In a spelling competition, competitors were required to spell 108

Question

In a spelling competition, competitors were required to spell 108 diﬀerent words. How many of these words contained the letter T but did not contain the letter U?

(1) Every word that contained the letter Q also contained the letter U, and every word that did not contain the letter Q did contain the letter T.

(2)  \frac{1}{3}  of the words contained the letter U.

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Rucha.Shukla · Answer

IMO E

Because we don't know if every word that did not contain the letter Q but did contain the letter T, did not contain the letter U

Meaning, Option 1 only says words with Q have U but doesn't clarify that words without Q are also without U

Sukrit_08 · Answer

imo E

1) Says if a letter has Q it has U, and if a letter does not have Q does, it has T

Insufficient as does not give any information about number of letters that have Q,U or T

2) 1/3 of the words have U but does not give any information about T

Insufficient alone

1) + 2)

We know that since 1/3 of the words have U that means there are exactly 1/3 of the words that have Q => 2/3 of the words don't contain Q => at least 2/3 words have T because it may be possible that the letters that do have Q also have T because the condition is that if it does not have Q it has T but not if it has Q it does not have T.

There fore, both together insufficient.

stne · Answer

(1) Every word that contained the letter Q also contained the letter U, and every word that did not contain the letter Q did contain the letter T.

If we only knew the number of words with the letter Q we could answer the question. But we don't.

INSUFF.

(2)  \frac{1}{3}  of the words contained the letter U.

Thus  \frac{2}{3}  of  108   or  72  words did not contain U. But we know nothing of T.

INSUFF.

1+2

Total words = words with Q  +  words without Q

Words with Q = Words with U  = 36 ...from statement (2)

Words without Q and U  = 108-36 = 72   -> Contained T

SUFF.

Ans C

Hope   it helped.

ribbons · Answer

Cant it be that it doesnt contain Q but contains U aka both U and T?

Madhvendrasinh · Answer

But statement 1 talks only about the words that conatin letter Q. It does not talk about every word with letter U. A word that contains letter Q must have a letter U but that doesn't mean that every word containing letter U will have letter Q.

I mean there can be words that contain letter U but not letter Q. Since the total words with letter U is not clear IMO E

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stne · Answer

Statement 1 talks of two things:

1) Words with Q
2) Words without Q

Total words: Words with Q + Words without Q

(a)Words with Q will contain the letter U
(b)Words without Q will contain the letter T.

You are taking of words that will NOT contain Q but will have to contain both T and U.

I think from point (a) and (b) above the question states the conditions for "U" and "T" to exist in a word.

Hence IMO "U " and "T" cannot exist together. Hope it helps.

Vivek1707 · Answer

This question is involves more verbal skills (specifically conditional reasoning) than math skills

Some back ground

When we write  If x then y , think of  x  as a  smaller circle  which is inside the  larger circle which is y .

" Every  " ---> x part

For instance, if I say  every  guitarist is a musician ---> it would mean that  if  someone is a  guitarist  ---->  then  that person is a  musician

So here Statement 1

"  Every  word that contained the letter Q also contained the letter U, and  every  word that did not contain the letter Q did contain the letter  T."

If Q ----> U implying that Q is the smaller circle inside U (the larger circle). This also implies that  region outside Q must have T

So, this just tells us that the region  outside Q contains T  but we have  no way of finding out how many words  are there which contain T but not U  Not sufficient.

Statement 2 ---> This just tells us that 36 words contain U again 72 words don't contain U. But we don't know out of those 72 how many contain T  Not sufficient

When you combine ---> since we know that Q is inside U and the region outside Q contains T, we know that since 72 is the region outside U ---> it MUST contain T ---> hence 72 is the number words that contain T but not U  sufficient . i.e. Answer is C

gullyboy09 · Answer

Hi  Bunuel  can you please help with this question. One without Q will have T, but they can have U also. So 36 words with U will be distributed in Q with U and NOT Q with T and U.

egmat · Answer

Hi gullyboy09,

You're right about one thing:  statement (1) never forbids a non-Q word from also containing U.  A word without Q must have T, but nothing stops it from carrying a U as well. So your instinct isn't wrong.

Here's the key move, though:  it doesn't matter for what we're counting.  The question asks for words with  T but not U . That group lives entirely inside the  non-U  words - and the non-U count is locked down.

Why the non-U group is fully determined: 
- Statement (2):  36  words have U, so  72  words do  not  have U.
- Statement (1): Q → U. So any word  without  U cannot have Q. That means all  72  non-U words are non-Q.
- Non-Q words all contain T. So all  72  non-U words have  T and not U .

Notice we never needed to know how the  36  U-words split up. Let me show that with your own scenario.

Case A - all U-words are Q-words: 
- Q-words (all with U) =  36 , non-Q =  72 , all with T, none with U.
- Words with T but not U =  72 .

Case B - some non-Q words also have U (your worry): 
- Q-words (all with U) =  30 ; plus  6  non-Q words that have both U and T.
- U total =  30  +  6  =  36  ✓, non-U =  72 , all with T.
- Words with T but not U = still  72 .

Same answer both ways. The  6  non-Q words that grabbed a U simply fall into the  U-group  - they were never candidates for "T but not U" in the first place, so they can't change our count.

Takeaway:  when a distribution you can't pin down (the U-words) sits entirely  outside  the group you're counting, it can't create a second answer - so the statements together are  sufficient .

Answer: C

In a spelling competition, competitors were required to spell 108

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In a spelling competition, competitors were required to spell 108

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615 to 715 in 15 Days (Ria's Strategies)

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My Rewards