`20%` of the students in a class failed in an examination. Out of the students who failed, `75%` were males. Male students who failed constitute `90%` of the economically poor students in the class. What is the ratio of the number of economically poor students to the number of students in the class?

Options

A. `1 : 4`
B. `1 : 5`
C. `5 : 6`
D. `1 : 6`

Answer

To solve the problem step by step, we will first define the total number of students in the class and then use the given percentages to find the number of economically poor students.

### Step 1: Define the total number of students in the class
Let the total number of students in the class be \( x \).

### Step 2: Calculate the number of students who failed
According to the problem, \( 20\% \) of the students failed the examination. Therefore, the number of students who failed is:
\[ \text{Number of students who failed} = 20\% \text{ of } x = \frac{20}{100} \times x = \frac{x}{5} \]

### Step 3: Calculate the number of male students who failed
Out of the students who failed, \( 75\% \) were males. Thus, the number of male students who failed is:
\[ \text{Number of male students who failed} = 75\% \text{ of } \left(\frac{x}{5}\right) = \frac{75}{100} \times \frac{x}{5} = \frac{3x}{20} \]

### Step 4: Calculate the number of economically poor students
The problem states that the male students who failed constitute \( 90\% \) of the economically poor students in the class. Let \( y \) be the number of economically poor students. Therefore, we have:
\[ \frac{3x}{20} = 90\% \text{ of } y = \frac{90}{100} \times y = \frac{9y}{10} \]

### Step 5: Solve for \( y \)
We can set up the equation:
\[ \frac{3x}{20} = \frac{9y}{10} \]
To eliminate the fractions, we can multiply both sides by \( 20 \):
\[ 3x = 18y \]
Now, we can solve for \( y \):
\[ y = \frac{3x}{18} = \frac{x}{6} \]

### Step 6: Find the ratio of economically poor students to the total number of students
We have found that the number of economically poor students \( y \) is \( \frac{x}{6} \). Therefore, the ratio of economically poor students to the total number of students in the class is:
\[ \text{Ratio} = \frac{y}{x} = \frac{\frac{x}{6}}{x} = \frac{1}{6} \]

### Final Answer
The ratio of the number of economically poor students to the number of students in the class is \( 1:6 \).

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