Let's break down the problem step by step and formulate the required equations.

Problem Breakdown

1. Total number of vehicles:

   • Cars: 125
   • Motorcycles: $3p$
   • Lorries: $2q$
   • Buses: 20

   Therefore, the total number of vehicles in the car park is: $\text{Total Vehicles} = 125 + 3p + 2q + 20 = 145 + 3p + 2q$

2. Probability that the vehicle is a motorcycle:

   • Given probability: $\frac{3}{40}$

   The probability of selecting a motorcycle is given by: $\frac{\text{Number of motorcycles}}{\text{Total number of vehicles}} = \frac{3p}{145 + 3p + 2q}$ Equating this to the given probability: $\frac{3p}{145 + 3p + 2q} = \frac{3}{40}$

3. Probability that the vehicle is a bus:

   • Given probability: $\frac{1}{10}$

   The probability of selecting a bus is given by: $\frac{\text{Number of buses}}{\text{Total number of vehicles}} = \frac{20}{145 + 3p + 2q}$ Equating this to the given probability: $\frac{20}{145 + 3p + 2q} = \frac{1}{10}$

Solving the Equations

Now let's solve these two equations to find the values of $p$ and $q$.

First Equation:

$\frac{3p}{145 + 3p + 2q} = \frac{3}{40}$

Cross-multiply to solve for $3p$: $3p \times 40 = 3 \times (145 + 3p + 2q)$ $120p = 435 + 9p + 6q$ Bring all terms involving $p$ and $q$ to one side: $120p - 9p - 6q = 435$ $111p - 6q = 435$ Divide the entire equation by 3: $37p - 2q = 145 \quad \text{(Equation 1)}$

Second Equation:

$\frac{20}{145 + 3p + 2q} = \frac{1}{10}$

Cross-multiply to solve for 20: $20 \times 10 = 145 + 3p + 2q$ $200 = 145 + 3p + 2q$ Bring all terms involving $p$ and $q$ to one side: $3p + 2q = 200 - 145$ $3p + 2q = 55 \quad \text{(Equation 2)}$

Solving the System of Equations

Now we have two equations:

1. $37p - 2q = 145$
2. $3p + 2q = 55$

Add these two equations to eliminate $q$: $(37p - 2q) + (3p + 2q) = 145 + 55$ $40p = 200$ $p = \frac{200}{40} = 5$

Substitute $p = 5$ into Equation 2: $3(5) + 2q = 55$ $15 + 2q = 55$ $2q = 40$ $q = \frac{40}{2} = 20$

Final Answer

• $p = 5$
• $q = 20$

Would you like further details on any part of this solution or have any questions?

Here are some additional questions related to this problem:

1. How would the equation change if the number of cars increased?
2. What if the probability of selecting a lorry was given instead?
3. How can we verify the solution using the values of $p$ and $q$?
4. What is the probability of selecting a lorry given these values of $p$ and $q$?
5. How would the problem change if another type of vehicle was added?

Tip: When forming equations from word problems, ensure each term in the equation accurately reflects the scenario described, especially in terms of probabilities and total quantities.