In two types of stainless steel, the ratio of chromium and steel are `2 : 11` and `5 : 21`,respectively. In what proportion should the two types be mixed, so thta the ratio of chromium to steel in the mixed type become `7 : 32`?

Options

A. `1 : 2`
B. `1 : 3`
C. `2 : 3`
D. `3 : 4`

Answer

To solve the problem, we need to find the proportion in which two types of stainless steel should be mixed so that the ratio of chromium to steel in the mixture becomes \(7 : 32\).

### Step-by-Step Solution:

1. **Define the Ratios**:
- For the first type of stainless steel, the ratio of chromium to steel is \(2 : 11\).
- For the second type, the ratio is \(5 : 21\).

2. **Let the Quantities be x and y**:
- Let the first type (with ratio \(2 : 11\)) be mixed in quantity \(x\).
- Let the second type (with ratio \(5 : 21\)) be mixed in quantity \(y\).

3. **Calculate Chromium and Steel Content**:
- In the first type (for quantity \(x\)):
- Chromium = \(\frac{2}{2 + 11} \cdot x = \frac{2}{13} x\)
- Steel = \(\frac{11}{2 + 11} \cdot x = \frac{11}{13} x\)

- In the second type (for quantity \(y\)):
- Chromium = \(\frac{5}{5 + 21} \cdot y = \frac{5}{26} y\)
- Steel = \(\frac{21}{5 + 21} \cdot y = \frac{21}{26} y\)

4. **Set Up the Equation for the Mixture**:
- The total chromium in the mixture = \(\frac{2}{13} x + \frac{5}{26} y\)
- The total steel in the mixture = \(\frac{11}{13} x + \frac{21}{26} y\)

5. **Form the Ratio**:
- We want the ratio of chromium to steel in the mixture to be \(7 : 32\):
\[ \frac{\frac{2}{13} x + \frac{5}{26} y}{\frac{11}{13} x + \frac{21}{26} y} = \frac{7}{32} \]

6. **Cross Multiply to Eliminate the Fraction**:
\[ 32 \left( \frac{2}{13} x + \frac{5}{26} y \right) = 7 \left( \frac{11}{13} x + \frac{21}{26} y \right) \]

7. **Multiply Through by the Common Denominator (26)**:
\[ 32 \left( 4x + 5y \right) = 7 \left( 22x + 21y \right) \]
This simplifies to:
\[ 128x + 160y = 154x + 147y \]

8. **Rearranging the Equation**:
\[ 128x - 154x + 160y - 147y = 0 \]
\[ -26x + 13y = 0 \]

9. **Solving for the Ratio**:
\[ 26x = 13y \implies \frac{x}{y} = \frac{13}{26} = \frac{1}{2} \]

10. **Final Ratio**:
- Therefore, the ratio of the two types of stainless steel to be mixed is \(x : y = 1 : 2\).

### Conclusion:
The two types of stainless steel should be mixed in the ratio of \(1 : 2\).