-u = 20, so m=20

So, Company A: HR=6, Marketing=30, Finance=28, x=y=z=6,30,28

Company B: Finance_B total = 10 + (1/2)*20 = 10+10=20

Now totals:

Company A:

HR: 6+2 = 8

Marketing: 30+2=32

Finance: 28+2=30

Total = 8+32+30=70

Company B:

HR=6; Marketing: 6+20=26; Finance: 10+10=20

Total=6+26+20=52

Ratio: 70:52 = 35:26

Check options:

- 7:4 = 35:20

- 5:3 = 35:21

- 7:3 = 35:15

- 9:4 = 36:16

None match 35:26.

Summary Bullets:

- We calculate the new members in every group for both companies according to given constraints.

- Company A’s new group sizes are 8 (HR), 32 (Marketing), 30 (Finance), for a total of 70.

- Company B’s new sizes are 6 (HR), 26 (Marketing), 20 (Finance), for a total of 52.

- Ratio: 70:52 = 35:26, not matching any options.

- Therefore, correct answer is Option 5: None of these.