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 that the ratio of chromium to steel in the mixed type becomes 7: 32 ?

Options

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

Answer

To solve the problem of mixing two types of stainless steel in the given ratios of chromium to steel, we can use the method of alligation. Here's a step-by-step solution:

### Step 1: Understand the Ratios
We have two types of stainless steel:
- **Type 1**: The ratio of chromium to steel is 2:11.
- **Type 2**: The ratio of chromium to steel is 5:21.

### Step 2: Calculate the Fraction of Chromium in Each Type
For Type 1:
- Total parts = 2 (chromium) + 11 (steel) = 13
- Fraction of chromium = \( \frac{2}{13} \)

For Type 2:
- Total parts = 5 (chromium) + 21 (steel) = 26
- Fraction of chromium = \( \frac{5}{26} \)

### Step 3: Calculate the Fraction of Chromium in the Desired Mixture
The desired ratio of chromium to steel in the mixture is 7:32.
- Total parts = 7 (chromium) + 32 (steel) = 39
- Fraction of chromium in the mixture = \( \frac{7}{39} \)

### Step 4: Set Up the Alligation
Now, we will use the alligation method to find the required proportions of Type 1 and Type 2.

- Place the fractions in a line:
```
Type 1: 2/13
Mixture: 7/39
Type 2: 5/26
```

### Step 5: Calculate the Differences
- Difference between the mixture and Type 1:
\[ \frac{7}{39} - \frac{2}{13} \]
To calculate this, find a common denominator (which is 39):
\[ \frac{2}{13} = \frac{6}{39} \]
So,
\[ \frac{7}{39} - \frac{6}{39} = \frac{1}{39} \]

- Difference between Type 2 and the mixture:
\[ \frac{5}{26} - \frac{7}{39} \]
Find a common denominator (which is 78):
\[ \frac{5}{26} = \frac{15}{78}, \quad \frac{7}{39} = \frac{14}{78} \]
So,
\[ \frac{15}{78} - \frac{14}{78} = \frac{1}{78} \]

### Step 6: Set Up the Ratio
Now we have the differences:
- From Type 1: \( \frac{1}{39} \)
- From Type 2: \( \frac{1}{78} \)

The ratio of the two types is:
\[ \text{Ratio} = \frac{1/39}{1/78} = \frac{78}{39} = 2:1 \]

### Step 7: Conclusion
Thus, the two types of stainless steel should be mixed in the ratio of **2:1**.

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