Let’s solve the question step-by-step in easy and copyable format.
Given:
-
Milk : Water ratios in three vessels:
- Vessel 1 → 3 : 1
- Vessel 2 → 5 : 3
- Vessel 3 → 3 : 5
-
Capacities of the three vessels: 2 : 3 : K
-
After mixing, overall Milk : Water = 21 : 19
We are to find the GCD of K², 2K, and 2K
Step 1: Assume total capacities of the three vessels
Let’s take actual capacities:
- Vessel 1 = 2x
- Vessel 2 = 3x
- Vessel 3 = Kx
Now calculate milk and water in each vessel.
Step 2: Find milk and water amounts
Vessel 1 → 3:1 → 3/4 milk, 1/4 water
- Milk = (3/4) × 2x = 1.5x
- Water = (1/4) × 2x = 0.5x
Vessel 2 → 5:3 → 5/8 milk, 3/8 water
- Milk = (5/8) × 3x = 15x/8
- Water = (3/8) × 3x = 9x/8
Vessel 3 → 3:5 → 3/8 milk, 5/8 water
- Milk = (3/8) × Kx = 3Kx/8
- Water = (5/8) × Kx = 5Kx/8
Step 3: Total milk and water after mixing
Total Milk = 1.5x + 15x/8 + 3Kx/8
Total Water = 0.5x + 9x/8 + 5Kx/8
Let’s simplify:
Milk:
1.5x = 12x/8
So total milk = (12x + 15x + 3Kx)/8 = (27x + 3Kx)/8
Water:
0.5x = 4x/8
So total water = (4x + 9x + 5Kx)/8 = (13x + 5Kx)/8
Given ratio:
Milk : Water = 21 : 19
So,
(27x + 3Kx) / (13x + 5Kx) = 21 / 19
Cross-multiply:
19 × (27x + 3Kx) = 21 × (13x + 5Kx)
Multiply:
513x + 57Kx = 273x + 105Kx
Bring terms together:
513x - 273x = 105Kx - 57Kx
240x = 48Kx
Cancel x:
240 = 48K → K = 5
Step 4: Now find GCD of K², 2K, 2K
K = 5
- K² = 25
- 2K = 10
- 2K = 10
Now find GCD of 25, 10, 10:
GCD(25, 10) = 5
✅ Final Answer: 5 ✅
Let me know if you want this full solution in Telugu also.
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