Inductive and Deductive Reasoning (THEA Handout #9) The mathematics geometry section of the THEA test contains a few questions which require the use of inductive and deductive reasoning. These questions do not require advanced math skills, but demand good logical thinking. This handout will show you how to solve some examples of these types of questions. Also included are tips on how to improve your reasoning skills and examples for you to solve. Inductive Reasoning is used when a sequence of individual pieces of information is generalized into a conclusion that relates to those pieces of information (e.g. After several cakes baked in the same cake pan came out burned, Carl concluded that if he bakes a cake in that particular cake pan it will probably come out burned). On the THEA test, an inductive reasoning question will take the form of a sequence of numbers or figures that exhibit a trend (increasing, decreasing, decreasi ng, alternating, or a combination of these). By recognizing the trend, you must draw a conclusion about the next (or missing) number or figure in the sequence. Deductive Reasoning is used when a generalization relating pieces of information is known, and a conclusion about a specific piece of information is desired. (e.g. The baseball coach knows the opposing batter hits fast-balls well, so she instructs her pitcher not to throw a fast-ball). THEA test deductive reasoning questions describe locations, time sequences, memberships in groups, or cause and effect relationships among a few people, objects, or events. Based on these descriptions, you must be able to answer a question about a particular person, object, or event. Here are a few examples to illustrate these two types of problems. INDUCTIVE REASONING EXAMPLES EXAMPLE #1:
?
a)
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b)
c)
d)
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Example #1 is an inductive reasoning problem involving a sequence of shapes. A systematic approach for solving it involves noticing which shapes are present and what happens to those shapes when moving from figure to figure. Notice if figures change in size and/or in position. Look for aspects of the figures that alternate between two or more possibilities. In the first figure, a square is given, with a black dot in the middle. In the next figure, notice that the square has remained in the same place, but now the black dot has disappeared, and a diamond is drawn inside the original square. The following step shows that the black dot is present again, so it must be alternating. Also, a square is drawn inside the previous diamond, so squares and diamonds alternate, each being added inside the previous one. Answer (a) seems tempting, but it is incorrect because the black dot should not be present in the figure. Hence, figure (c) is also incorrect. That leaves (b) and (d). Apparently, figure (b) is missing the next diamond that should have been inscribed according to the pattern. Hence, the answer is (d). It does not contain a black dot in the middle, and it does have the extra diamond needed. EXAMPLE #2: What is the missing number in this sequence? a) 26
b) 27
19, 23, __, 31, 37 c) 21
d) 29
This is another inductive reasoning problem, but this one uses a sequence of numbers. To determine the missing number, notice the general trend of the sequence, and then look for a specific pattern that relates the sequence’s members. For a sequence of numbers like what you’ll face for a THEA question, think about simple patterns. Usually, there’s one relationship that completely explains how to go from one number to the next. Remember, whatever relation(s) you find connecting the sequence of numbers (or figures), you must be able to explain the presence of all the numbers (or figures). If you think you’ve found the pattern(s), but you can’t account for all the numbers (or figures), consider other patterns. If you can’t seem to see a relation between the numbers (or figures), look at each answer choice one at a time and see which one fits best into the given sequence. Even in the worst case, this should give you enough clues to disregard some of the answer choices. Whether the sequence has numbers or figures, look for consistent patterns between successive members of the sequence. This procedure will help you to decide what the next (or missing) member should look like. In Example #2, the sequence of numbers follows an increasing order. Answer (c) can be immediately disregarded because 21 is less than 23. Notice that 19 and 23 differ by 4, while 31 and 37 differ by 6. Finding a number that differs from 23 by 3 is a tempting answer choice to try to establish a pattern of adding 4, then 3, then 2, etc. to each successive number in the sequence. But, don’t be too hasty! Notice that if 26 is chosen, the sequence becomes 19, 23, 26, 31, 37 and the differences between successive numbers become 4, 3, 5, 6. Answer (a) is not going to establish a consistent relationship. In fact, there seems to be no consistent relationship of addition, subtraction, division, or multiplication between the numbers, so they must be linked by a special relationship. A more careful analysis leads to the conclusion that
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the series corresponds to the prime numbers, starting with 19. The number 27 is definitely not prime, and the next largest prime number greater than 23 is 29, so the answer is (d).
EXAMPLE #3:
?
a)
c)
b)
d)
This problem is similar to Example #1. Use the same type of thinking to solve it. The first figure gives a small black circle with a bracket under it. Careful study of the sequence shows that the circle is always getting larger, so (a) cannot be the answer, because this circle is too small. Pay attention to the brackets, which alternate from top to bottom and appear in every other figure. With this in mind, you can assume that the next figure should not have a bracket, because the last one shown in the sequence did. Therefore, (b) can be disregarded as an answer. You can conclude that the next circle should be white, and not black, because a black circle appears every third figure only. This leaves choices (c) and (d). This sequence seems to have a circular pattern changing from a black circle to a white circle, then to two circles, and finally back to a black circle. Since the last figure was a white circle, the next figure should have two circles, and thus (c) is the answer. DEDUCTIVE REASONING EXAMPLES EXAMPLE #4: In the Tour de France bicycle race, Franz led Henry by 3 miles, while Javier lagged behind Eduardo by 2 miles. Eduardo was ahead of Henry by 1 mile. Halfway through the race, Franz and Eduardo exchanged places (they remained the same distance apart as they were before); then Javier pulled even with Franz. In the last position change before the finish line, Javier dropped back by one mile and Henry passed Eduardo. Who finished third? a) Franz
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b) Javier
c) Henry
d) Eduardo
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Example #4 is a deductive reasoning problem requiring organization of the given information and careful attention to a series of changes. Drawing the situation on paper to keep track of all the positional changes is very helpful in solving this kind of problem. The first sentence establishes the position of each cyclist in relation to only one other cyclist (lead cyclist on the right): Henry <-------- 3 miles --------> Franz and Javier <---- 2 miles ----> Eduardo The second sentence allows the positions of all the cyclists to be linked this way: Henry <-------- 3 miles --------> Franz Javier <-- 1 mile --> Henry <-- 1 mile --> Eduardo The entire problem gives distances in 1 mile increments. Therefore, it is convenient to continue to represent the cyclists’ positions with a horizontal scale where each position is 1 mile from the position on either side of it. Because Eduardo and Franz are separated by 2 miles, the series of dashes represents an empty slot that is one mile ahead of Eduardo and one mile behind Franz: Javier
Henry
Eduardo
----
Franz
Halfway through the race, Franz and Eduardo exchange their positions, and then Javier pulls even with Franz, so the new order is as follows:
Henry
Javier Franz
----
Eduardo
Just before the end of the race, Javier drops back by one mile. This results in: Javier Henry
Franz
----
Eduardo
Finally, Henry passes Eduardo, making the final order: Javier
Franz
----
Eduardo
Henry
So Franz finished third. Therefore, the answer is (a). EXAMPLE #5: In a small hospital, there was only 1 room left, which had space for 2 patients. The nurses were instructed to put patients in the remaining room according to these rules: 1. A rubella patient cannot be put with a tuberculosis patient, unless they are both female. 2. If a rubella patient is male, then he can be put with a female emphysema patient. 3. A cancer patient cannot be put with an emphysema patient if one of them is male and (2571) 11/99 revised 6/05
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the other female. 4. A tuberculosis patient cannot be put with an emphysema patient if they are the same sex. Which of the following patients could be put in the room if a female emphysema patient was already accepted into the room? a) female rubella patient b) female tuberculosis patient c) male cancer patient d) male tuberculosis patient A couple of different approaches could be used to solve Example #5 – another deductive reasoning problem. Each answer choice could be checked against each rule, one at a time, discarding the answer choices that violate any rule. That may require going through each rule up to four times – once with each of the four answer choices. Although that method will work, another method involves using each of the rules once to identify all the possible combinations of patients who are allowed in the room, then picking the answer choice that matches one of them. The first rule says that unless a rubella patient and a tuberculosis patient are both female, they cannot be put together. So the allowable combination is: rubella(F) and tuberculosis(F) The second rule states that if a rubella patient is male, then he can be put with a female emphysema patient. Hence, another allowable combination is: rubella(M) and emphysema(F) The third rule demands that a cancer patient cannot be put in the same room with an emphysema patient unless they are both of the same sex. This gives two more allowable combinations: cancer(M) and emphysema(M) cancer(F) and emphysema(F) The last rule orders that a tuberculosis patient should only be placed with an emphysema patient if they are of opposite sexes. Again, this gives two more allowable combinations: tuberculosis(M) and emphysema(F) tuberculosis(F) and emphysema(M) Now, find an answer choice that is compatible with the patient who is already in the room– a female with emphysema. Rule two allows a male rubella patient, but that is not one of the answer choices. Rule three allows a female cancer patient, but that is not an answer choice either. The last rule allows a female emphysema patient to room with a male tuberculosis patient, so the answer is d) Here are a couple of problems for you to try:
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SAMPLE PROBLEMS A. Two women, Anissa and Crystin, and two men, Brett and Donald, are athletes. One is a runner, a second is a skater, a third is a gymnast, and a fourth is a tennis player. On day they were seated around a square table. 1. 2. 3. 4.
The runner sat on Anissa’s left. The gymnast sat across from Brett. Crystin and Donald sat next to each other. A woman sat on the skater’s left.
Who is the tennis player? a) Anissa
B.
b) Brett
What is the next number in this sequence? a) 96
b) 100
c) Crystin
d) Donald
-4, 6, 22, 44, 72, ...? c) 106
d) 111
Answers: A. a) ; B. c)
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