Questions: A new study company plans to spend 2 million on 20 new vehicles. Each van will cost 50,000, and each truck costs 100,000 and each pickup truck costs 40,000. Pari operations plan has: This company can buy □ vans, trucks, and □ small trucks, and □ large trucks.

A new study company plans to spend 2 million on 20 new vehicles. Each van will cost 50,000, and each truck costs 100,000 and each pickup truck costs 40,000. Pari operations plan has:

This company can buy

□ vans, trucks, and

□ small trucks, and

□ large trucks.

Transcript text: A new study company plans to spend $2 million on 20 new vehicles. Each van will cost $50,000, and each truck costs $100,000 and each pickup truck costs $40,000. Pari operations plan has: This company can buy □ vans, trucks, and □ small trucks, and □ large trucks.

Solution

Solution Steps

Step 1: Define Variables

Let $ v $ represent the number of vans, $ t $ represent the number of trucks, and $ p $ represent the number of pickup trucks.

Step 2: Set Up Equations

We have two main conditions given in the problem:

The total number of vehicles is 20: \[ v + t + p = 20 \]
The total cost of the vehicles is $2 million: \[ 50000v + 100000t + 40000p = 2000000 \]

Step 3: Simplify the Cost Equation

Divide the cost equation by 10,000 to simplify: \[ 5v + 10t + 4p = 200 \]

Step 4: Solve the System of Equations

We now have the system of equations:

$ v + t + p = 20 $
$ 5v + 10t + 4p = 200 $

Let's solve this system using substitution or elimination. First, solve the first equation for $ v $: \[ v = 20 - t - p \]

Substitute $ v = 20 - t - p $ into the second equation: \[ 5(20 - t - p) + 10t + 4p = 200 \]

Simplify: \[ 100 - 5t - 5p + 10t + 4p = 200 \]

Combine like terms: \[ 5t - p = 100 \]

Step 5: Solve for $ t $ and $ p $

From the equation $ 5t - p = 100 $, express $ p $ in terms of $ t $: \[ p = 5t - 100 \]

Substitute $ p = 5t - 100 $ back into the equation $ v + t + p = 20 $: \[ v + t + (5t - 100) = 20 \]

Simplify: \[ v + 6t - 100 = 20 \]

Solve for $ v $: \[ v = 120 - 6t \]

Final Answer

The system of equations is:

$ v = 120 - 6t $
$ p = 5t - 100 $

The solution depends on the integer values of $ t $ that satisfy both $ v \geq 0 $ and $ p \geq 0 $. The specific values of $ v $, $ t $, and $ p $ will depend on additional constraints or specific integer solutions. However, the general form of the solution is:

\[ \boxed{v = 120 - 6t, \quad t = t, \quad p = 5t - 100} \]

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