A mixture P is formed by removing a certain amount of coffee from a coffee jar and replacing the same amount with cocoa powder. The same amount is again removed from mixture P and replaced with same amount of cocoa powder to form a new mixture Q. If the ratio of coffee and cocoa in the mixture Q is 16 : 9, then the ratio of cocoa in mixture P to that in mixture Q is
Explanation:
Let the initial quantity of coffee in the jar be 100 kg and r kg is replaced each time.
Since r kg out of 100 kg is removed, fraction of coffee removed = r/100 ∴ fraction of coffee remaining = 1-r100
⇒ Quantity of coffee remaining after first replacement = 100 × 1-r100 And quantity of cocoa after first replacement = r kgs.
⇒ Similarly, quantity of coffee remaining after second replacement = 100 × 1-r1002
Now, after 2nd replacement coffee and cocoa are in the ratio of 16 : 9 ⇒ Quantity of coffee left after 2nd replacement = 1616+9 × 100 = 64 kg
⇒ 100 × 1-r1002 = 64
⇒ 1-r1002= 64100
⇒ 1-r100 = 810 = 45
⇒ r = 20 kg
∴ 20 kg of cocoa is added after 1st replacement. Also, quantity of cocoa after 2nd replacement = 100 – 64 = 36 kgs
⇒ Required ratio = 20 : 36 = 5 : 9.
Hence, option (d).
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