Question

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:

• A. \(2.5\)
• B. \(5\)
• C. \(4\)
• D. \(6\)

Hint:

When mixtures of the same two ingredients are given with different compositions and costs, set up linear equations using the percentage of each ingredient and solve for the individual prices. Then use those prices to find the composition of any new mixture.

Answer 1

The Correct Option is B

Let the cost per kg of coffee be \(C_f\) and that of cocoa be \(C_c\). Step 1: Determine the individual costs of coffee and cocoa. From the first mixture, which contains \(16\%\) coffee and \(84\%\) cocoa and costs Rs 240 per kg, we get \[ 0.16C_f + 0.84C_c = 240. \tag{1} \] From the second mixture, which contains \(36\%\) coffee and \(64\%\) cocoa and costs Rs 320 per kg, we have \[ 0.36C_f + 0.64C_c = 320. \tag{2} \] Subtracting equation (1) from equation (2), \[ 0.20C_f - 0.20C_c = 80. \] Dividing by \(0.20\), \[ C_f - C_c = 400 \Rightarrow C_f = C_c + 400. \] Substituting this value in equation (1), \[ 0.16(C_c + 400) + 0.84C_c = 240, \] \[ 0.16C_c + 64 + 0.84C_c = 240, \] \[ C_c = 176. \] Hence, \[ C_f = 176 + 400 = 576. \] Step 2: Find the composition of the new mixture. Let the fraction of coffee in the new mixture be \(x\), so the fraction of cocoa is \(1 - x\). The cost of the mixture is Rs 376 per kg, hence \[ 576x + 176(1 - x) = 376. \] Simplifying, \[ 576x + 176 - 176x = 376, \] \[ 400x = 200, \] \[ x = \frac{1}{2}. \] Thus, the new mixture contains \(50\%\) coffee. Step 3: Calculate the amount of coffee in 10 kg of the mixture. \[ \text{Coffee} = \frac{1}{2} \times 10 = 5 \text{ kg}. \] Therefore, the quantity of coffee in 10 kg of the new mixture is \(5\) kg.