Question

James is making concrete using cement, sand and gravel in the ratio 1 : 2 : 3
James has:
63kg cement
112kg sand
210kg gravel
What is the maximum amount of concrete that James can make?

Answer 1

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.

1. **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.

2. **Calculate the Required Amounts**:
- If \(x\) is the amount of cement in kg, then:
- Amount of sand needed \(= 2x\)
- Amount of gravel needed \(= 3x\)

3. **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: \(3x \leq 210 \Rightarrow x \leq \frac{210}{3} = 70\)

4. **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 \(x \leq 56\).

5. **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**.