Let's break down the problem step by step.

Step 1: Defining Variables

Let $P$ represent the total number of participants in the race.

• $\frac{1}{5}$ of the participants were adults, so the number of adults is $\frac{1}{5}P$.
• The remaining participants were children, so the number of children is $P - \frac{1}{5}P = \frac{4}{5}P$.
• $\frac{4}{7}$ of the adults were men. Therefore, the number of men is $\frac{4}{7} \times \frac{1}{5}P = \frac{4}{35}P$.
• We are told there are 51 boys, so the number of boys is 51.

Step 2: Analyzing Remaining Participants

• After all the men and boys left, $\frac{2}{5}$ of the participants remained. This means $\frac{3}{5}$ of the participants left the race.
• The participants who left were all the men and boys. So, the number of people who left is the sum of the number of men and boys, i.e., $\frac{4}{35}P + 51$.
• Since $\frac{3}{5}$ of the total participants left, we have the equation: $\frac{3}{5}P = \frac{4}{35}P + 51$

Step 3: Solving the Equation

Let's solve for $P$:

1. Multiply both sides by 35 to eliminate the fractions: $35 \times \frac{3}{5}P = 35 \times \left( \frac{4}{35}P + 51 \right)$ Simplifying both sides: $21P = 4P + 1785$
2. Subtract $4P$ from both sides: $17P = 1785$
3. Divide both sides by 17: $P = \frac{1785}{17} = 105$

So, there were 105 participants in total.

Step 4: Fraction of Participants that were Men

• We already know that the number of men is $\frac{4}{35}P$.
• Substituting $P = 105$ into this: $\text{Number of men} = \frac{4}{35} \times 105 = 12$ Thus, the fraction of the total participants that were men is: $\frac{12}{105} = \frac{4}{35}$

Final Answers:

1. The fraction of the total participants that were men is $\frac{4}{35}$.
2. The total number of participants was 105.

Would you like more details on any part of the solution?

Here are 5 related questions to extend this topic:

1. What fraction of the participants were boys?
2. How many children were there in total?
3. What fraction of the participants were women?
4. How many women participated in the race?
5. How many girls were there among the participants?

Tip: When solving word problems, defining variables early can help clarify complex relationships and make equations easier to set up.