9.4 Arithmetical Verbal Problems, Percentages & Verbal Logic Tests

9.4 Arithmetical Verbal Problems, Percentages & Verbal Logic Tests

The final section of the Pakistan Army Medical Cadet (AMC) Initial Verbal Intelligence Test combines numerical word problems with formal verbal logic tests. These questions assess quantitative reasoning, mental arithmetic speed, and formal logical deduction without requiring advanced mathematics.

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1. Arithmetical Verbal Problems & Age Equations

Arithmetical word problems present math scenarios in prose form. Setting up concise linear equations quickly is critical under the 20-second time limit.

Age Relationship Problems

Age problems compare the ages of two or more individuals across different time frames (present, past, and future).

The Standard 3-Step Age Algorithm:

1. Define Present Variables: Let the current age of the younger person be $S$ and the older person be $F$.
2. Express Future/Past Ages:
   • Age $k$ years ago $= S - k$ or $F - k$.
   • Age $k$ years from now $= S + k$ or $F + k$.
3. Formulate & Solve the Equation: Substitute the ratio given in the prompt.

Example: Prompt: A father is currently 3 times as old as his son. In 10 years, he will be twice as old as his son. How old is the son now? Formulation:

• Present: $F = 3S$
• In 10 years: $F + 10 = 2(S + 10)$
• Substitute $F = 3S$: $3S + 10 = 2S + 20 \implies 3S - 2S = 20 - 10 \implies S = 10$
• Son's present age $= \mathbf{10\text{ years}}$. Father's present age $= 30\text{ years}$.

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2. Percentage Verbal Logic & Examination Thresholds

Percentage questions in the AMC verbal section test basic fractional conversions, pass-mark thresholds, and percentage increase/decrease calculations.

Pass Mark & Total Score Threshold Problems

In these problems, a candidate obtains a certain mark, fails by a specified margin, and the pass percentage is provided.

$\text{Passing Marks} = \text{Obtained Marks} + \text{Deficit Marks}$

$\text{Total Maximum Marks } (M) = \frac{\text{Passing Marks}}{\text{Pass Percentage}} \times 100$

Example: A candidate needs 40% to pass an exam. He scores 60 marks and fails by 20 marks. What are the total maximum marks?

• Passing Marks $= 60 + 20 = 80\text{ marks}$.
• $40% \text{ of } M = 80 \implies 0.40 M = 80 \implies M = \frac{80}{0.40} = \mathbf{200\text{ marks}}$.

Mental Percentage Conversion Benchmarks

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3. Speed, Time & Distance Verbal Problems

Speed and distance word problems require fundamental physics formulas and specific average speed calculations.

Basic Motion Formulas

$\text{Distance } (D) = \text{Speed } (S) \times \text{Time } (T)$

$\text{Speed } (S) = \frac{D}{T}, \quad \text{Time } (T) = \frac{D}{S}$

The Average Speed Trap (Harmonic Mean)

When an object travels equal distances at two different speeds ($v_1$ and $v_2$), the average speed for the entire journey is NOT the simple arithmetic average $\frac{v_1 + v_2}{2}$. You must use the Harmonic Mean Formula:

$S_{avg} = \frac{2 v_1 v_2}{v_1 + v_2}$

Worked Example: A cadet drives to the military academy at 30 km/h and returns along the same route at 60 km/h. What is his average speed for the round trip?

• Incorrect Method: $\frac{30 + 60}{2} = 45\text{ km/h}$.
• Correct Harmonic Formula: $S_{avg} = \frac{2 \times 30 \times 60}{30 + 60} = \frac{3600}{90} = \mathbf{40\text{ km/h}}$

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4. Categorical Syllogisms & Logical Deductions

A syllogistic test consists of two or more premise statements followed by conclusions. Your task is to determine whether the conclusions follow logically from the premises, regardless of real-world truth.

Universal & Particular Quantifiers

• Universal Affirmative ("All A are B"): Set A is completely contained within Set B ($A \subseteq B$).
• Universal Negative ("No A are B"): Set A and Set B are completely disjoint ($A \cap B = \emptyset$).
• Particular Affirmative ("Some A are B"): Set A and Set B overlap ($A \cap B \neq \emptyset$).

      All A are B                  No A are B                Some A are B
    +---------------+          +---+     +---+              +---+---+
    | Set B         |          | A |     | B |              | A | X | B |
    |   +-------+   |          +---+     +---+              +---+---+
    |   | Set A |   |
    |   +-------+   |
    +---------------+

Syllogism Rules of Validity

1. Transitive Chain: If All A are B and All B are C, then All A are C is a valid conclusion.
2. Negative Premise Rule: If one premise is negative, the conclusion must be negative.
3. Two Particular Premises: No valid conclusion can be drawn from two particular premises (e.g., "Some A are B" and "Some B are C").

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5. Comprehensive AMC Test Execution Strategy

The AMC Initial Verbal Test presents dozens of (commonly reported high-volume) questions in 30 minutes. Follow this operational plan during the test:

1. Phase 1: Rapid First Pass (Questions 1 to 90):
   • Answer all instant verbal analogies, classification, and letter series questions immediately (5 to 10 seconds per question).
   • Do not spend more than 25 seconds on any single complex arithmetic word problem.
2. Phase 2: Skip-and-Mark Protocol:
   • If an age or distance problem requires scratchpad work that exceeds 20 seconds, select your best educated guess, flag it, and move forward immediately.
3. Zero Negative Marking Advantage:
   • Pakistan Army Initial Computerized Tests do not apply negative marking. Never leave any question unattempted when the timer runs out.