Logical Problems notes

MDCAT Analytical and Logical Reasoning

Logical problems use mathematics, language, patterns, relationships and formal statements to reach a definite conclusion. This chapter covers work and time, speed, ratios, counting, coding, family relations, geometry, trigonometry, sets and syllogistic reasoning.

Work, Time and Efficiency

Work problems compare the amount of work completed with the time and number of workers. If a person completes a job in T days, the work done in one day is 1/T of the job. When people work together, add their individual daily rates.

For the same amount of work, workers and time are inversely proportional. Increasing the number of workers decreases the required time, provided all workers have equal efficiency. If speed changes, time changes in the opposite direction.

If A completes a job in 12 days, A's one-day work is 1/12. If B completes it in 18 days, B's one-day work is 1/18.
Together, A and B complete 1/12 + 1/18 = 5/36 of the job per day. Their total time is 36/5 = 7.2 days.
If A completes a job in 20 days and B in 10 days, their one-day work is 1/20 + 1/10 = 3/20.
For 5 men working 20 days, total work is 5 × 20 = 100 man-days. In 10 days, required men = 100/10 = 10.
If 1/3 of a hedge is trimmed in 2 hours, one gardener's rate is 1/6 hedge per hour. Two equal gardeners trim 1/3 per hour, so the whole hedge takes 3 hours.
Work formula: Total work = number of workers × time × individual efficiency.

Speed, Production and Rate Changes

Rate means the amount of work completed in one unit of time. For a fixed distance, time is inversely proportional to speed. A decrease in speed causes a proportionate increase in time, not an equal decrease.

Production questions use the same principle. If the number of workers is reduced but total production must remain unchanged, each remaining worker must work for a longer time.

If a van makes 3 trips in 6 hours, its rate is 3 trips in 6 hours. A second van working at 80% speed makes 2.4 trips in 6 hours, so the total is 5.4 trips if fractional trips are allowed.
If the intended question treats both vans as making complete scheduled trips at the same route capacity, the stated answer may be 6 trips. Always check whether the problem assumes whole trips or uses exact rates.
A 20% decrease in speed leaves 80% of the original speed. New time = old time ÷ 0.8.
For a 1-hour delivery, new time = 1 ÷ 0.8 = 1.25 hours, or 1 hour 15 minutes.
A cleaner needs 2 hours for one floor. If cleaning time doubles, one floor takes 4 hours. Three floors take 3 × 4 = 12 hours.
A worker repairing 2 bikes per hour needs 20 ÷ 2 = 10 hours for 20 bikes. If event work is 3 hours longer than regular work, regular time = 10 − 3 = 7 hours.
If a window cleaner works at 4 windows per hour and speed rises by 50%, new rate = 4 × 1.5 = 6 windows per hour. Time for 30 windows = 30 ÷ 6 = 5 hours.

Ratios, Proportion and Counting

A ratio compares quantities in the same units. First express the quantities in the form of the given ratio. If boys to girls is 3 to 2, the total number of parts is 5.

Counting questions require careful treatment of the endpoints. In particular, ask whether 0 or 100 is included and whether repeated appearances of a digit are counted separately.

If boys:girls = 3:2 and there are 30 boys, one part is 30 ÷ 3 = 10. Girls = 2 × 10 = 20, so total students = 50.
Direct proportion: if one quantity increases by a factor, the other also increases by that factor.
Inverse proportion: if one quantity increases by a factor, the other decreases by the same factor.
The digit 9 appears once in each unit position from 1 to 9, 19 to 29, and so on, giving 10 appearances in the units position from 1 to 99 and 10 in the tens position from 90 to 99. The number 99 contributes two appearances. Therefore, there are 20 appearances of 9 from 1 to 100 if counting every occurrence.
The number of integers containing at least one 9 from 1 to 100 is different from the number of appearances. There are 19 such integers if the question counts numbers rather than individual digits, depending on the exact wording and convention used.
Always distinguish between number of 9s as digit occurrences and number of numbers containing 9.

Coding, Alphabet Values and Interfaces

Coding questions replace letters or words according to a hidden rule. Compare each original letter with its coded letter. Check forward or backward shifts, position numbers, alternating changes, reversal and grouping.

Alphabet-value questions assign A = 1, B = 2, through Z = 26. Add the values of the letters unless another operation is stated.

Alphabet positions are A = 1, B = 2, C = 3, D = 4, E = 5, and Z = 26.
EAR has value E + A + R = 5 + 1 + 18 = 24.
CAT has value C + A + T = 3 + 1 + 20 = 24. If the expected answer is 25, the question is using an unstated or different coding rule and should be checked for a printing error.
For a code such as HEART becoming IGSQU, write both words in two rows and compare each corresponding position.
The coded form supplied for LIVER in the given pattern is MJWDS. In an examination, use the exact transformation established by the example rather than applying a random shift.
A user interface is the means by which a user interacts with a computer or device. Common types include command-line, menu-driven, graphical user interface and touch interface.
Design interface is not normally classified as a standard type of user interface in this list.

Family Relations and Direction Sense

Family-relation questions are solved by drawing a small family tree. Mark gender and generation before deciding the final relationship. Do not rely on the order in which the information is written.

Direction questions can be represented on a coordinate plane. North is positive vertical direction, east is positive horizontal direction, south is negative vertical direction and west is negative horizontal direction.

If A is the father of B, B is the brother of C, C is the daughter of D, and D is the wife of A, then D is B's mother.
Brother indicates a male sibling. Daughter indicates a female child. Wife indicates a female spouse.
For a movement of 6 km north followed by 8 km west, the shortest distance is the hypotenuse of a right triangle.
Using Pythagoras, distance = √(6² + 8²) = √100 = 10 km.
The final direction in the 6 km north and 8 km west example is north-west.
A family tree should show spouses on the same generation and children one level below their parents.

Coordinate Geometry and Parallelograms

Coordinates locate points using an ordered pair (x,y). In a parallelogram, opposite sides are parallel and equal, and the diagonals bisect one another.

For points P, Q, R and S written in order, the diagonals are PR and QS. Therefore, the midpoint of PR equals the midpoint of QS.

Midpoint of (x1,y1) and (x2,y2) is ((x1 + x2)/2, (y1 + y2)/2).
For parallelogram PQRS, S = P + R − Q when P, Q, R, S are consecutive vertices.
Equivalently, S = (xP + xR − xQ, yP + yR − yQ).
For P(1,2), Q(4,6), and R(5,7), the consecutive-vertex formula gives S = (1 + 5 − 4, 2 + 7 − 6) = (2,3).
Other point orders give different fourth vertices. Thus, the order of the named vertices must be known before selecting an answer.
The stated answer (2,4) for these coordinates does not satisfy the usual parallelogram formula when P, Q, R, S are consecutive vertices.

Trigonometric Equations

A trigonometric equation is solved by finding all angles that give the required trigonometric value. General solutions include an integer n because trigonometric functions repeat after complete revolutions.

For sine, the period is 2π radians. The equation sin x = 1 occurs at the top of the unit circle.

1 − sin x = 0 can be rearranged to sin x = 1.
The general solution of sin x = 1 is x = π/2 + 2nπ.
Here n is any integer, positive, negative or zero.
In degrees, the same solution is x = 90° + 360°n.
A complete revolution is 2π radians or 360°.
Use radians when the question gives answers in terms of π.

Statements, Conclusions and Syllogisms

In statement-conclusion questions, accept only conclusions that must be true from the given statements. Do not add information from common sense or assume that a category has members unless existence is stated.

Words such as all, some and no have precise meanings. An argument is valid only when the conclusion follows in every possible situation allowed by the statements.

All A are B means every member of A is also a member of B. It does not mean every B is A.
Some A are B means at least one object belongs to both A and B.
No A is B means A and B do not overlap.
From all windows are rods and some rods are frames, no conclusion follows about windows and frames. The rods that are frames may be different rods from the windows.
From all artists are painters, all actors are gentle, and all gentle are painters, it does not follow that artists and actors overlap or that all painters are actors.
Some pictures are frames and some frames are idols, with all idols as curtains. It follows that some frames are curtains, but it does not necessarily follow that some pictures are curtains.
If all players are spectators, some spectators are theatres, and some theatres are dramas, none of the stated overlaps must involve players. Therefore, none of the listed conclusions follows.
In a statement that Ramzan is always successful and no fool is always successful, Ramzan cannot be a fool. The conclusion that Ramzan is not a fool follows.

Sets and Circles

A set is a well-defined collection of objects. Set notation describes membership and relationships clearly. A circle can be described as the set of all points at a fixed distance from a fixed centre.

The symbol P(x,y) represents a point with coordinates (x,y). The expression |CP| means the distance between centre C and point P.

The set notation of a circle with centre C and radius r is S(C,r) = {P(x,y): |CP| = r}.
The colon means such that.
The equation of a circle with centre (h,k) and radius r is (x − h)² + (y − k)² = r².
A point inside the circle has distance from the centre less than r. A point on the circle has distance exactly r.
A point outside the circle has distance from the centre greater than r.
Union combines elements in either set. Intersection contains only elements common to both sets.

Key terms

Rate: The amount of work or production completed in one unit of time.
Efficiency: The amount of work completed by one worker or machine in a given time.
Inverse proportion: A relationship in which an increase in one quantity causes a corresponding decrease in another.
Ratio: A comparison of two quantities expressed in the same units.
Alphabet position: The numerical value assigned to a letter, with A = 1 through Z = 26.
Coding: A rule-based replacement of letters, numbers, words or symbols.
Syllogism: A logical argument in which conclusions are tested from given statements.
Universal statement: A statement using words such as all or no to describe an entire category.
Particular statement: A statement using some to indicate at least one member of a category.
Parallelogram: A quadrilateral whose opposite sides are parallel and equal.
Midpoint: The point exactly halfway between two points.
Hypotenuse: The side opposite the right angle in a right triangle.
General solution: A form that gives every value satisfying an equation, usually using an integer parameter.
User interface: The method through which a user interacts with a computer or electronic device.
Set: A well-defined collection of distinct objects.
Circle: The set of all points at a fixed distance from a fixed centre.

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Analytical and Logical Reasoning shortcuts

Choosing the stronger supporting argument

Choose the option that directly explains why the claim is true. Prefer relevant evidence with a clear link to the claim, not a statement that is merely related.

Identify the claim and ask, “Why would this make the claim true?”
Reject personal opinions, unrelated facts and extreme statements.
Example: “Teachers should be paid more” is supported by “They shape future generations.”

This shortcut does not apply when the option gives direct statistical or experimental evidence that is stronger than a general reason.

Finding the pattern in letter and number series

Check letters and numbers separately. Look for equal increases, alternating changes, multiplication, or a repeating pattern.

Convert letters to positions if needed, such as A = 1 and D = 4.
Find the change in the letters and in the numbers independently.
Example: A2, D4, G6, J8 has letters increasing by 3 and numbers by 2, so the answer is M10.

Do not assume a simple addition pattern if alternating or multiplication patterns fit all the given terms.

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