We know that roses smell better than geraniums and that geraniums smell worse than lilacs. That means that both roses and lilacs smell better than geraniums. There is no information, though, about whether lilacs or roses smell best. Therefore, the third sentence is uncertain.

Although Diego is the best athlete in his high school, Jocelyn is the best high school athlete in the entire city. This means both that she must go to a different high school and that she is a better athlete than Diego. Therefore, the statement that Diego is a better athlete than both Jocelyn and Allison is false.

From the information in only the first two sentences, we know that blue is the most popular color of the three; green is the next most popular; and purple is the least popular of these three colors. Orange is only introduced in the third sentence, so we have no way of knowing, looking at only the first two sentences, if purple is more popular than orange.

We know that Anika's house is less expensive than both Helena's house and Stephen's house. From the first two sentences only, however, we have nothing to compare Stephen and Helena's houses. We only know that they are both more expensive than Anika's house. Therefore, the third sentence is undetermined.

Let's look at the problem visually by putting the three dogs on a spectrum. The ugliest dog should be on the right, and the cutest dog is on the left. If Boomer is cuter than Pepper, the spectrum would look like this:

Boomer----Pepper

If Pepper is not as cute as Princess, Princess goes to the left of Pepper. We don't know where Princess is on the spectrum compared to Boomer, so we'll put them together temporarily.

Boomer & Princess ---- Pepper

Since the third sentence says that BOTH Boomer and Princess are cuter than Pepper, we don't need to know where they are on the spectrum compared to each other. The third sentence is true.

Let's look at this problem mathematically.  Where soccer players = s; basketball players = b; and football players = f, the equations would like:

It requires twice as many soccer players to equal the number of basketball players. It requires twice as many basketball players to equal the number of football players. To determine if the third sentence is true, we must multiply the first equation by 2 which gives us:

We can now determine that:

Divide each side by 4 to isolate s.

This means that  times as many athletes play high school football as soccer, so the third sentence is true.

Let's look at the problem visually by putting all four people on a spectrum. The youngest person should be on the right, and the oldest person is on the left. If Alicia is older than Frank and Simon, the spectrum looks like:

Alicia----Frank & Simon

If Frank and Simon are older than Anthony, the spectrum looks like:

Alicia----Frank & Simon----Anthony

The third sentence is true: Anthony is younger than Alicia.

Let's look at the problem visually by putting the three vegetables on a spectrum. The least popular vegetable should be on the right, and the most popular vegetable is on the left. If corn is more popular than broccoli, it goes to the left of broccoli on the spectrum:

corn----broccoli

If broccoli is more popular than asparagus, asparagus should be placed to the right of broccoli on the spectrum.

corn----broccoli----asparagus

The third sentence stating that asparagus is more popular than corn is false.

For this problem, we have to create two spectrum: one for the number of Russian-speaking students at each high school and one for the number of Spanish-speaking students at each high school. The school with the most Russian or Spanish-speaking students will be placed on the left of the spectrum, while the school with the least Russian or Spanish-speaking students will be placed on the right of the spectrum.

The first half of the first sentence tells us that Kennedy HS has more Russian-speaking students than Lincoln HS. So, our Russian spectrum looks like:

Russian-speaking students:

Kennedy----Lincoln

The second half of the first sentence tells us that Kennedy HS has fewer Spanish-speaking students than Roosevelt HS. Our Spanish spectrum looks like:

Spanish-speaking students:

Roosevelt----Kennedy

The first half of the second sentence tells us that Roosevelt HS has more Spanish-speaking students than Lincoln HS. We don't know whether Kennedy HS or Lincoln HS has more Spanish-speaking students, so we'll put them together on the spectrum for now.

Spanish-speaking students:

Roosevelt----Kennedy & Lincoln

The second half of the second sentence tells us that Roosevelt HS has more Russian-speaking students than Kennedy HS. They'll be placed to the left of Kennedy on the Russian spectrum. After the second sentence, the two spectrums look like this:

Russian-speaking students:

Roosevelt----Kennedy----Lincoln

Spanish-speaking students:

Roosevelt---Kennedy & Lincoln

Therefore, the third sentence, stating that Roosevelt HS has more Russian-speaking students AND more Spanish-speaking students than either Kennedy HS or Lincoln HS is true.

We know that Sandra has more shoes than either Ellen or Joan and that Ellen has more shoes than Kerri. We do not know, however, how many shoes Ellen has compared to Joan. This is the key to determining whether Kerri has more shoes than Joan. Therefore, the third sentence is uncertain.