In three different vessels A, B and C, syrup and water are mixed in the ratios 2:3, 3:5 and 5:7 respectively. If 15 liters of mixture is taken from A, 16 liters from B and x liters from C and mixed, the resulting mixture has syrup and water in the ratio 2:3. Find x=.

Given:

Mixture ratios in vessels:

A = 2:3 → Syrup = 2/5, Water = 3/5

B = 3:5 → Syrup = 3/8, Water = 5/8

C = 5:7 → Syrup = 5/12, Water = 7/12

From A: 15 litres, From B: 16 litres, From C: x litres

Final ratio (Syrup:Water) = 2:3

Formula used:

Total Syrup = Syrup from A + Syrup from B + Syrup from C

Total Water = Water from A + Water from B + Water from C

Calculations:

Syrup from A = 15 × 2/5 = 6

Water from A = 15 × 3/5 = 9

Syrup from B = 16 × 3/8 = 6

Water from B = 16 × 5/8 = 10

Syrup from C = x × 5/12

Water from C = x × 7/12

Total Syrup = 6 + 6 + (5x/12) = 12 + 5x/12

Total Water = 9 + 10 + (7x/12) = 19 + 7x/12

Given: Syrup : Water = 2:3

⇒ \(\frac{12 + \frac{5x}{12}}{19 + \frac{7x}{12}} = \frac{2}{3}\)

⇒ 3(12 + 5x/12) = 2(19 + 7x/12)

⇒ 36 + (5x/4) = 38 + (7x/6)

⇒ 5x/4 - 7x/6 = 2

⇒ x = 24

∴ The correct answer is x = 24 litres