A mixture of coffee and cocoa, 16% of which is coffee, costs Rs 240 per kg. Another mixture of coffee and cocoa, of which 36% is coffee, costs Rs 320 per kg. If a new mixture of coffee and cocoa costs Rs 376 per kg, then the quantity, in kg, of coffee in 10 kg of this new mixture is

The correct answer is 5.

To find the quantity of coffee in the new mixture, we must first determine the individual costs of pure coffee and pure cocoa per kg.

Let the cost of pure coffee be $C$ per kg, and the cost of pure cocoa be $K$ per kg.

According to the problem, the first mixture consists of 16% coffee and 84% cocoa, and its cost is Rs 240 per kg. We can write this as a linear equation:

$0.16C+0.84K=240$

Multiplying this entire equation by 100 to remove decimals gives:

$16C+84K=24000$

Dividing by 4 gives our first simplified equation:

$4C+21K=6000$

The second mixture consists of 36% coffee and 64% cocoa (since 100% - 36% = 64%), and its cost is Rs 320 per kg. This gives the second equation:

$0.36C+0.64K=320$

Again, multiplying by 100 gives:

$36C+64K=32000$

Dividing by 4 gives our second simplified equation:

$9C+16K=8000$

Now, we solve the system of equations. Multiply the first simplified equation by 9 and the second by 4 to eliminate $C$:

$36C+189K=54000$

$36C+64K=32000$

Subtracting the second equation from the first, we get:

$125K=22000$

$K=\frac{22000}{125}=176$

So, the cost of pure cocoa is Rs 176 per kg. We can now substitute $K=176$ into the first simplified equation to find $C$:

$4C+21\left(176\right)=6000$

$4C+3696=6000$

$4C=6000-3696=2304$

$C=\frac{2304}{4}=576$

Thus, the cost of pure coffee is Rs 576 per kg.

Now we consider the new mixture which costs Rs 376 per kg. Let the proportion of coffee in this mixture be $x$. Consequently, the proportion of cocoa will be $1-x$. The cost equation for the new mixture is:

$576x+176(1-x)=376$

Expanding and solving for $x$:

$576x+176-176x=376$

$400x=376-176$

$400x=200$

$x=0.5$

This means that 50% of the new mixture is coffee. Finally, we must find the quantity of coffee in a 10 kg batch of this new mixture:

$\text{Quantity of coffee}=0.5\times 10=5\text{ kg}$

This mathematically confirms the correct answer is 5.