Brass is an alloy of copper and zinc. Batch A of brass weighs 200 kg of which 70% is copper. Batch B weighs 120 kg of which 50% is copper. Batch C is 45% copper. When all three batches are combined, we get an alloy that is 55% copper. What is the weight of Batch C?
Step-by-step solution
To find: y, the weight of Batch C
Idea: Concentration of copper in the mixture = quantity of copper in the mixturequantity of the mixture. Write this with the unknown weight y, set it equal to 0.55 and solve the linear equation.
- Copper in A = 200 × 0.7 = 140 kg; copper in B = 120 × 0.5 = 60 kg; copper in C = 0.45y kg.½ mark
- Total copper = 140 + 60 + 0.45y = 200 + 0.45y; total weight = 200 + 120 + y = 320 + y.½ mark
- 0.55 = 200 + 0.45y320 + y, so 0.55(320 + y) = 200 + 0.45y.1 mark
- 176 + 0.55y = 200 + 0.45y ⇒ 0.1y = 24 ⇒ y = 240. Batch C weighs 240 kg.1 mark
Check: Copper = 140 + 60 + 0.45 × 240 = 140 + 60 + 108 = 308 kg; total = 200 + 120 + 240 = 560 kg; 308560 = 0.55 = 55% ✓.
Answer to write in the exam
Let the weight of Batch C = y kg
0.55 = 200 × 0.7 + 120 × 0.5 + y × 0.45200 + 120 + y
0.55(320 + y) = 140 + 60 + 0.45y
176 + 0.55y = 200 + 0.45y
0.1y = 24 ⇒ y = 240
∴ Weight of Batch C = 240 kg
Common mistakes that cost marks
- Leaving y out of the denominator: the total weight is 320 + y, not 320.
- Averaging the percentages 70, 50 and 45 directly, ignoring the weights.
- Arithmetic slip in 0.55 × 320 = 176 or in moving terms: 0.55y − 0.45y = 0.1y.
How this can come in the exam
150 kg of an alloy with 60% copper is melted with 50 kg of an alloy with 20% copper. The copper content of the new alloy is
- 40%
- 50%
- 45%
- 55%
Show answer
(B) 50%
150 × 0.6 + 50 × 0.2200 = 90 + 10200 = 0.5 = 50%.
Alloy X weighs 50 kg and is 40% tin. Alloy Y is 30% tin. When they are melted together the alloy is 34% tin. Find the weight of Y.
Show answer
0.34(50 + y) = 20 + 0.3y (1 mark) ⇒ 17 + 0.34y = 20 + 0.3y ⇒ 0.04y = 3 ⇒ y = 75 kg (1 mark).Try one yourself
Batch P of an alloy weighs 80 kg and is 25% zinc. Batch Q is 20% zinc. Together they make an alloy that is 22% zinc. Find the weight of Batch Q.
Show answer
0.22(80 + q) = 20 + 0.2q ⇒ 17.6 + 0.22q = 20 + 0.2q ⇒ 0.02q = 2.4 ⇒ q = 120 kg.
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