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Palestrina’s music is more beautiful than that of Tallis. Gregorian chant is more beautiful than Palestrina’s music. Tallis’ music is less beautiful than Gregorian chant. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is more beautiful than,” and “less than” (<) will mean “is less beautiful than.”
(1) Palestrina’s music is more beautiful than that of Tallis: Palestrina > Tallis
(2) Gregorian chant is more beautiful than Palestrina’s music: Gregorian chant > Palestrina
Thus, we can combine them to get:
Gregorian chant > Palestrina > Tallis
This is the same thing as saying that Tallis' music is less beautiful than Gregorian chant.
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Tallis’ music is more enjoyable than Byrd’s. Haydn’s music is less enjoyable than Byrd’s. Tallis’ music is less enjoyable Haydn’s. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is more enjoyable than,” and “less than” (<) will mean “is less enjoyable than.”
(1) Tallis’ music is more enjoyable than Byrd’s: Tallis > Byrd
(2) Haydn’s music is less enjoyable than Byrd’s: Haydn < Byrd
This could be rewritten: Byrd > Haydn
Thus, combining everything, we would have:
Tallis > Byrd > Haydn
It is false to say that Tallis' music is less enjoyable than Haydn's.
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Diego is the best athlete in his high school, but Jocelyn is the best high school athlete in the entire city. Allison is not an athlete, but is an outstanding concert pianist. Diego is a better athlete than both Jocelyn and Allison. If the first two sentences are true, the third sentence is ______________.
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.
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Jeremy is taller than Jean. John is shorter than Jeremy. John is shorter than Jean. If the first two sentences are true, the third is __________.
Let's use symbols from math to help us understand this:
(1) Jeremy is taller than Jean: Jeremy > Jean
(2) John is shorter than Jeremy: John < Jeremy
This could be rewritten: Jeremy > John
Thus, we know that Jeremy is taller than Jean and John; however, we do not know anything about the relationship between John and Jean.
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Thomas is smarter than Luca. Luca is smarter than John. John is smarter than Thomas. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this.
(1) Thomas is smarter than Luca: Thomas > Luca
(2) Luca is smarter than John: Luca > John
We could string these together as well:
Thomas > Luca > John
John is not smarter than Thomas; therefore, this is false.
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Baruch is weirder than Frederick. Frederick is weirder than Arthur. Baruch is weirder than Arthur. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this.
(1) Baruch is weirder than Frederick: Baruch > Frederick
(2) Frederick is weirder than Arthur: Frederick > Arthur
These can all be combined to give us:
Baruch > Frederick > Arthur
It is indeed the case that (3) Baruch > Arthur.
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Godfrey is more unknown than Henry. Henry is better known than Giles. Giles is more unknown than Godfrey. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. "Greater than" (>) will mean "more known," and "less than" (<) will mean "more unknown."
(1) Godfrey is more unknown than Henry: Godfrey < Henry
This could be rewritten: Henry > Godfrey
(2) Henry is better known than Giles: Henry > Giles
Thus, we know that Henry is more known than both Giles and Godfrey; however, this does not tell us anything about the relationship between Giles and Godfrey. Therefore (3) is uncertain.
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Teddy has more fun than Jack. Franklin has less fun than Jack. Teddy has more fun than Franklin. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. "Greater than" (>) will mean "has more fun," and "less than" (<) will mean "has less fun."
(1) Teddy has more fun than Jack: Teddy > Jack
(2) Franklin has less fun than Jack: Franklin < Jack
This could also be written: Jack > Franklin
Therefore, we could string all of these together into:
Teddy > Jack > Franklin
Therefore, (3) Teddy > Franklin does hold.
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Andrew is taller than Abraham. Abraham is shorter than Charles. Charles is taller than Andrew. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “taller,” and “less than” (<) will mean “shorter.”
(1) Andrew is taller than Abraham: Andrew > Abraham
(2) Abraham is shorter than Charles: Abraham < Charles
This could be rewritten: Charles > Abraham
Another way of looking at these two would be:
(1) Abraham < Andrew
(2) Abraham < Charles
This tells us that Abraham is shorter than Andrew and Charles. It tells us nothing about the relationship between Charles and Andrew.
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Physics is more difficult than chemistry. Chemistry is easier than philosophy. Philosophy is easier than physics. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “more difficult,” and “less than” (<) will mean “easier.”
(1) Physics is more difficult than chemistry: Physics > chemistry
This could be rewritten: Chemistry < physics
(2) Chemistry is easier than philosophy: Chemistry < philosophy
Therefore, this tells us that chemistry is easier than the other two subjects. However, it does not tell us anything about the relationship between philosophy and physics.
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It is easier to drive in snow than it is in ice. It is harder to drive in snow than in rain. It is easier to drive in rain than in ice. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “harder to drive in,” and “less than” (<) will mean “easier to drive in.”
(1) It is easier to drive in snow than it is in ice: snow < ice
(2) It is harder to drive in snow than in rain: snow > rain
This could be rewritten: rain < snow.
All of these could be combined to give us: rain < snow < ice.
This means that rain is indead easier to drive in that ice.
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Robert runs more quickly than Nicole. Nicole runs more quickly than Fred. Fred runs more slowly than Robert. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “runs more quickly,” and “less than” (<) will mean “runs more slowly.”
(1) Robert runs more quickly than Nicole: Robert > Nicole
(2) Nicole runs more quickly than Fred: Nicole > Fred
This could give us a combined statement:
Robert > Nicole > Fred
Now, (3) could be rewritten: Fred < Robert. This is the same thing as Robert > Fred, which we do see above as being true.
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Nuclear power is more efficient than coal power. Coal power is less efficient than gas power. Gas power is more efficient than nuclear power. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “more efficient than,” and “less than” (<) will mean “less efficient than.”
(1) Nuclear power is more efficient than coal power: nuclear > coal
This could be rewritten as: coal < nuclear
(2) Coal power is less efficient than gas power: coal < gas
Thus, we are told that coal is less efficient than both nuclear and gas; however, we are told nothing about the relationship between nuclear and gas power.
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The tuba has a more beautiful tone than the trumpet. The flute has a less beautiful tone than the trumpet. The tuba has a less beautiful tone than the flute. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “has a more beautiful tone,” and “less than” (<) will mean “has a less beautiful tone.”
(1) The tuba has a more beautiful tone than the trumpet: tuba > trumpet
(2) The flute has a less beautiful tone than the trumpet: flute < trumpet
This could be rewritten: trumpet > flute
Therefore, we could combine these to get:
tuba > trumpet > flute
Therefore, it is not true to say that the tuba has a less beautiful tone than the flute.
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Valerie is kinder than Carol. Patricia is much nastier than Carol. Patricia is kinder than Valerie. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is kinder than,” and “less than” (<) will mean “is nastier than.”
(1) Valerie is kinder than Carol: Valerie > Carol
(2) Patricial is much nastier than Carol: Patricia < Carol
This can be rewritten: Carol > Patricia
Thus, we can combine all of these into:
Valerie > Carol > Patricia
Therefore, it is not true to say that Patricia is kinder than Valerie.
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Corn is a more popular vegetable than broccoli. Broccoli is more popular than asparagus. Asparagus is more popular than corn. If the first two sentences are true, then the third sentence is ___________.
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.
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Sandra has more shoes than Ellen and Joan. Ellen has more shoes than Kerri. Kerri has more shoes than Joan. If the first two sentences are true, than the third sentence is ___________.
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.
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There are more cabs than trucks on the roads in New York City. There are more motorcycles than there are trucks on the roads in New York City. There are more cabs than motorcycles on the roads in New York City. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “more on the road,” and “less than” (<) will mean “less on the road.”
(1) There are more cabs than trucks on the roads in New York City: cabs > trucks
This could be rewritten as: trucks < cabs
(2) There are more motorcycles than there are trucks on the roads in New York City: motorcycles > trucks
This could be rewritten as: trucks < motocycles
Thus, we are actually only being told something about trucks—there are fewer of them than both cabs and motorcycles (in New York City). Thus, we cannot say anything about the relationship between the number of cabs and the number of motorcycles.
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Gremlins are more frightening than ogres. Skeletons are less frightening than ogres. Gremlins are more frightening than skeletons. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “more frightening,” and “less than” (<) will mean “less frightening.”
(1) Gremlins are more frightening than ogres: gremlins > ogres
(2) Skeletons are less frightening than ogres: skeletons < ogres
This could be rewritten as: ogres > skeletons
All together, we can say:
gremlins > ogres > skeletons
Therefore, it is indeed the case that gremlins are more frightening than skeletons.
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George has a greater salary than John. Filbert has a lower salary than George. John’s salary is less than Filbert's. If the first two sentences are true, the third is __________.
Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “has a greater salary than,” and “less than” (<) will mean “has a lower salary than.”
(1) George has a greater salary than John: George > John
(2) Filbert has a lower salary than George: Filbert < George
This could be rewritten as: George > Filbert
This means that we are being told that George has a greater salary than both John and Filbert; however, it tells us nothing about the relation between these latter two men.
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