SOLUTION: I am having a lot of trouble with word problem questions. If someone could help me that would be great. If you could show your work so that I can understand how you got the answer

You need to memorize this for the conditional:

When you have "IF p THEN q"   

1. "IF p THEN q" is written as "p -> q"
2. p, the object of "IF", is the antecedent and the q, the object of 
"THEN", is the consequence.
3. The converse of p -> q is gotten by reversing the antecedent and 
the consequence and getting q -> p
4. The inverse of p -> q is gotten by negating the antecedent and 
the consequence and getting ~p -> ~q.
4. The contrapositive of p -> q is gotten by both reversing the 
antecedent and the consequence and negating them ~q -> ~p

Remember this:

"IF ANTECEDENT, THEN CONSEQUENCE"{ 


(a) If I don’t go to the movie, I’ll study my math. 

Always look for an IF and a THEN, even when they are only understood.
Here the THEN is understood:

IF I don’t go to the movie, THEN I’ll study my math.

Here p = "I don't go to the movie" = the antecedent
     q = "I'll study my math"      = the consequence

and the statement is p -> q

CONVERSE: q -> p    (switch them but don't negate them)

If I’ll study my math, then I don’t go to the movie.

Now we know what that means, but people don't talk like that,
so we have to re-write it by changing the tenses of the verbs, say, as:

If I end up having studied my math, then I won’t have gone to the movie.

INVERSE: ~p -> ~q   (negate them but don't switch them")

If I’ll NOT study my math, then I don’t NOT go to the movie.

Again, we know what that means, but, as we said before, people don't 
talk like that, so we have to re-write it by changing the tenses of the 
verbs, and removing the double negative "don't NOT" 

If I do NOT study my math, then I will go to the movie.

CONTRAPOSITIVE: ~q -> ~p  (negate them and switch them)

If I NOT study my math, then I don’t NOT go to the movie.

We know what that means, but people don't talk like that,
so we have to re-write it, say, as:

If I don't study my math, then I will go to the movie.

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(b) Your car won’t start if you don’t have gasoline in the tank. 

This one you have to rewrite the original with the IF clause FIRST:


IF you don’t have gasoline in the tank, THEN your car won't start.

So the antecedent is "you don’t have gasoline in the tank" and
the consequence is "your car won't start".

To get the converse, just switch them:

IF your car won't start THEN you don’t have gasoline in the tank.

To get the inverse just negate them only.

IF you don’t NOT have gasoline in the tank, THEN your car won't NOT start.  

Now get rid of the double negatives "don't NOT" and "wont NOT":

IF you have gasoline in the tank, THEN your car will start.

To get the contrapositive, switch them and also negate them:

IF your car won't NOT start THEN you don’t NOT have gasoline in the tank.

Now get rid of the double negatives:

IF your car will start THEN you do have gasoline in the tank.

Edwin