SSC & PO Exam Participant Analysis

The question asks us to find the percentage of boys who participated only in the PO exam compared to the boys who participated only in the SSC exam, based on the given ratios and relationships between participants in SSC, PO, and both exams.

Let's break down the information provided:

• Participants are divided into Boys and Girls for SSC, PO, Both, Only SSC, and Only PO events.
• Ratio of Boys to Girls in SSC (Total): 4 : 3
• Ratio of Boys to Girls in PO (Total): 8 : 13
• Ratio of Boys to Girls in Both (SSC & PO): 2 : 1
• Ratio of Boys to Girls in Only PO: 1 : 2
• Number of Boys who participate only in SSC = Number of Girls who participate only in SSC + 10
• Number of Boys who participate in Both = 20% of the Number of Boys who participate only in SSC.

Let's define variables for the number of participants in each category:

• B_{Only SSC}: Boys who participated only in SSC
• G_{Only SSC}: Girls who participated only in SSC
• B_{Only PO}: Boys who participated only in PO
• G_{Only PO}: Girls who participated only in PO
• B_Both: Boys who participated in both SSC and PO
• G_Both: Girls who participated in both SSC and PO

From the given information, we can write the following relationships:

1. B_Both = 20% of B_{Only SSC} ⇒ B_Both = 0.20 × B_{Only SSC}

2. Ratio of Boys to Girls in Both is 2:1 ⇒ B_Both / G_Both = 2 / 1 ⇒ G_Both = B_Both / 2

3. B_{Only SSC} = G_{Only SSC} + 10

4. Ratio of Boys to Girls in SSC (Total) is 4:3 ⇒ (B_{Only SSC} + B_Both) / (G_{Only SSC} + G_Both) = 4 / 3

5. Ratio of Boys to Girls in PO (Total) is 8:13 ⇒ (B_{Only PO} + B_Both) / (G_{Only PO} + G_Both) = 8 / 13

6. Ratio of Boys to Girls in Only PO is 1:2 ⇒ B_{Only PO} / G_{Only PO} = 1 / 2 ⇒ G_{Only PO} = 2 × B_{Only PO}

Step-by-Step Calculation

Let's use equations to find the number of participants in each category.

Substitute (1) and (2) into (4):

$ \frac{B_{Only SSC} + 0.20 \times B_{Only SSC}}{G_{Only SSC} + (0.20 \times B_{Only SSC})/2} = \frac{4}{3} $

$ \frac{1.20 \times B_{Only SSC}}{G_{Only SSC} + 0.10 \times B_{Only SSC}} = \frac{4}{3} $

$ 3 \times (1.20 \times B_{Only SSC}) = 4 \times (G_{Only SSC} + 0.10 \times B_{Only SSC}) $

$ 3.60 \times B_{Only SSC} = 4 \times G_{Only SSC} + 0.40 \times B_{Only SSC} $

$ 3.60 \times B_{Only SSC} - 0.40 \times B_{Only SSC} = 4 \times G_{Only SSC} $

$ 3.20 \times B_{Only SSC} = 4 \times G_{Only SSC} $

$ G_{Only SSC} = \frac{3.20}{4} \times B_{Only SSC} = 0.80 \times B_{Only SSC} $

Now substitute this into equation (3):

$ B_{Only SSC} = (0.80 \times B_{Only SSC}) + 10 $

$ B_{Only SSC} - 0.80 \times B_{Only SSC} = 10 $

$ 0.20 \times B_{Only SSC} = 10 $

$ B_{Only SSC} = \frac{10}{0.20} = \frac{1000}{20} = 50 $

So, the number of Boys who participated only in SSC is 50.

Now we can find other values:

• G_{Only SSC} = B_{Only SSC} - 10 = 50 - 10 = 40
• B_Both = 0.20 × B_{Only SSC} = 0.20 × 50 = 10
• G_Both = B_Both / 2 = 10 / 2 = 5

Now use equation (5) and (6) to find B_{Only PO} and G_{Only PO}:

$ \frac{B_{Only PO} + B_{Both}}{G_{Only PO} + G_{Both}} = \frac{8}{13} $

Substitute B_Both = 10, G_Both = 5, and G_{Only PO} = 2 × B_{Only PO}:

$ \frac{B_{Only PO} + 10}{2 \times B_{Only PO} + 5} = \frac{8}{13} $

$ 13 \times (B_{Only PO} + 10) = 8 \times (2 \times B_{Only PO} + 5) $

$ 13 \times B_{Only PO} + 130 = 16 \times B_{Only PO} + 40 $

$ 130 - 40 = 16 \times B_{Only PO} - 13 \times B_{Only PO} $

$ 90 = 3 \times B_{Only PO} $

$ B_{Only PO} = \frac{90}{3} = 30 $

So, the number of Boys who participated only in PO is 30.

Also, G_{Only PO} = 2 × B_{Only PO} = 2 × 30 = 60.

Let's summarize the number of participants in each category:

The ratios check out:

• Total SSC Boys : Girls = 60 : 45 = 4 : 3 (Correct)
• Total PO Boys : Girls = 40 : 65 = 8 : 13 (Correct)
• Both Boys : Girls = 10 : 5 = 2 : 1 (Correct)
• Only PO Boys : Girls = 30 : 60 = 1 : 2 (Correct)
• Boys Only SSC (50) = Girls Only SSC (40) + 10 (Correct)
• Boys Both (10) = 20% of Boys Only SSC (50) ⇒ 10 = 0.20 * 50 (Correct)

Calculating the Final Percentage

The question asks: The number of Boys who participated only in PO exam is what percent of Boys who participated only in SSC events?

Percentage = $ \frac{\text{Number of Boys who participated only in PO}}{\text{Number of Boys who participated only in SSC}} \times 100 $

Percentage = $ \frac{B_{Only PO}}{B_{Only SSC}} \times 100 $

Percentage = $ \frac{30}{50} \times 100 $

Percentage = $ \frac{3}{5} \times 100 $

Percentage = $ 0.6 \times 100 $

Percentage = 60%

Therefore, the number of Boys who participated only in PO exam is 60% of the Boys who participated only in SSC events.