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 the 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:

Question

Accepted Answer

Let the total initial quantity of coffee be **25 units** (as derived from the ratio $$16:9$$, where total = $$16+9=25$$).

#### Step 1: First Replacement
Let $$x$$ units of coffee be removed and replaced with cocoa powder. After the first replacement, the quantities are:
- Coffee = $$25 - x$$
- Cocoa = $$x$$

This forms **Mixture P**.

#### Step 2: Second Replacement
Now, $$x$$ units of Mixture P are removed. Since Mixture P has both coffee and cocoa, the proportion of coffee and cocoa removed is proportional to their respective quantities in Mixture P. The amount of coffee removed is:
$$
	ext{Coffee removed} = \frac{(25 - x)}{25} 	imes x
$$
The remaining coffee after the second replacement is:
$$
	ext{Remaining Coffee} = (25 - x) - \frac{(25 - x)}{25} 	imes x = (25 - x)\left(1 - \frac{x}{25}ight)
$$

This is given to equal **16 units** (as per the ratio in Mixture Q). Solving for $$x$$:
$$
(25 - x)^2 = 16 	imes 25
$$
$$
25 - x = 20 \quad \Rightarrow \quad x = 5
$$

#### Step 3: Cocoa Quantities in Mixtures P and Q
- In Mixture P, the amount of cocoa added is $$x = 5$$.
- In Mixture Q, the final amount of cocoa is $$9$$.

Thus, the required ratio of cocoa in Mixture P to that in Mixture Q is:
$$
	ext{Ratio} = \frac{5}{9} = 5:9
$$

Hence, the correct answer is **C) 5:9**.

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