2.7 Combining Three or More Mixtures in Given Ratios

When three (or more) vessels of milk-water (or any two-component) mixtures are combined, the safest and always-correct method is: convert each vessel’s contents into actual quantities of each component, then add them up.

Worked Example 2.7.1: Three vessels of equal volume contain milk and water mixed in the ratios 1 : 2, 2 : 3 and 4 : 1 respectively. If the contents of all three are poured into one large vessel, find the resultant ratio of milk to water.

Solution: Take a common volume for each vessel using the LCM of the sums of ratio terms: sums are 3, 5, 5; LCM =15. Let each vessel hold 15 units. - Vessel 1 (1:2, sum 3): milk =15×(1)/(3)=5, water =15×(2)/(3)=10. - Vessel 2 (2:3, sum 5): milk =15×(2)/(5)=6, water =15×(3)/(5)=9. - Vessel 3 (4:1, sum 5): milk =15×(4)/(5)=12, water =15×(1)/(5)=3.

Total milk = 5+6+12=23. Total water =10+9+3=22. Milk : Water = 23:22

This method generalises cleanly to any number of vessels and any ratios, and is far safer under exam pressure than trying to average ratios directly.

Worked Example 2.7.2 (unequal volumes): Container A holds 10 L of milk and water in the ratio 1 : 1. Container B holds 15 L in the ratio 2 : 1. Container C holds 20 L in the ratio 3 : 2. All three are mixed together. Find the resultant ratio of milk to water.

Solution: Since the volumes are unequal, compute actual quantities directly (there is no shortcut — this is exactly why the “convert to actual quantities” method is the universal, always-safe approach): - Container A: milk =5 L, water =5 L. - Container B: milk =15×(2)/(3)=10 L, water =15×(1)/(3)=5 L. - Container C: milk =20×(3)/(5)=12 L, water =20×(2)/(5)=8 L.

Total milk =5+10+12=27 L. Total water =5+5+8=18 L. Total volume check: 27+18=45=10+15+20 ✓. Milk : Water = 27:18 = 3:2

Worked Example 2.7.3 (reverse — combined ratio given, find the unknown ratio inside ONE vessel): Vessel A contains 20 L of a milk-water mixture in the ratio 3 : 2. Vessel B contains 30 L of a milk-water mixture in an unknown ratio. When both are mixed, the resultant 50 L mixture has milk and water in the ratio 2 : 3. Find the ratio of milk to water in vessel B.

Solution: Vessel A: milk =20×(3)/(5)=12 L, water =8 L. Combined mixture (50 L, ratio 2:3): total milk =50×(2)/(5)=20 L, total water =30 L. So vessel B’s milk =20-12=8 L, and vessel B’s water =30-8=22 L (which correctly sums to B’s 30 L). Milk : Water in B = 8:22 = 4:11 Check: 12+8=20 (total milk ✓), 8+22=30 (total water ✓), 20+30=50 (total volume ✓), and 20:30=2:3 ✓.

Worked Example 2.7.4 (four vessels of unequal volumes, find the final percentage): Four vessels contain milk-water mixtures: Vessel 1 has 10 L at 80% milk, Vessel 2 has 15 L at 60% milk, Vessel 3 has 20 L at 50% milk, and Vessel 4 has 5 L at 90% milk. All four are poured together. Find the percentage of milk in the resulting mixture.

Solution: Milk from each: 10(0.8)=8, 15(0.6)=9, 20(0.5)=10, 5(0.9)=4.5. Total milk =8+9+10+4.5=31.5 L. Total volume =10+15+20+5=50 L. Milk% = (31.5)/(50)×100 = 63%