Analyzing Participants in SSC and PO Exams

The problem provides information about participants (Boys and Girls) in SSC and PO exams, classifying them based on whether they participate in SSC alone, PO alone, or both exams. We are given ratios and relationships between the number of boys and girls in these different categories. Our goal is to find the ratio of the total number of participants who took only the SSC exam to the total number of participants who took only the PO exam.

Defining Variables

Let's define the variables to represent the number of boys and girls in each category:

• B_SSC_only: Boys participating only in SSC
• G_SSC_only: Girls participating only in SSC
• B_PO_only: Boys participating only in PO
• G_PO_only: Girls participating only in PO
• B_Both: Boys participating in both SSC and PO
• G_Both: Girls participating in both SSC and PO

Given Information

From the problem statement, we have the following facts:

• Ratio of Boys to Girls in SSC (Total SSC) = 4 : 3
• Ratio of Boys to Girls in PO (Total PO) = 8 : 13
• Ratio of Boys to Girls in Both = 2 : 1
• Ratio of Boys to Girls in PO Only = 1 : 2
• B_SSC_only = G_SSC_only + 10
• B_Both = 20% of B_SSC_only

Step-by-Step Calculation

Using the Relationship Between Both and SSC Only Participants

We are given that Boys in Both is 20% of Boys in SSC Only.

B_Both = 0.20 × B_SSC_only

Let B_SSC_only = S.

Then, B_Both = 0.2S.

The ratio of Boys to Girls in Both is 2:1. So, G_Both = (1/2) × B_Both = (1/2) × 0.2S = 0.1S.

Also, we know that G_SSC_only = B_SSC_only - 10 = S - 10.

Calculating Participants in SSC

Total Boys in SSC = B_SSC_only + B_Both = S + 0.2S = 1.2S

Total Girls in SSC = G_SSC_only + G_Both = (S - 10) + 0.1S = 1.1S - 10

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

\(\frac{\text{Total Boys in SSC}}{\text{Total Girls in SSC}} = \frac{1.2S}{1.1S - 10} = \frac{4}{3}\)

Cross-multiplying:

\(3 \times (1.2S) = 4 \times (1.1S - 10)\)

\(3.6S = 4.4S - 40\)

\(40 = 4.4S - 3.6S\)

\(40 = 0.8S\)

\(S = \frac{40}{0.8} = \frac{400}{8} = 50\)

Now we can find the number of participants in SSC Only and Both:

• B_SSC_only = S = 50
• G_SSC_only = S - 10 = 50 - 10 = 40
• Total SSC Only = B_SSC_only + G_SSC_only = 50 + 40 = 90
• B_Both = 0.2S = 0.2 × 50 = 10
• G_Both = 0.1S = 0.1 × 50 = 5
• Total Both = B_Both + G_Both = 10 + 5 = 15

Calculating Participants in PO

The ratio of Boys to Girls in PO Only is 1:2.

Let B_PO_only = m.

Then, G_PO_only = 2m.

We already found B_Both = 10 and G_Both = 5.

Total Boys in PO = B_PO_only + B_Both = m + 10

Total Girls in PO = G_PO_only + G_Both = 2m + 5

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

\(\frac{\text{Total Boys in PO}}{\text{Total Girls in PO}} = \frac{m + 10}{2m + 5} = \frac{8}{13}\)

Cross-multiplying:

\(13 \times (m + 10) = 8 \times (2m + 5)\)

\(13m + 130 = 16m + 40\)

\(130 - 40 = 16m - 13m\)

\(90 = 3m\)

\(m = \frac{90}{3} = 30\)

Now we can find the number of participants in PO Only:

• B_PO_only = m = 30
• G_PO_only = 2m = 2 × 30 = 60
• Total PO Only = B_PO_only + G_PO_only = 30 + 60 = 90

Finding the Final Ratio

We need to find the ratio of participants who participated only in SSC to participants who participated only in PO.

Total participants only in SSC = 90

Total participants only in PO = 90

Ratio (Only SSC) : (Only PO) = 90 : 90

Simplifying the ratio by dividing both sides by 90:

Ratio = 1 : 1

Therefore, the ratio of participants who participated only in SSC to only in PO is 1:1.