UMAT Sample Questions

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Construct 1 - Logical Reasoning and Problem Solving

 

Question 1

The following table shows the number and rate of deaths due to transport accidents and assaults for young Australians.

 

 

 

With respect to young Australians in 2004, which of the following can be best concluded from the table above?

 

A. A transport accident is a more common occurrence than an assault.

B. About 25% of the persons aged 18-24 dying from a transport accident were female.

C. 0.093% of females aged 18-24 died as a result of a transport accident.

D. Transport accidents and assaults are a major cause of death amongst the young Australian cohort.

 

Answer: B

Explanation: A is incorrect because although the number of deaths may be higher due to transport accidents, this does not imply that transport accidents occur more often (instead, assaults may occur more often but only lead to minor injuries, not death). C is incorrect because if 9.3 females die per 100,000 young females, then the percentage is 0.0093% (9.3/100000 = 0.000093 = 0.0093%). D is incorrect because this table does not make any comparison of transport accidents/assaults to other causes of death amongst young people, so it is impossible to conclude how major a cause these are. B is correct because there are 88 females in the 340 people aged 18-24 who died in a transport accident, which is equivalent to roughly 25%.

Source: Young Australians: their health and wellbeing 2007 Australian Institute of Health and Welfare Canberra

 

 

Questions 2-5

This unit describes a process for creating a number sequence.

A Szabo sequence is created by finding the sum of the digits of a whole number, then adding this sum to the whole number to get the next term in the sequence.

For example, the next term after 403 is 410, since 403 + (4 + 0 + 3) = 410. 

By repeating this process, we get successive terms of the sequence: 

403, 410, 415, 425, 436, 449 ...

In this sequence, 410 is referred to as the term immediately prior to 415.

Any whole number can be used as the first term, or seed number, of a Szabo sequence. The seed number of the standard Szabo sequence is 1: 

1, 2, 4, 8, 16, 23, 28, 38, 49, 62...

 

Question 2

Each of the numbers from 91 to 95 inclusive is used as the seed number of five different Szabo sequences.

The three seed numbers which produce sequences that have 107 as a member are

 

(A)     91, 92 and 94. 

(B)     91, 94 and 95.

(C)     91, 92 and 95. 

(D)     92, 94 and 95.

 

The correct answer is A

91: 101, 103, 107

92: 103, 107

94: 107

95: 109

 

Question 3

What is the sum of the digits of the term immediately prior to 704 in a Szabo sequence?

 

(A)     11

(B)     15

(C)     18

(D)     19

 

The correct answer is D

We are looking for what number is added to the previous number to get 704. Subtract the answer choice from 704 then add up the digits, if the answer given is the same as the answer choice then it is correct.

(A) is incorrect as this would put the previous number as 693 which adds up to 18 instead of 11.

(B) is incorrect as this would put the previous number as 689 which adds up to 23 instead of 15. 

(C) is incorrect as this would put the previous number as 686 which adds up to 20 instead of 18.

(D) is correct as this would put the previous number as 685 which adds up to 19.

 

The following additional information refers to questions 4-5.

 

Some numbers only appear in Szabo sequences if they are the seed number, because no number can immediately precede them.

For example, we can prove that 53 has no term immediately prior to it.

A search for it starts in the low fifties, but because 50 — the smallest number in the fifties — is followed by 55, we won't be able to reach 53 from 50 or any larger number. 

From the forties we find that 44 immediately precedes 52, that 45 immediately precedes 54, but that 53 misses out.

The largest number in the thirties, 39, is followed by only 51. No smaller number needs to be tried.

Therefore 53 has no term immediately prior to it, which means that 53 can appear in a Szabo sequence only as the seed number.

Also, there are numbers which have two different terms immediately prior to them.

For example, 505 can be reached from 500 and 491. This is called merging.

Any number which is not a multiple of three produces a Szabo sequence which eventually merges with the standard Szabo sequence.

 

Question 4

The Szabo sequence which has 64 as its seed number merges with the standard Szabo sequence at

 

(A)     115.

(B)     122.

(C)     127.

(D)     128.

 

The correct answer is A

 

 

 

Question 5

A merge graph shows the standard Szabo sequence in rectangles and other sequences in circles.

Which of these merge graphs correctly shows how a sequence containing 178 and another sequence containing 187 merge with the standard Szabo sequence?

 

  

 

The correct answer is B

178: 194, 208, 218

187: 203, 208, 218

The two sequences merge at 208, (eliminate C) and both sequence have a single step between the seed number and the merging so the answer is B.

 

Question 6

Computer Scientist: It is inevitable that a computer program will eventually be able to defeat a human opponent at chess every time they play. Current computers can plot out millions of move permutations every second, and future computers will have even greater computational power. Humans, unable to match the computers' computational abilities, will not be able to prevail in a single match.

 

Chess Player: Your conclusion is incorrect. "While computers may possess vast computational abilities, they lack the capacity for insight and innovation demonstrated by all great chess players.

 

Which of the following can be best inferred from the statements of the chess player above?

A.   A computer with a capacity for insight and innovation could defeat any human player.

B.   Computers may defeat all human chess players consistently, but they are unable to enjoy either the game or the victory.

C.   Even the best computer chess programs are limited by the abilities of their programmers.

D.   Computational power alone is not sufficient to win consistently at the game of chess.

 

The correct answer is D

The chess player states that the scientist's conclusion — that a computer program will be able to consistently defeat human players because of its superior computational ability — is incorrect, and then he states that computers do not have the capacity for insight and innovation. The clear implication is that insight and innovation are necessary in order to be great at chess, and therefore it follows that computational ability alone is not sufficient. There is insufficient support for any of the other statements: Option A is unsupported because it is not specified whether this computer would have the computational power of other computers; option B contradicts the chess player’s statements as the chess player suggests that computers won’t beat great human chess players consistently; Option C is incorrect because the programmers are never discussed.