Solving School Ratio Problems: Boys and Girls Admission

This problem involves understanding and manipulating ratios to find the number of students needed to achieve a specific ratio after changes in student numbers. We start with an initial ratio of boys to girls and a total number of students, then account for new admissions to find the required number of additional students of one type.

Step-by-Step Solution to the Ratio Problem

1. Calculate the Initial Number of Boys and Girls

The total number of students in the school is 640. The initial ratio of boys to girls is 5 : 3.

• The ratio 5 : 3 means the total number of students is divided into $5 + 3 = 8$ parts.
• Each part represents $\frac{640}{8} = 80$ students.
• Initial number of boys = $5 \text{ parts} \times 80 \text{ students/part} = 400$ boys.
• Initial number of girls = $3 \text{ parts} \times 80 \text{ students/part} = 240$ girls.

We can verify this: $400 + 240 = 640$ (total students) and the ratio $400 : 240$ simplifies to $40 : 24$, which further simplifies to $5 : 3$. This confirms our initial calculation is correct.

2. Account for the Admission of More Girls

30 more girls are admitted to the school.

• New number of girls = Initial girls + 30 = $240 + 30 = 270$ girls.
• The number of boys is still the initial number at this point, which is 400.

3. Determine the Number of Boys to Admit for the New Ratio

Let 'x' be the number of more boys that should be admitted. The new ratio of boys to girls should become 14 : 9.

• New number of boys = Initial boys + x = $400 + x$.
• New number of girls = 270.
• The new ratio is $(400 + x) : 270$.

According to the problem, this new ratio is equal to 14 : 9. We can write this as an equation:

$\frac{\text{New number of boys}}{\text{New number of girls}} = \frac{14}{9}$

$\frac{400 + x}{270} = \frac{14}{9}$

4. Solve the Equation for 'x'

To solve for 'x', we can cross-multiply:

$9 \times (400 + x) = 14 \times 270$

Expand the left side:

$3600 + 9x = 14 \times 270$

Calculate the right side:

$14 \times 270 = 3780$

So, the equation becomes:

$3600 + 9x = 3780$

Subtract 3600 from both sides:

$9x = 3780 - 3600$

$9x = 180$

Divide by 9:

$x = \frac{180}{9}$

$x = 20$

Therefore, 20 more boys should be admitted.

Verification

If 20 more boys are admitted:

• New number of boys = $400 + 20 = 420$.
• New number of girls = 270.
• New ratio = $420 : 270$.

Let's simplify this ratio by dividing both numbers by their greatest common divisor. Both are divisible by 10:

$420 : 270 \rightarrow 42 : 27$

Both 42 and 27 are divisible by 3:

$42 \div 3 = 14$

$27 \div 3 = 9$

So, the simplified ratio is 14 : 9, which matches the desired ratio. The calculation is correct.

Conclusion

To achieve the new ratio of 14 : 9 after 30 girls are admitted, 20 more boys should be admitted to the school.

Revision Table: School Ratio Changes

Additional Information: Understanding Ratios

A ratio is a comparison of two quantities. It tells us how much of one thing there is compared to another. Ratios can be written in several ways:

• Using a colon (e.g., 5 : 3)
• As a fraction (e.g., $\frac{5}{3}$)
• Using the word "to" (e.g., 5 to 3)

When dealing with ratio problems involving total quantities, it's often helpful to think of the ratio as parts. If a ratio is A : B, the total is divided into A + B parts. The value of each part can be found by dividing the total quantity by the total number of parts.

When quantities change, the ratio also changes unless adjusted. In problems like this, setting up an equation based on the new ratio and the changed quantities is key to finding the unknown value.