A can contains a mixture of two liquids A and B in the ratio 7 : 5. When 9 litres of mixture are drawn off and the can is refilled with B, the ratio of A to B becomes 7 : 9. How many litres of liquid A were in the can initially?

Problem restatementFrom a can containing a well-mixed liquid A and B in the ratio 7 : 5, 9 L are drawn out and replaced by pure B. The new ratio becomes 7 : 9. Find the original litres of A.

Given data

• Initial ratio A : B = 7 : 5 (i.e., A = (7/12) of total, B = (5/12) of total)
• Drawn off and replaced volume = 9 L (drawn from the mixture, refilled with pure B)
• Final ratio A : B = 7 : 9

Concept/ApproachWhen a well-mixed quantity is withdrawn, each component reduces proportionally. Let the can's capacity be T litres. After removing 9 L, the remaining amounts of A and B are each multiplied by (T − 9)/T. Then 9 L of pure B is added. Set up the final ratio and solve for T, then compute initial A = (7/12)T.

Step-by-step calculationInitial A = (7/12)T,   Initial B = (5/12)TAfter removing 9 L (proportionally):A_rem = (7/12)T − (7/12)×9 = (7/12)(T − 9)B_rem = (5/12)T − (5/12)×9 = (5/12)(T − 9)After adding 9 L of pure B:   B_new = (5/12)(T − 9) + 9Final ratio given:   A_rem : B_new = 7 : 9⇒ 9 × (7/12)(T − 9) = 7 × (5/12)(T − 9) + 9Multiply by 12:   63(T − 9) = 35(T − 9) + 75628(T − 9) = 756   ⇒   T − 9 = 27   ⇒   T = 36Initial A = (7/12) × 36 = 21 litres

Verification/AlternativeWith T = 36:   After removal, remaining total = 27 L.A_rem = (7/12)×27 = 15.75 L;   B_rem = (5/12)×27 = 11.25 L.After adding 9 L of B:   B_new = 11.25 + 9 = 20.25 L.Ratio = 15.75 : 20.25 = (divide by 2.25) = 7 : 9 — matches perfectly.

Common pitfalls

• Forgetting that the 9 L removed is a mixture (so both A and B reduce proportionally).
• Adding 9 L to both A and B (only B increases because the refill is pure B).
• Setting up the ratio as totals rather than component-wise after each operation.

Final Answer21 litres