Analytical Reasoning from Statements and Relationships

Letter substitution coding rules

A letter substitution code replaces each letter of an original word with another letter or number according to a rule that the question states. The examiner does not expect you to remember any code. The skill being tested is applying a fully stated rule accurately, letter by letter. A reliable method has four steps: identify the type of rule, check the rule against any sample pair given in the question, apply it to every letter in order, and read the result in the direction the question asks for. Most errors come from skipping a letter, shifting the wrong direction, or miscounting positions, so write out the intermediate letters rather than trying to do the whole word in your head.

The first type is class-based substitution, in which the rule depends on whether a letter is a vowel or a consonant. Vowels form the cycle A-E-I-O-U, and a vowel is replaced by the next vowel in this cycle, with U wrapping around to A. Consonants are replaced by the previous consonant in the alphabet, which means vowels are skipped. For example, the letter before V is U, a vowel, so the previous consonant of V is T; the letter before P is O, a vowel, so P becomes N. Keeping the two directions and the skipping rule separate prevents most mistakes. For example, TEAM becomes SIEL, because T gives S, E gives I, A gives E and M gives L, and PUSH becomes NARG. Words in this type of question must be checked letter by letter, because vowels and consonants move differently even within the same word.

• next vowel (cycle) — the vowel that follows in the sequence A-E-I-O-U, with U returning to A. For example, O becomes U and U becomes A.
• previous consonant — the nearest earlier letter that is not a vowel, so vowels between two consonants are ignored.
• wrap-around — the rule that continues from the end of a list back to its beginning, as U to A in the vowel cycle.

The second type assigns numerical values to letters. Each letter takes its alphabetical position, with A = 1 and Z = 26. A code word is then often the sum of these values, optionally with a fixed amount added for each letter. Adding one to each letter's value is the same as adding the number of letters in the word. For example, CAT has the values 3, 1 and 20, which sum to 24, and adding one for each of its three letters gives 27. Equivalently, (3+1)+(1+1)+(20+1)=27. Always test the stated rule on the sample code before using it, because a question may add a different constant, or none at all.

The third type shifts each letter by an amount that depends on its position in the word. A common version uses odd positions and even positions with opposite shifts. For example, if the odd positions move forward one place and the even positions move back one place, then BOAT becomes CNBS, since B moves to C, O moves to N, A moves to B and T moves to S. Shifts of two places work in the same way, but the shifts are often not symmetric, so the direction and size of each shift must be confirmed on the sample pair. Once the sample has fixed the pattern, apply the same shift to each position of the new word and check that the result has the same length. A mistake in a single letter, such as taking the wrong shift for the seventh position, changes the whole answer.

The fourth type rearranges letters in blocks, combining the rearrangement with a shift on some letters. The typical operations are swapping the first two letters, reversing a group of letters, and shifting a middle group by one place. For example, if the first two letters are interchanged and the last three letters are reversed, PLANET becomes LPATEN, since PL becomes LP and NET becomes TEN. A block may also be left unchanged, so a question can apply one operation to some letters and no operation to others. Check each block against the sample separately, because the operations on different blocks are independent of one another.

The fifth type applies different rules to vowels and consonants in the opposite sense from the first type. Here a vowel moves back to the previous letter of the full alphabet, and a consonant moves forward to the next letter. For example, BLUE becomes CMTD, since B and L move forward to C and M, and U and E move back to T and D. Such a question often asks for a letter at a particular position counted from the right, so practice the counting method. In an n-letter word, the k-th letter from the right is the (n-k+1)-th letter from the left. For a six-letter coded word, the second letter from the right is therefore the fifth from the left. Compute the whole coded word first and then count, because counting directly on the original word gives the wrong letter.

Kinship relation terms

Kinship questions give a chain of family statements and ask how two named people are related, or which relations are possible. The standard method is to draw a family tree and fix the generation of every person first, because most kinship terms are determined by generation. A parent is one generation above a child, siblings share the same generation, and the spouse of a sibling or child usually belongs to that same generation. After the generation is fixed, the term is refined by side and by sex. Maternal means the relation runs through the mother, and paternal means it runs through the father. Sex determines whether a term reads as grandfather or grandmother, uncle or aunt, and so on. If the statement does not fix the sex, both gendered options may be possible.

The following terms are frequently tested.

• Maternal and paternal grandparent: a parent of your parent. A parent of your mother is your maternal grandparent, whose sex may be left open, so both grandfather and grandmother can be possible.
• Grandchild: a child of your child. If a person is the child of your child, the relation is reversed when read from the other side.
• Sibling: a person who shares at least one parent with you. Siblings are in the same generation.
• Uncle and aunt: a sibling of a parent is a maternal or paternal uncle (male) or aunt (female), depending on the side. The husband of an aunt is an uncle by marriage, which is not a blood relation.
• Father and brother: these terms exclude a sibling of a parent, because such a person is one generation above you and is not your parent or your sibling.

When a question asks which relation is possible, you list every option that is consistent with the statements. When it asks which relation is necessary, the conclusion must hold in every arrangement that fits the data. Counting family members with gender totals is often the key step. Totals of males and females fix how many spouses exist and which married child has a spouse of which sex. An unmarried child has no children, and an only child has no siblings in the family. The pigeonhole principle then forces a conclusion, because if m items are placed into k groups with m > k, at least one group contains two or more items. Thus, if three granddaughters are split between two sons, at least one son has two daughters.

Overlapping set counting with Venn constraints

Venn constraint problems give the sizes of groups and their overlaps, and the task is to count the total number of people or objects that belong to at least one group. The main tool is the inclusion-exclusion principle for three sets:

n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)

If a pairwise intersection is stated as zero, the two groups are disjoint, so that term vanishes, and the triple intersection is then also zero. For example, if n(A)=30, n(B)=20, n(C)=10, n(A \cap B)=5, and the other intersections are zero, then the union is 30+20+10-5=55.

The phrase 'at least one' is usually handled by complement counting. The number who belong to at least one group equals the total minus the number who belong to none of them. In a pass-or-fail table, if 15 out of 100 candidates failed every subject, then 100-15=85 candidates passed at least one subject. Read the words carefully. 'Failed in all three' and 'passed in at least one' are different events, and only the first is given directly.

A useful check is to add the regions. The exactly-one, exactly-two and all-three regions must sum to the number who belong to at least one group. In the pass-or-fail data, the counts of candidates who passed in only one subject and in all three subjects must be consistent with the count of those who passed in exactly two subjects.

In figure-based problems, each region is identified by the set of outlines that enclose it. 'Belonging to only two shapes' means the region lies inside exactly two outlines, not inside one or three of them. Numbers placed in regions inside a single shape, or inside three or more shapes, are excluded from the sum. The required total is the sum of the numbers in the exactly-two regions.

Occupation assignment constraints

Occupation assignment constraints are logic puzzles in which several people must each take a different role, and each clue rules out some of the possible matches. The reliable method is to draw a grid with people along one side and roles along the other, record every clue as a yes or no, and keep eliminating until exactly one role remains for each person. Because no two people may share a role, once a role is fixed for one person it is removed from everyone else's options.

Clues about relatives and neighbours are usually negative. If one person is said to be a cousin of, or to live next door to, the holder of a role, the two people must be different, so the named person cannot hold that role. Comparative clues work the same way: a statement such as 'A knows more than the teacher' means that A is not the teacher. A clue that A must travel a long way to reach the home of a neighbour of the plumber means that A is not the plumber, because the plumber lives next door to that neighbour. Each clue of this kind removes one cell from the grid.

Comparative puzzles about height or rank work with chains of 'taller than' statements. Two statements such as A is taller than B and B is taller than C can be merged into A is taller than C, and the person with no one above them in the chain is the tallest. A role that is stated not to be held by the tallest person excludes that person from the role, and a role whose holder must be shorter than another role-holder cannot go to the tallest person.

When a puzzle leaves more than one consistent arrangement, test each case separately. If every consistent arrangement gives the same answer to the question, that answer is determined by the clues.

Example: Ravi, Sita and Tara take the roles of nurse, pilot and chef. The clues are that Sita is the sister of the nurse, Tara lives beside the chef, and Sita is not the chef. Sita is therefore neither the nurse nor the chef, so she is the pilot. Tara is not the chef and the pilot role is taken, so Tara is the nurse, which leaves Ravi as the chef.

Word-pattern substitution in invented languages

Word-pattern substitution in invented languages is a coding and decoding task in which a made-up language is shown through short coded phrases, and the reader must find the code of a new phrase. The first step is to compare phrases. A word that occurs in two or more coded sentences has a code common to all of those sentences, so a shared English word can be matched to a shared code.

Elimination is the second step. When a coded sentence contains a word whose code is already known, the code that remains in that sentence must belong to the one English word that is still unknown. In this way a single unknown word can be found by subtraction, once the other words have been identified.

Every deduced pair should be checked against all the coded sentences. A code that matches a word in one sentence but conflicts with the meaning of another sentence is wrong, so the final answer is accepted only when each word in the requested phrase has a code that is consistent throughout.

Invented languages can also code letters rather than words. In such a code the rule is applied position by position: a letter may be swapped with a letter in another position, moved forward or back in the alphabet by a fixed count, or left unchanged. To find the rule, write the original word and its code letter by letter, then confirm the rule on a second pair before applying it to a new word.

Example: Under a rule that swaps the first two letters and advances the third to fifth letters by one place each, the word PLANT becomes LPBOU.

Conditional promotion rules and spending thresholds

Conditional promotion rules and spending thresholds describe offers in which a benefit depends on meeting stated conditions. Such a rule has two parts: a threshold, such as a minimum amount to be spent, and a qualifying action, such as buying a certain quantity of a particular item. The benefit is given only when the qualifying action is met, so a benefit cannot be received unless the condition behind it is satisfied.

Words such as 'minimum' and 'at least' mean equal to or more than. A purchase of exactly the required amount qualifies, but a purchase below it does not. A customer who spends less than the threshold falls outside the offer, so receiving no gift is consistent with the terms. Quantity conditions work in the same way: buying one unit fewer than the stated quantity does not earn the reward.

Example: A bookshop gives a free bookmark to customers who buy at least four books and spend at least ₹1,000 in total. A customer who buys five books for ₹900 receives no bookmark, which is consistent with the terms, whereas a customer who buys two books and receives a bookmark breaks the rule.

When a question asks which outcome is not possible, check each outcome against the terms of the offer. The impossible outcome is a benefit received without its qualifying action, or a benefit received when the threshold has not been reached. Outcomes that respect every stated condition are possible, even if they involve no reward at all.

Monty Hall problem probability

The Monty Hall problem is a probability puzzle in which one of three doors hides a prize, and a host who knows where the prize is opens an unchosen door that holds no prize before offering the chance to switch. Many people judge that the two remaining doors are equally likely, so switching makes no difference and the chance is one half. This is the naive error, because it treats the host's opening of a door as a random event, when in fact the host's choice depends on where the prize is.

The correct analysis uses the full set of possible positions for the prize. With three equally likely positions, the door first chosen has probability \frac{1}{3} of hiding the prize, so the other two doors together have probability \frac{2}{3}. Because the host always reveals a door without the prize, the whole \frac{2}{3} passes to the single unopened door. Switching therefore wins with probability \frac{2}{3}, which is double the probability \frac{1}{3} of staying.

The result can be checked by listing the cases. Suppose the first choice is door 1. If the prize is behind door 1, the host opens door 2 or door 3, and switching loses. If the prize is behind door 2, the host must open door 3, and switching to door 2 wins. If the prize is behind door 3, the host must open door 2, and switching to door 3 wins. Over three equally likely cases, switching wins two times in three.

The host's action is not independent of the prize location, which is why intuition tends to go wrong. Solutions should state the assumptions that the host knows where the prize is and always opens a door that hides no prize. Under these assumptions, switching is the better choice.

Score ordering among students

Score ordering problems give several pairwise comparisons, often spread across two papers, and ask who is highest or lowest in one of them. Read 'A scores less than B' as A < B and 'A scores more than B' as A > B, and keep this direction when you join statements. The relation is transitive, so P > Q and Q > R give P > R; statements that share a name can therefore be linked into one chain. Only the paper named in the question matters. A statement about the other paper cannot be combined with these, and such clues are often distractors. A student is the highest only if no chain places anyone above him, so check every candidate against all comparisons in that paper. For a lowest-score question, reverse the test and look for the student who is never on the greater side. For example, if K > L, M > K and N < L hold in one test, then M > K > L > N, so M is the highest and N is below L.

Wage bill constraint equations

Wage bill problems are systems of two linear equations in the numbers of workers in two categories. Let x be the number of workers in one group; if N is the fixed total headcount, the other group has N - x workers. The total wage bill gives a second equation, ax + b(N - x) = T, where a and b are the monthly wage rates of the two groups and T is the total bill. Solve by substitution: (a - b)x = T - bN, so x = (T - bN)/(a - b). Check the result by substituting back into both the headcount and the total wage equations, and confirm that the answer is a non-negative whole number. Keep the time unit consistent, since monthly and daily rates cannot be mixed in one equation. Example: a workshop has 20 staff, junior pay is 500 per day, senior pay is 800 per day, and the daily bill is 13,000. With s seniors, 500(20 - s) + 800s = 13000 gives s = 10, and the check is 10 \times 800 + 10 \times 500 = 13000.

Part-whole relationships

Part-whole analogies pair a whole with one of its parts or constituent units, and the order of the pair matters. If the given pair runs from whole to part, the answer pair must also run from whole to part; a pair written in reverse does not hold the same relation. Useful vocabulary includes 'composed of', 'made up of' and 'consists of' for the whole, and 'constituent', 'component' and 'unit' for the parts. Material-to-product pairs, such as a raw substance and an object made from it, form a different relation, and an ingredient-to-finished-product pair should not be mistaken for a whole and its parts. Two forms of the same substance are related by similarity rather than by composition, so they do not match a part-whole pair. Example: a book is composed of chapters, so 'book : chapter' has the same whole-to-unit relation as 'forest : tree'.

Machine input-output rearrangement steps

In rearrangement problems, a machine applies one fixed rule at each step, and the rule must be read from the example steps before it is applied to new input. A common rule places the alphabetically first remaining word at the next free position from the left, and moves the largest remaining number to the right end, while all other items keep their relative order. Verify the rule on every given step; if one step does not follow from the previous one, the rule is wrong. Write out the whole line after each step, since one misplaced item shifts the position counts of every other item. To convert a position counted from the right, use the fact that for n items the kth item from the right is the (n - k + 1)th item from the left. If your traced position matches none of the listed options, recheck every step; in some published items only the nearest listed option is keyed, so such items should be verified against the official key.

Digit-pattern odd one out

Digit-pattern odd-one-out questions give several numbers that share one rule between their digit positions, and one number that breaks it. Write each number as d_1 d_2 d_3 d_4 d_5 and test relations between positions, such as d_3 = d_1 + 2, d_4 = d_2 + 3, or a last digit that stays the same across all terms. A digit that is constant across all terms does not distinguish any number, so it can be ignored when looking for the odd one out. Some sequences follow a progression, such as the first digit rising by 1 from one term to the next, which shows how the set is built. Do not stop at the first relation you check: a number may satisfy one relation and break another, so test each position against all terms. Find the relation that four numbers satisfy, then find the one that violates it; that number is the odd one out. For example, in 21436, 32547, 43658, 54769 and 65870 the fourth digit is always the second digit plus 2, so a number such as 65890 would be the odd one out.

Counting items held by each person

A counting question asks you to compare how many items each person holds, so the first task is to fix what counts as holding. A person holds an item when it is in his or her possession at the time the question refers to, and this includes an item that has just been bought or received. Each item is counted once for each person who has it, and an item that no one is said to hold does not enter any total.

Next, tally the items person by person rather than by kind. Two different items make two, even when they belong to the same category, so a person holding one mango and one lemon has two fruits. Once each total is written down, compare them. The phrase 'the least number of' refers to the smallest total among the people being compared, and if two people share that smallest total, the question has no single answer. Most errors come from forgetting a newly acquired item or from attributing an item to the wrong person.

• hold / be in possession of — to own or carry something at the reference time. e.g., 'She holds two tickets now.'
• has just bought — a fresh purchase is part of the current holdings and must be counted. e.g., 'He has just bought a jacket, so his total rises by one.'
• the least number of — the smallest count within the group being compared. e.g., 'Which student has the fewest pens?'

Example: Ravi has two pens and one notebook, and Meena has one pen. Ravi holds three items and Meena holds one, so Meena has the least number of items.

Directional facing relations

Directional facing questions describe people seated around a table, usually a square one, and each person faces the centre. The rule that links direction and seat is simple: a person who faces a given direction sits on the side opposite to that direction. So a person facing South sits on the North side, and a person facing West sits on the East side. Partners who sit opposite each other therefore face opposite directions, while people on adjacent sides face directions at right angles to one another.

To answer a question, place the person with the given direction first, then use the partner rule to fill the opposite seat. Once one pair is fixed, the other pair occupies the two remaining sides, and the person on each side faces the centre, that is, the side opposite to his or her own seat. A useful check is that every stated direction matches the seat the person has been given; if one does not, the arrangement is wrong.

• face towards — the direction in which a seated person looks, toward the centre of the table. e.g., 'She faces towards the door.'
• opposite — the seat directly across the table, whose occupant faces the reverse direction.
• side of the board — the seat on one edge of the table, identified by the direction the occupant faces.

Only people named in the arrangement can be given a direction, so a name that does not appear among the seated players is never the answer.

Travel time and arrival calculation

Travel-time questions ask for an arrival or departure clock time, and they are solved by combining a clock time with a duration. The arrival time is the departure time plus the travel time, and the arithmetic is done on hours and minutes separately, with 60 minutes forming one hour. If the minutes add up to 60 or more, subtract 60 from the minutes and carry 1 to the hours. A sum such as 13 h 90 min is not a valid clock time and must be rewritten as 14 h 30 min.

Example: a bus leaves at 7:25 and travels for 3 h 45 min. The minutes give 25 + 45 = 70 = 1 h 10 min, and the hours give 7 + 3 + 1 = 11, so the bus arrives at 11:10.

The wording also signals which quantity is which. 'Starts from' and 'leaves' mark the departure time, 'reaches' and 'arrives at' mark the arrival time, and 'time taken' or 'duration' describes the length of the trip.

In 24-hour notation, such as 14:30 hours, the clock reads 2:30 pm, so no am or pm adjustment is needed. In 12-hour notation, check whether the addition crosses noon or midnight. To verify an answer, subtract the departure time from the arrival time and confirm that the difference equals the stated travel time.

Age ordering relations

Age-ordering questions give a chain of comparisons between people together with one known age, and the task is to find an unknown age. Translate every comparison into an equation before computing. 'x is n years younger than y' means x = y - n, and 'x is n years older than y' means x = y + n, so 'older' always raises the value and 'younger' always lowers it. A phrase such as 'one year older than' is the case n = 1, so the relation becomes x = y + 1.

Start from the person whose age is given and move along the relations one step at a time, computing each age from one already known. When a value is found, use it in every relation in which that person appears, because the same person is often described more than once in a problem. If one relation gives a second value for the same person, check that both values agree; if they differ, the data are inconsistent.

• younger than — having a smaller age, so the difference is subtracted. e.g., 'Ben is three years younger than Asha' gives B = A - 3.
• older than — having a larger age, so the difference is added. e.g., 'Chen is five years older than Dev' gives C = D + 5.
• the age of — the unknown being sought, which is reported as a whole number of years.

Example: Maya is four years older than Tara, and Tara is 7, so Maya is 7 + 4 = 11. If Kiran is two years younger than Lila and Lila is 9, then Kiran is 7, and Mohan, who is one year older than Kiran, is 8.

Operator substitution arithmetic

Operator-substitution questions give a rule that replaces each arithmetic symbol with another, and then ask for the value of the rewritten expression. Apply the whole substitution at once, reading each symbol from the original expression, and never replace a symbol that was itself produced by an earlier replacement. After substitution, evaluate the expression with the usual order of operations: multiplication and division are done before addition and subtraction, and operations of equal rank are done from left to right.

For example, if + means \times and \times means +, then 2 + 3 \times 4 becomes 2 \times 3 + 4, and evaluating the multiplication first gives 6 + 4 = 10. If \times means - and - means +, then 6 \times 2 - 3 becomes 6 - 2 + 3 = 7.

Results often come out as fractions. Keep the value exact during the computation, for instance \frac{56}{40} = \frac{7}{5}, and convert only at the end. A mixed number such as 7\frac{2}{5} equals \frac{37}{5}, which is 7.4.

• substitute / replace — to exchange one symbol for another under a stated rule.
• order of operations — the precedence rule, often remembered as BODMAS or PEMDAS.
• mixed number — a whole number and a proper fraction together, so that 3\frac{1}{4} means \frac{13}{4}.

Arithmetic progression nth term

An arithmetic progression (AP) is a sequence in which each term differs from the previous term by a fixed constant called the common difference d. If the first term is a, the terms are a, a+d, a+2d, \ldots, and the nth term is a_n = a + (n-1)d. The factor (n-1) is the most common source of error, because the first term is a itself with no d added. The formula a + nd gives the term after the nth position, so it is off by one step.