Let  represent the sets of people who like apples, bananas, etc. , respectively. Let  represent Smith.

Assume Statement 1 alone. 

If , then . Contrapositively, if  , then . Therefore,  cannot be in both  and ;  is in one, the other, or neither. By similar reasoning,  is in at most one of  and  and at most one of  and . 

Suppose  and . Then  is in three of , and either  falls in both  and  or both  and ; since both are impossible, it follows that  or , but not both. Therefore, Smith likes exactly one of bananas and figs, but not both. Since, from Statement 1, Smith doesn't like bananas, he must like figs.

Assume Statement 2. By similar reasoning, Smith likes cranberries or dates, but not both. Since he does not like cranberries, he likes dates. However, without further information, it cannot be determined what else he likes or dislikes.

Assume Statement 1 alone. Let  represent the sets of Beeps, Deeps, Meeps, and Veeps. Then  and, since all Veeps are not Deeps, . This is represented by the Venn diagram below:

Note that the set  (Meeps) need not fall completely inside the set  (Veeps). Therefore, Harpy being a Meep does not necessarily make him a Veep or not a Veep.

Assume Statement 2 alone. Every Beep is a Deep, but no Veep is a Deep. If Harpy is a Beep, then he is a Deep. If Harpy were a Veep, then he cannot be a Deep, so, by contradiction, Harpy cannot be a Veep.

Assume Statement 1 alone.  is the intersection of the complements of all three of , , and  - the sets of people who do not like Neil Young, people who do not like Prince, and people who do not like Marvin Gaye. Any person who is in this intersection does not like any of these three. , so Phillip does not like Neil Young, Prince, or Marvin Gaye.

Assume Statement 2 alone.  is the union of all three , , and . Any person who is in this union likes any one, two, or three of these musicians. However, , which is the complement of this union - the set of people who like none of Neil Young, Prince, or Marvin Gaye. Phil is in this set.

For statement (1), she spent

on rent, and

on clothes.

So she spent  in total.

Therefore, she has  left.

For statement (2), we can set Clara’s salary to be , then we have

Then simplify the equation:

Therefore, .

Now we can just follow what we did for the first statement to calculate the money he had left.

Think of this in terms of "greenhouses per minute", not "minutes per greenhouse". Converting hours and minutes to just minutes, Steve can paint  greenhouses per minute; Phil can paint  greenhouses per minute. 

If we let  be the time in minutes that it takes to paint the greenhouse, then Steve and Phil paint   and  greenhouses, respectively; since one greenhouse total is painted, then we can add the labor and set up this equation:

Simplify and solve:

or 2 hours, 38 minutes.

A work problem is actually a rate problem in disguise.

If you know that an object working alone can do a job in  hours, then you know that the object works at a rate of  jobs per hour. After  hours, the object accomplishes  of a job. Similarly, the other object working alone does a job in  hours, and therefore does  of a job. Together, the objects do one whole job, so solve this equation 

for .

Statement 1 alone gives us half the picture; , but  is unknown.

Statement 2 alone tells us that  is 75% of . But  is unknown.

From Statement 1 and 2 together, we know

 is 75% of  - this allows us to calculate :

75% of  is , so . Since we have both  and , we have the complete equation

and we can calculate .

A work problem is actually a rate problem in disguise.

If you know that an object working alone can do a job in  hours, then you know that the object works at a rate of  jobs per hour. After  hours, the object accomplishes  of a job. Similarly, the other object working alone does a job in  hours, and therefore does  of a job. Together, the objects do one whole job, so solve this equation 

for .

Statement 1 alone tells us that , and Statement 2 alone tells us that .

Therefore, each statement alone gives us only half the picture, but together, they give us the equation

,

which can be solved to yield the answer. 

A work problem is actually a rate problem in disguise.

If you know that David working alone can do a job in  hours, then you know that the object works at a rate of  jobs per hour. After  hours, the object accomplishes  of a job. Similarly, Eddie and Floyd, working without David, do a job in  hours, and therefore does  of a job. Together, the objects do one whole job, so solve this equation 

for .

Statement 1 alone tells us that , and Statement 2 alone tells us that ; each one alone leaves the other value unknown. However, if both statements are given, the equation 

can be solved for  to yield the correct answer.

Since the group is now without the help of Mrs. Smith, we look at Mrs. Smith's contribution to the work as a whole, and the sum of the other two ladies' contribution as a whole; Statement 2, which deals with Mrs. Hume alone, is irrelevant and unhelpful.

Since the three ladies together wrote 400 invitations in 120 minutes, we can infer that they would have spent three-fourths of this time, or 90 minutes, writing 300 invitations.

If Statement 1 is true, then Mrs. Smith, who can write 100 invitations in 90 minutes, would take three times this, or 270 minutes, to write 300 invitations. 

A work problem is a rate problem in disguise. Think in terms of "jobs per hour", and take the reciprocal of each "hours per job". The three ladies together would have done  job in one hour, and Mrs. Smith alone would have done  job in one hour.

Therefore, in one hour today, the two ladies will do

jobs, and in  hours today, they will do

job.

Solve for  in this equation. 

This proves that Statement 1 alone allows us to find the answer - but not Statement 2.

The total time is calculated by the equation . Statement 2 provides the rate, but we have no information regarding distance, therefore, the quesiton is impossible to solve without more information.