Logical Problems: MDCAT Logical Reasoning notes

How to approach logical problems

These are short word problems that need careful reading and simple arithmetic, not advanced mathematics. Write down what is given, what is asked, and draw a quick sketch (a row, a family tree or a compass) when positions or relations are involved.

1. Unitary method and rates

Find the value of one unit, then scale up.

Example: 30 pencils cost Rs. 300 and 20 pens cost Rs. 600. One pencil = Rs. 10, one pen = Rs. 30, so 3 pencils cost the same as 1 pen.

Example: A cake has 8 slices and feeds 4 people (2 slices each). For 100 people you need 25 cakes = 200 slices.

Net weight: 4 crates weigh 60 kg; each empty crate is 2 kg. Contents = $60-4\times2=$ 52 kg.

2. Percentages and discounts

• Price after a $d\%$ discount $=$ original $\times\left(1-\dfrac{d}{100}\right)$.
• To find the original price, divide: original $=\dfrac{\text{sale price}}{1-d/100}$.

Example: After a 20% discount an item costs Rs. 800. Original $=800/0.8=$ Rs. 1000 (not $800+160$).

3. Ranking in a row

• Total $=$ (position from left) $+$ (position from right) $-1$.
• If $k$ students are below someone in a class of $N$, his rank from the top is $N-k$.
• Number of people between ranks $a$ and $b$ $=|a-b|-1$.

Example: A girl is 8th from the left and 15th from the right: total $=8+15-1=$ 22.

Example: In a class of 40, Zara is 5th; 10 students are below Hina, so Hina is 30th. Between them: $30-5-1=$ 24.

4. Direction sense

Track east-west and north-south separately, then combine. If both remain, use Pythagoras: distance $=\sqrt{x^2+y^2}$.

Example: Walk 6 m north, 8 m east, 6 m south. North and south cancel; you are 8 m east of the start. If instead you walked 3 m north and 4 m east, distance $=\sqrt{9+16}=5$ m.

5. Blood relations

Draw a family tree: use + for male, - for female, horizontal lines for siblings and vertical lines for generations.

• "The only child of my grandmother" is one of my parents.
• If A and B are the parents of C, and B is C's mother, then A is C's father.
• A father's or mother's brother is an uncle; their child is a cousin.

6. Data sufficiency

Decide whether the statements give enough information; you do not always need to calculate.

Example: "How many students are in the class? I. Sana teaches one-third of them. II. Sana teaches 20 students." Neither alone is enough; together total $=60$. Answer: both needed.

7. Number puzzles and ages

• Translate words into equations: "5 more than twice the son's age" $=2s+5$.
• Respect range limits: "greater than 11 and less than 24" excludes 11 and 24.
• Smallest prime in 11 to 24 (exclusive) is 13; largest multiple of 3 there is 21.

Common MDCAT traps

• Forgetting to subtract 1 when combining positions from both ends.
• Finding an original price by adding the discount to the sale price.
• Counting the boundary values when the question says "greater than" or "less than".
• Forgetting the weight of containers when asked for contents only.

Quick revision

• Row total = left + right - 1.
• Between two ranks = difference - 1.
• Original price = sale price ÷ (1 - discount).
• Opposite direction legs cancel.
• Draw a family tree for relation questions.