Finding the Ratio of Boys to Girls in College

Let's break down this college ratio problem step-by-step. We are given a relationship between the number of boys and the number of girls in a college and asked to find the ratio of boys to girls.

Let:

• $B$ represent the number of boys in the college.
• $G$ represent the number of girls in the college.

According to the question, 15% of the boys are equal in number to one-third of the girls. We can write this relationship as an equation.

First, let's express the given percentages and fractions mathematically:

• 15% of boys can be written as $\frac{15}{100} \times B$. This simplifies to $\frac{3}{20} \times B$.
• One-third of girls can be written as $\frac{1}{3} \times G$.

Now, we can set up the equation based on the problem statement:

$\frac{3}{20} \times B = \frac{1}{3} \times G$

We need to find the ratio of the number of boys to that of girls, which is $B : G$ or $\frac{B}{G}$. To find this ratio from our equation, we need to isolate the term $\frac{B}{G}$.

Let's rearrange the equation:

Start with: $\frac{3}{20} B = \frac{1}{3} G$

To get $\frac{B}{G}$ on one side, we can divide both sides of the equation by $G$ (assuming $G \neq 0$):

$\frac{\frac{3}{20} B}{G} = \frac{\frac{1}{3} G}{G}$

$\frac{3}{20} \times \frac{B}{G} = \frac{1}{3}$

Now, to isolate $\frac{B}{G}$, we need to multiply both sides of the equation by the reciprocal of $\frac{3}{20}$, which is $\frac{20}{3}$:

$\left(\frac{20}{3}\right) \times \left(\frac{3}{20} \times \frac{B}{G}\right) = \left(\frac{20}{3}\right) \times \left(\frac{1}{3}\right)$

On the left side, $\frac{20}{3}$ and $\frac{3}{20}$ cancel out, leaving $\frac{B}{G}$.

On the right side, we multiply the numerators and the denominators:

$\frac{B}{G} = \frac{20 \times 1}{3 \times 3}$

$\frac{B}{G} = \frac{20}{9}$

This means the ratio of the number of boys to the number of girls is $20:9$.

Let's verify with an example. If $B = 200$ and $G = 90$, then the ratio is $200:90 = 20:9$.

• 15% of boys = $0.15 \times 200 = 30$.
• One-third of girls = $\frac{1}{3} \times 90 = 30$.

Since $30 = 30$, the condition in the question is met, confirming our ratio $20:9$ is correct.

Summary of Ratio Calculation Steps

1. Define variables for the quantities (Boys $B$, Girls $G$).
2. Translate the word problem into an algebraic equation: $15\% \text{ of } B = \frac{1}{3} \text{ of } G$.
3. Convert percentage to a fraction or decimal: $0.15 B = \frac{1}{3} G$ or $\frac{3}{20} B = \frac{1}{3} G$.
4. Rearrange the equation to find the ratio $\frac{B}{G}$.
5. Simplify the resulting fraction to get the final ratio.

Final Ratio Result

The ratio of the number of boys to that of girls is $20 : 9$. Looking at the options provided, this matches one of them.

Revision Table: Key Concepts for Ratio Problems

Additional Information on Percentages, Fractions, and Ratios

Understanding how percentages, fractions, and ratios relate to each other is crucial for solving many math problems.

• Percentages and Fractions: A percentage can always be written as a fraction with a denominator of 100. For example, 25% is $\frac{25}{100}$, which simplifies to $\frac{1}{4}$. Similarly, a fraction can be converted to a percentage by multiplying by 100%. For example, $\frac{1}{2} = \frac{1}{2} \times 100\% = 50\%$.
• Fractions and Ratios: A fraction like $\frac{a}{b}$ directly represents the ratio $a:b$. This is why when we found $\frac{B}{G} = \frac{20}{9}$, we could immediately state the ratio $B:G$ is $20:9$. The numerator corresponds to the first part of the ratio, and the denominator corresponds to the second part.
• Ratios and Proportions: A proportion is an equation stating that two ratios are equal, like $\frac{a}{b} = \frac{c}{d}$. The problem we solved is essentially setting up a proportion based on the given information and solving for an unknown ratio.

Being comfortable converting between these forms helps in setting up and solving such problems efficiently.