In a school `(1)/(10)` of the boys are same in number as `(1)/(4)` of the girls and `(5)/(8)` of the girls are same in numbers as `(1)/(4)` of the boys. The ratio of the boys to girls in that school is

Options

A. `2:1`
B. `5:2`
C. `4:3`
D. `3:2`

Answer

To find the ratio of boys to girls in the school, we will set up equations based on the information provided in the question.

Let:
- \( B \) = number of boys
- \( G \) = number of girls

### Step 1: Set up the first equation
According to the question, \( \frac{1}{10} \) of the boys is equal to \( \frac{1}{4} \) of the girls. We can express this as:
\[ \frac{1}{10}B = \frac{1}{4}G \]

### Step 2: Cross-multiply to eliminate the fractions
To eliminate the fractions, we can cross-multiply:
\[ 4 \times \frac{1}{10}B = 1 \times G \]
This simplifies to:
\[ \frac{4}{10}B = G \]
or
\[ G = \frac{2}{5}B \]

### Step 3: Set up the second equation
The question also states that \( \frac{5}{8} \) of the girls is equal to \( \frac{1}{4} \) of the boys. We can express this as:
\[ \frac{5}{8}G = \frac{1}{4}B \]

### Step 4: Substitute \( G \) in the second equation
Now, we can substitute \( G \) from Step 2 into this equation:
\[ \frac{5}{8} \left(\frac{2}{5}B\right) = \frac{1}{4}B \]
This simplifies to:
\[ \frac{5 \times 2}{8 \times 5}B = \frac{1}{4}B \]
or
\[ \frac{1}{8}B = \frac{1}{4}B \]

### Step 5: Solve for the ratio of boys to girls
To solve for the ratio, we can equate the two expressions:
\[ \frac{1}{8}B = \frac{1}{4}B \]
Cross-multiplying gives:
\[ 4B = 8B \]
This indicates that we need to find a common ratio.

### Step 6: Find the ratio
From the equations we derived:
1. \( G = \frac{2}{5}B \)
2. Rearranging gives \( B = \frac{5}{2}G \)

Thus, the ratio of boys to girls is:
\[ B : G = 5 : 2 \]

### Final Answer
The ratio of boys to girls in the school is \( 5 : 2 \).

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