Word problem I can’t understand

Question

We are planning to have a beach day and need to get 54 people there using cars c which can take 4 people each and vans v which can take 6 people each.

Answer 1

Likely what's happening is that you need to set up two algebraic equations that describe the word problem in order to find the number of cars c and vans v needed to take all the people to the beach.

The first equation will allow you to relate the number of vans and cars required to get all 54 people to the beach (if you don't know the exact number of each).

To make it simpler, let's say we ONLY had vans to take the 54 people to the beach. What equation would you set up? You would say 6v = 54 because that's how you get the number of vans needed to transport the 54 people, which is just 9. But, we're now adding cars to the mix. Some mix of cars and vans will transport the 54 people to the beach. So, the first equation is just:

(1) 4c + 6v = 54

We need a second equation that relates the number of cars c to the number of vans v because we can't solve one equation with 2 unknowns. In order to find the second equation to set up, we need the total number of cars and vans (even though we don't know the exact number of each). For the sake of argument, let's just say there are 30 vehicles in total consisting of vans and cars. The second equation would then be:

(2) c + v = 30

Now, you have two equations and two unknowns, which is solvable using Substitution or Elimination.

Hope this was helpful.

Answer 2

Prior to having you set up and solve a certain type of word problem, your teacher may be focusing on the steps required. This should make it easier than having you work on a full problem to start with.

All you are being asked to do is to take the wording of one of the equations and convert it into equation form.

In this case, let us start by establishing the variables.

Let c = the number of cars needed;

and let v = the number of vans needed.

Since the cars in this problem hold at most 4 people, we can say 4c to mean the number of people each car can transport.

Similarly, 6v represents the number of people each van can transport.

So 4c + 6v represents the number of people that a certain number of cars and trucks can transport. In this case, we want this to be 54.

So, this information can be represented by the equation:

4c + 6v = 54

The SAT exam often gives you a word problem and tests you to see if you can set up the equation(s) correctly. Students often have a difficult time mastering this process, so it has become more common to teach the steps first rather than all of the problem techniques at the same time.

That is what I think your teacher is doing. You would need a second equation to be able to solve this "system".

I am sure you have graphed two lines before to find their intersection point (if one exists). The above equation is linear. Meaning that if you graphed it, it would be a straight line. It has some restrictions. You cannot have fractional nor negative people, so the only possible solutions are whole numbers.

But to find a unique solution along that line, you need another equation which intersects it.

So, your next step, after practicing setting up one equation in 2 unknowns, is to set up 2 equations in 2 unknowns.

Then you will be given the enire problem; setting up and solving 2 (linear) equations on 2 unknowns. This step-by -step process may seem slow to you, but you will better for it.

Answer 3

Since there wasn't a mention of the total number of vehicles, there are several answers.

If 4c+6v =54 then c =(27-3v)/2. From this, c can't be negative and c needs to be a whole number since we can't put a half a person in a car. So v is in the range [0,9] containing integers only.

Therefore combinations of whole people in vans and cars is:

v c

1 12

3 9

5 6

7 3

9 0