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 of mixing two types of stainless steel with given chromium to steel ratios, we will follow these steps:

### Step 1: Identify the ratios of chromium to steel in both types of stainless steel.
- For Type 1: The ratio of chromium to steel is \(2:11\).
- For Type 2: The ratio of chromium to steel is \(5:21\).

### Step 2: Convert these ratios into fractions.
- For Type 1, the fraction of chromium in the mixture is:
\[ \text{Fraction of chromium in Type 1} = \frac{2}{2 + 11} = \frac{2}{13} \]
- For Type 2, the fraction of chromium in the mixture is:
\[ \text{Fraction of chromium in Type 2} = \frac{5}{5 + 21} = \frac{5}{26} \]

### Step 3: Convert the desired ratio of chromium to steel in the mixture into a fraction.
- The desired ratio of chromium to steel is \(7:32\).
- Thus, the fraction of chromium in the mixture is:
\[ \text{Fraction of chromium in mixture} = \frac{7}{7 + 32} = \frac{7}{39} \]

### Step 4: Set up the alligation formula.
- We will use the alligation method to find the ratio in which the two types should be mixed. The formula is:
\[ \text{Type 1} \quad \text{Type 2} \quad \text{Mixture} \]
\[ \frac{2}{13} \quad \frac{5}{26} \quad \frac{7}{39} \]

### Step 5: Calculate the differences.
- Calculate the difference between the fraction of chromium in Type 2 and the mixture:
\[ \frac{5}{26} - \frac{7}{39} \]
To perform this subtraction, we need a common denominator. The least common multiple of 26 and 39 is 78.
- Convert \(\frac{5}{26}\) to \(\frac{15}{78}\) (by multiplying numerator and denominator by 3).
- Convert \(\frac{7}{39}\) to \(\frac{14}{78}\) (by multiplying numerator and denominator by 2).
- Now, perform the subtraction:
\[ \frac{15}{78} - \frac{14}{78} = \frac{1}{78} \]

- Now, calculate the difference between the fraction of chromium in the mixture and Type 1:
\[ \frac{7}{39} - \frac{2}{13} \]
Again, convert \(\frac{2}{13}\) to \(\frac{12}{78}\) (by multiplying numerator and denominator by 6).
- Now, perform the subtraction:
\[ \frac{14}{78} - \frac{12}{78} = \frac{2}{78} \]

### Step 6: Set up the ratio.
- The ratio of the two types of stainless steel is given by the differences calculated:
\[ \text{Type 1 : Type 2} = \frac{1}{78} : \frac{2}{78} = 1 : 2 \]

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

---