To determine the maximum amount of concrete James can make in the ratio of 1:2:3 for cement, sand, and gravel, we first need to consider the quantities he has and how they fit into that ratio.
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Ratio Components:
- For every 1 part of cement, he needs 2 parts of sand and 3 parts of gravel.
- This means the total parts = \(1 + 2 + 3 = 6\) parts.
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Calculate the Required Amounts:
- If \(x\) is the amount of cement in kg, then:
- Amount of sand needed \(= 2x\)
- Amount of gravel needed \(= 3x\)
- If \(x\) is the amount of cement in kg, then:
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Set Up the Inequalities: We have the amounts of materials James possesses:
- Cement: 63 kg
- Sand: 112 kg
- Gravel: 210 kg
Based on the ratio:
- For cement: \(x \leq 63\)
- For sand: \(2x \leq 112 \Rightarrow x \leq \frac{112}{2} = 56\)
- For gravel: \(3x \leq 210 \Rightarrow x \leq \frac{210}{3} = 70\)
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Find the Limiting Factor: The limiting quantity for \(x\) is the smallest value obtained:
- From cement: \(x \leq 63\)
- From sand: \(x \leq 56\)
- From gravel: \(x \leq 70\)
Therefore, the limiting ingredient is sand, giving \(x \leq 56\).
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Calculate the Maximum Amount of Concrete: The total amount of concrete can now be calculated using the ratio:
- Total parts = \(6\) (from the ratio of 1:2:3)
- Maximum \(x = 56\) kg (cement part)
Total concrete made: \[ \text{Total concrete} = 6x = 6 \times 56 = 336 \text{ kg} \]
Thus, the maximum amount of concrete that James can make is 336 kg.
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