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 option is 5.

Let us break down the problem step-by-step to find the total quantity of coffee in 10 kg of the new mixture.

Step 1: Understand the composition and cost of pure components.
Let C be the cost of pure coffee per kg, and K be the cost of pure cocoa per kg.

We are given two initial mixtures:

1. First Mixture:
It contains 16% coffee and 84% cocoa. Its cost is Rs 240 per kg.
Writing this as a linear equation for the cost per kg:

$0.16C+0.84K=240$

2. Second Mixture:
It contains 36% coffee and 64% cocoa. Its cost is Rs 320 per kg.
Writing this as a linear equation for the cost per kg:

$0.36C+0.64K=320$

Step 2: Solve for the cost of coffee (C) and cocoa (K).
To simplify, let us multiply both equations by 100 to eliminate decimals:

(Equation 1) $16C+84K=24000$

(Equation 2) $36C+64K=32000$

We can simplify Equation 1 by dividing by 4:

$4C+21K=6000$

We can simplify Equation 2 by dividing by 4:

$9C+16K=8000$

Now subtract 4 × (Equation 2) from 9 × (Equation 1) to eliminate C, or solve the system directly:
From $4C+21K=6000$, we get $4C=6000-21K$.

Multiply the second simplified equation by 4:
$36C+64K=32000$
Substitute $9\times \left(4C\right)+64K=32000$:
$9(6000-21K)+64K=32000$
$54000-189K+64K=32000$
$54000-125K=32000$
$125K=22000$
$K=176$

So, the cost of cocoa is Rs 176 per kg.

Now substitute K = 176 back into $4C+21\left(176\right)=6000$:
$4C+3696=6000$
$4C=2304$
$C=576$

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

Step 3: Determine the proportion of coffee in the new mixture.
Let x be the fraction of coffee in 1 kg of the new mixture, which costs Rs 376 per kg.

$x·C+(1-x)·K=376$

Substitute the values of C = 576 and K = 176 into the equation:

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

$576x+176-176x=376$

$400x=200$

$x=\frac{200}{400}=0.5$

Thus, coffee makes up 50% (or 0.5 fraction) of the new mixture.

Step 4: Calculate the quantity of coffee in 10 kg of the new mixture.
Quantity of coffee = $10\times 0.5=5\text{kg}$.

Hence, the quantity of coffee in 10 kg of the new mixture is 5 kg.