In school X, the number of boys is more than that of the girls by `40%.` In school Y, the number of girls is more than that of boys by `50%.` If `50%` boys in school X is equal to `70%` of girls in school Y, what is the ratio between the number of students of school X and School Y?

Options

A. `24:25`
B. `16:174`
C. `3:4`
D. `36:25`

Explanation

To solve the problem step by step, we will define variables for the number of boys and girls in each school and then set up equations based on the information provided.

### Step 1: Define Variables
Let:
- The number of girls in School X = \( G_X \)
- The number of boys in School X = \( B_X \)
- The number of boys in School Y = \( B_Y \)
- The number of girls in School Y = \( G_Y \)

### Step 2: Set Up Equations for School X
According to the problem, the number of boys in School X is more than that of girls by 40%. Therefore, we can express this as:
\[ B_X = G_X + 0.4 \cdot G_X = 1.4 \cdot G_X \]

### Step 3: Set Up Equations for School Y
In School Y, the number of girls is more than the number of boys by 50%. Thus, we can express this as:
\[ G_Y = B_Y + 0.5 \cdot B_Y = 1.5 \cdot B_Y \]

### Step 4: Relate Boys in School X to Girls in School Y
The problem states that 50% of boys in School X is equal to 70% of girls in School Y. Therefore, we can write:
\[ 0.5 \cdot B_X = 0.7 \cdot G_Y \]

### Step 5: Substitute \( B_X \) and \( G_Y \)
Substituting the expressions we derived for \( B_X \) and \( G_Y \):
\[ 0.5 \cdot (1.4 \cdot G_X) = 0.7 \cdot (1.5 \cdot B_Y) \]
This simplifies to:
\[ 0.7 \cdot G_X = 1.05 \cdot B_Y \]

### Step 6: Express \( G_X \) in terms of \( B_Y \)
Rearranging gives us:
\[ G_X = \frac{1.05}{0.7} \cdot B_Y = 1.5 \cdot B_Y \]

### Step 7: Substitute \( G_X \) back into \( B_X \)
Now substituting \( G_X \) back into the equation for \( B_X \):
\[ B_X = 1.4 \cdot (1.5 \cdot B_Y) = 2.1 \cdot B_Y \]

### Step 8: Calculate Total Students in Each School
Now we can calculate the total number of students in each school:
- Total students in School X:
\[ T_X = G_X + B_X = 1.5 \cdot B_Y + 2.1 \cdot B_Y = 3.6 \cdot B_Y \]
- Total students in School Y:
\[ T_Y = G_Y + B_Y = 1.5 \cdot B_Y + B_Y = 2.5 \cdot B_Y \]

### Step 9: Find the Ratio of Total Students
Now we find the ratio of the total number of students in School X to School Y:
\[ \text{Ratio} = \frac{T_X}{T_Y} = \frac{3.6 \cdot B_Y}{2.5 \cdot B_Y} = \frac{3.6}{2.5} = \frac{36}{25} \]

### Final Answer
Thus, the ratio between the number of students in School X and School Y is:
\[ \text{Ratio} = 36 : 25 \]

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