The ratio of the numbers of boys and girls of a school with 504 students is 13:11. What will be the new ratio if 12 more girls are admitted?

Options

A. `91: 81`
B. `81:91`
C. `9:10`
D. `10:9`

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understand the Given Ratio
The ratio of boys to girls in the school is given as 13:11. This means for every 13 boys, there are 11 girls.

### Step 2: Set Up the Equation
Let the number of boys be \(13x\) and the number of girls be \(11x\). The total number of students in the school is given as 504.

### Step 3: Write the Total Students Equation
The total number of students can be expressed as:
\[ 13x + 11x = 504 \]
This simplifies to:
\[ 24x = 504 \]

### Step 4: Solve for \(x\)
To find \(x\), divide both sides of the equation by 24:
\[ x = \frac{504}{24} = 21 \]

### Step 5: Calculate the Number of Boys and Girls
Now that we have \(x\), we can find the actual number of boys and girls:
- Number of boys:
\[ 13x = 13 \times 21 = 273 \]
- Number of girls:
\[ 11x = 11 \times 21 = 231 \]

### Step 6: Calculate the New Number of Girls After Admission
According to the problem, 12 more girls are admitted. Therefore, the new number of girls will be:
\[ 231 + 12 = 243 \]

### Step 7: Find the New Ratio
Now we can find the new ratio of boys to girls:
- Number of boys = 273
- Number of girls = 243

The new ratio can be expressed as:
\[ \text{New Ratio} = \frac{273}{243} \]

### Step 8: Simplify the Ratio
To simplify the ratio, we can divide both numbers by their greatest common divisor (GCD). The GCD of 273 and 243 is 9.
\[ \frac{273 \div 9}{243 \div 9} = \frac{30.33}{27} \text{ (not an integer)} \]
However, we can express it in the simplest form:
\[ \text{New Ratio} = 273:243 \text{ or } 91:81 \text{ (after dividing both by 3)} \]

### Final Answer
The new ratio of boys to girls after admitting 12 more girls is:
\[ \text{New Ratio} = 91:81 \]

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