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Question:

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 = $\left(1-\frac{\mathrm{r}}{100}\right)$

⇒ Quantity of coffee remaining after first replacement = 100 × $\left(1-\frac{\mathrm{r}}{100}\right)$
And quantity of cocoa after first replacement = r kgs.

⇒ Similarly, quantity of coffee remaining after second replacement = 100 × ${\left(1-\frac{\mathrm{r}}{100}\right)}^2$

Now, after 2nd replacement coffee and cocoa are in the ratio of 16 : 9
⇒ Quantity of coffee left after 2nd replacement = $\frac{16}{16+9}$ × 100 = 64 kg

⇒ 100 × ${\left(1-\frac{\mathrm{r}}{100}\right)}^2$ = 64

⇒ ${\left(1-\frac{\mathrm{r}}{100}\right)}^2$= $\frac{64}{100}$

⇒ $\left(1-\frac{\mathrm{r}}{100}\right)$ = $\frac{8}{10}$ = $\frac{4}{5}$

⇒ 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).