The ratio of the number of students in schools X and Y is 4:7. if 15 students from Y join X, then the ratio becomes 1:1. What is the total number of students in the schools X and Y?

Answer

To solve the problem step by step, we will denote the number of students in school X as \( X \) and the number of students in school Y as \( Y \).

### Step 1: Set up the initial ratio
The ratio of the number of students in schools X and Y is given as 4:7. This can be expressed mathematically as:
\[ \frac{X}{Y} = \frac{4}{7} \]
From this, we can express \( X \) in terms of \( Y \):
\[ X = \frac{4}{7}Y \]

### Step 2: Account for the transfer of students
According to the problem, if 15 students from Y join X, the new ratio of students becomes 1:1. This means:
\[ X + 15 = Y - 15 \]

### Step 3: Substitute \( X \) in the equation
Now, substitute \( X \) from Step 1 into the equation from Step 2:
\[ \frac{4}{7}Y + 15 = Y - 15 \]

### Step 4: Clear the equation
To eliminate the fraction, multiply the entire equation by 7:
\[ 4Y + 105 = 7Y - 105 \]

### Step 5: Rearrange the equation
Now, rearranging the equation to isolate \( Y \):
\[ 4Y + 105 + 105 = 7Y \]
\[ 4Y + 210 = 7Y \]
\[ 210 = 7Y - 4Y \]
\[ 210 = 3Y \]

### Step 6: Solve for \( Y \)
Now, divide both sides by 3:
\[ Y = \frac{210}{3} = 70 \]

### Step 7: Find \( X \)
Now that we have \( Y \), we can find \( X \) using the equation from Step 1:
\[ X = \frac{4}{7}Y = \frac{4}{7} \times 70 = 40 \]

### Step 8: Calculate the total number of students
Finally, the total number of students in both schools is:
\[ X + Y = 40 + 70 = 110 \]

### Final Answer
The total number of students in schools X and Y is **110**.

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