A wholesaler deals in two types of grain: Type X costing

Solution

Step 1: Cost price (CP) per kg of mixtures P and Q. Mixture P (X:Y = 1:2): Average CP = (1×40 + 2×60) / 3 = (40 + 120)/3 = 160/3 ≈ Rs 53.33 Mixture Q (X:Y = 3:1): Average CP = (3×40 + 1×60) / 4 = (120 + 60)/4 = 180/4 = Rs 45 Step 2: Find ratio in which P and Q are mixed to get mixture R at Rs 48/kg. Let P:Q = m:n. Then, 48 = (m × 160/3 + n × 45) / (m + n) Multiply both sides by (m + n): 48(m + n) = (160m/3 + 45n) Multiply by 3 to clear denominator: 144(m + n) = 160m + 135n 144m + 144n = 160m + 135n ⇒ 16m = 9n ⇒ m : n = 9 : 16 So P:Q = 9 : 16. Step 3: Composition of mixture R (in terms of X and Y). In P, X:Y = 1:2 ⇒ X = 1/3, Y = 2/3. In Q, X:Y = 3:1 ⇒ X = 3/4, Y = 1/4. Combine 9 parts of P and 16 parts of Q: Total (take “parts of mixture”) = 9 + 16 = 25 parts Amount of X in 25 parts: = 9 × (1/3) + 16 × (3/4) = 3 + 12 = 15 parts Amount of Y in 25 parts: = 9 × (2/3) + 16 × (1/4) = 6 + 4 = 10 parts So in mixture R, X : Y = 15 : 10 = 3 : 2 Step 4: In 80 kg of mixture R: X = (3/5) × 80 = 48 kg Y = (2/5) × 80 = 32 kg Let z kg of pure Y be added. Then, New X = 48 kg New Y = 32 + z kg Total weight = 80 + z kg Average CP after adding Y is given as Rs 50/kg: (48×40 + (32 + z)×60) / (80 + z) = 50 Compute numerator: 48×40 = 1920 (32 + z)×60 = 1920 + 60z Total CP = 1920 + 1920 + 60z = 3840 + 60z Equation: (3840 + 60z) / (80 + z) = 50 ⇒ 3840 + 60z = 50(80 + z) ⇒ 3840 + 60z = 4000 + 50z ⇒ 10z = 160 ⇒ z = 16 So, quantity of pure Y added = 16 kg.