Understanding the Exam Participant Problem

This problem involves analyzing the number of boys and girls participating in two different exams, SSC and PO. Participants can be in SSC only, PO only, or both exams. We are given various ratios and conditions relating the number of participants in these different categories. The goal is to find the ratio of the total number of boys in the SSC exam to the total number of girls in the PO exam.

Defining Variables for Participant Categories

Let's use variables to represent the number of participants in each category:

• B_S_only: Number of Boys who participate only in SSC.
• G_S_only: Number of Girls who participate only in SSC.
• B_P_only: Number of Boys who participate only in PO.
• G_P_only: Number of Girls who participate only in PO.
• B_Both: Number of Boys who participate in both SSC and PO.
• G_Both: Number of Girls who participate in both SSC and PO.

Formulating Equations from the Given Information

We translate the given ratios and conditions into mathematical equations:

• The ratio of Boys to Girls who participate only in the PO exams is 1 : 2. This means $\text{B\_P\_only} : \text{G\_P\_only} = 1 : 2$. We can represent this as $\text{B\_P\_only} = k$ and $\text{G\_P\_only} = 2k$ for some constant $k$.
• The ratio of Boys to Girls who participate in both the exams PO and SSC is 2 : 1. This means $\text{B\_Both} : \text{G\_Both} = 2 : 1$. We can represent this as $\text{B\_Both} = 2m$ and $\text{G\_Both} = m$ for some constant $m$.
• Boys who participate in both SSC and PO is 20% of the Boys who participate in SSC events alone. This means $\text{B\_Both} = 0.20 \times \text{B\_S\_only}$. Substituting $\text{B\_Both} = 2m$, we get $2m = 0.20 \times \text{B\_S\_only}$, which simplifies to $\text{B\_S\_only} = \frac{2m}{0.20} = 10m$.
• The participation of boys in the SSC exam alone is 10 more than the Girls who participate in it. This means $\text{B\_S\_only} = \text{G\_S\_only} + 10$. Substituting $\text{B\_S\_only} = 10m$, we get $10m = \text{G\_S\_only} + 10$, so $\text{G\_S\_only} = 10m - 10$.

Calculating Total Participants in Each Exam

The total number of participants in each exam is the sum of those participating only in that exam and those participating in both.

• Total Boys in SSC = $\text{B\_S\_only} + \text{B\_Both} = 10m + 2m = 12m$.
• Total Girls in SSC = $\text{G\_S\_only} + \text{G\_Both} = (10m - 10) + m = 11m - 10$.
• Total Boys in PO = $\text{B\_P\_only} + \text{B\_Both} = k + 2m$.
• Total Girls in PO = $\text{G\_P\_only} + \text{G\_Both} = 2k + m$.

Using Overall Exam Ratios to Find Variable Values

Now we use the overall ratios for SSC and PO to solve for the constants $m$ and $k$.

Ratio in SSC Exam

The ratio of Boys to Girls who participate in SSC is 4 : 3.

$\frac{\text{Total Boys in SSC}}{\text{Total Girls in SSC}} = \frac{4}{3}$

$\frac{12m}{11m - 10} = \frac{4}{3}$

Cross-multiplying gives:

$3 \times 12m = 4 \times (11m - 10)$

$36m = 44m - 40$

$40 = 44m - 36m$

$40 = 8m$

$m = \frac{40}{8} = 5$

Ratio in PO Exam

The ratio of Boys to Girls who participate in PO is 8 : 13.

$\frac{\text{Total Boys in PO}}{\text{Total Girls in PO}} = \frac{8}{13}$

$\frac{k + 2m}{2k + m} = \frac{8}{13}$

Substitute the value $m = 5$ into this equation:

$\frac{k + 2(5)}{2k + 5} = \frac{8}{13}$

$\frac{k + 10}{2k + 5} = \frac{8}{13}$

Cross-multiplying gives:

$13 \times (k + 10) = 8 \times (2k + 5)$

$13k + 130 = 16k + 40$

$130 - 40 = 16k - 13k$

$90 = 3k$

$k = \frac{90}{3} = 30$

Calculating Final Participant Counts

With the values $m=5$ and $k=30$, we can find the number of participants in each category:

• m = 5
• k = 30
• B_Both = 2m = 2 * 5 = 10
• G_Both = m = 5
• B_S_only = 10m = 10 * 5 = 50
• G_S_only = 10m - 10 = 10 * 5 - 10 = 40
• B_P_only = k = 30
• G_P_only = 2k = 2 * 30 = 60

Finding the Required Ratio

The question asks for the ratio of boys who participated in SSC to girls who participated in PO.

Total Boys in SSC = $\text{B\_S\_only} + \text{B\_Both} = 50 + 10 = 60$

Total Girls in PO = $\text{G\_P\_only} + \text{G\_Both} = 60 + 5 = 65$

The ratio is Total Boys in SSC : Total Girls in PO = 60 : 65.

To simplify the ratio, we divide both numbers by their greatest common divisor, which is 5.

$\frac{60}{5} : \frac{65}{5} = 12 : 13$

The ratio of boys who participated in SSC to girls who participated in PO is 12 : 13.