In an army selection process, the ratio of selected to unselected candidates was `4:1`. If 90 less had applied and 20 less were selected, the ratio of selected to unselected candidates would have been `5:1`. How many candidates had applied for the process?

Options

A. 1650
B. 3300
C. 825
D. 4950

Answer

To solve the problem step by step, we will denote the number of selected candidates as \( S \) and the number of unselected candidates as \( U \).

### Step 1: Set up the initial ratio
We know from the problem that the ratio of selected to unselected candidates is \( 4:1 \). This can be expressed as:
\[ \frac{S}{U} = \frac{4}{1} \]
From this, we can express the number of unselected candidates in terms of selected candidates:
\[ U = \frac{S}{4} \]

### Step 2: Express total candidates
The total number of candidates who applied can be expressed as:
\[ T = S + U \]
Substituting for \( U \):
\[ T = S + \frac{S}{4} = \frac{4S + S}{4} = \frac{5S}{4} \]

### Step 3: Set up the second scenario
According to the problem, if 90 less had applied and 20 less were selected, the new ratio of selected to unselected candidates would be \( 5:1 \). Thus:
- New number of selected candidates: \( S - 20 \)
- New number of unselected candidates: \( U - (90 - 20) = U - 70 \)

The new ratio can be expressed as:
\[ \frac{S - 20}{U - 70} = \frac{5}{1} \]
From this, we can express the number of unselected candidates in terms of the selected candidates:
\[ U - 70 = \frac{S - 20}{5} \]
Rearranging gives:
\[ U = \frac{S - 20}{5} + 70 \]

### Step 4: Substitute \( U \)
Now we have two expressions for \( U \):
1. \( U = \frac{S}{4} \)
2. \( U = \frac{S - 20}{5} + 70 \)

Setting these equal gives:
\[ \frac{S}{4} = \frac{S - 20}{5} + 70 \]

### Step 5: Clear the fractions
To eliminate the fractions, multiply through by 20 (the least common multiple of 4 and 5):
\[ 5S = 4(S - 20) + 1400 \]
Expanding this:
\[ 5S = 4S - 80 + 1400 \]
Simplifying:
\[ 5S - 4S = 1320 \]
\[ S = 1320 \]

### Step 6: Find \( U \)
Now substitute \( S \) back to find \( U \):
\[ U = \frac{S}{4} = \frac{1320}{4} = 330 \]

### Step 7: Find total candidates
Now we can find the total number of candidates who applied:
\[ T = S + U = 1320 + 330 = 1650 \]

### Final Answer
The total number of candidates who applied for the process is:
\[ \boxed{1650} \]