In a classroom, one-fifth of the boys leave the class and the ratio of the remaining boys to girls is 2:3. If further 44 girls leave the class, the ratio of boys to girls is 5:2. How many more boys should leave the class so that the number of boys equals that of girls ?

Options

A. 16
B. 24
C. 30
D. 36

Explanation

To solve the problem step by step, we will follow the information given in the question carefully.

### Step 1: Define Variables
Let:
- \( B \) = total number of boys in the classroom
- \( G \) = total number of girls in the classroom

### Step 2: Calculate Remaining Boys
According to the problem, one-fifth of the boys leave the class. Therefore, the number of remaining boys is:
\[ \text{Remaining Boys} = B - \frac{1}{5}B = \frac{4}{5}B \]

### Step 3: Set Up the First Ratio
The ratio of the remaining boys to girls is given as \( 2:3 \). This can be expressed mathematically as:
\[ \frac{\frac{4}{5}B}{G} = \frac{2}{3} \]
Cross-multiplying gives:
\[ 3 \cdot \frac{4}{5}B = 2G \]
Simplifying this, we get:
\[ \frac{12}{5}B = 2G \implies G = \frac{6}{5}B \]

### Step 4: Set Up the Second Ratio After Girls Leave
Next, we are told that if 44 girls leave the class, the new ratio of boys to girls becomes \( 5:2 \). The number of girls after 44 leave is \( G - 44 \). Thus, we can set up the equation:
\[ \frac{B - \frac{1}{5}B}{G - 44} = \frac{5}{2} \]
Substituting the remaining boys:
\[ \frac{\frac{4}{5}B}{G - 44} = \frac{5}{2} \]
Cross-multiplying gives:
\[ 2 \cdot \frac{4}{5}B = 5(G - 44) \]
Simplifying this:
\[ \frac{8}{5}B = 5G - 220 \]

### Step 5: Substitute \( G \) from Step 3
We already found that \( G = \frac{6}{5}B \). Substituting this into the equation gives:
\[ \frac{8}{5}B = 5\left(\frac{6}{5}B\right) - 220 \]
This simplifies to:
\[ \frac{8}{5}B = 6B - 220 \]
Multiplying through by 5 to eliminate the fraction:
\[ 8B = 30B - 1100 \]
Rearranging gives:
\[ 30B - 8B = 1100 \implies 22B = 1100 \implies B = 50 \]

### Step 6: Calculate Number of Girls
Now substituting \( B = 50 \) back into the equation for \( G \):
\[ G = \frac{6}{5} \cdot 50 = 60 \]

### Step 7: Determine How Many More Boys Should Leave
We need to find how many more boys should leave so that the number of boys equals the number of girls. Currently, there are 50 boys and 60 girls. Let \( x \) be the number of boys that need to leave:
\[ 50 - x = 60 \]
Solving for \( x \):
\[ -x = 60 - 50 \implies -x = 10 \implies x = 10 \]

### Final Answer
Thus, **10 more boys should leave the class so that the number of boys equals that of girls**.

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