Puzzles and Miscellaneous Problems — Ranking, Age, Profit/Loss, Partnership, Time-Work

This note covers miscellaneous quantitative and reasoning topics that appear frequently in Kerala PSC exams. Each section has shortcuts and solved examples.

1. Ranking and Position Problems

Key Formulas

Scenario	Formula
Total persons in a row	(Position from left) + (Position from right) - 1
Position from opposite end	Total + 1 - Given position
Two people interchange	After interchange, new positions swap

Solved Examples

Example 1: Arun is 12th from the left and 18th from the right in a row. How many persons are in the row?

Solution: Total = 12 + 18 - 1 = 29

Example 2: In a row of 40 students, Ravi is 15th from the left. What is his position from the right?

Solution: Position from right = 40 + 1 - 15 = 26th

Example 3: In a row, Meena is 8th from the left and Seema is 10th from the right. If they interchange, Meena becomes 15th from the left. How many persons are in the row?

Solution: After interchange, Meena is at Seema’s original position. Seema was 10th from right. So Meena’s new position from left = 15. Her position from right = 10. Total = 15 + 10 - 1 = 24

2. Age Problems

Key Concepts

Concept	Explanation
Age difference	Remains constant over time
Ratio of ages	Changes over time
Sum of ages	Increases by (number of people x years passed)

Shortcuts

• If the ratio of ages of A and B is a:b now, and was c:d years ago, set ages as ax and bx, then solve: (ax - years)/(bx - years) = c/d

Solved Examples

Example 1: The present ages of A and B are in the ratio 5:3. After 6 years, their ages will be in the ratio 3:2. Find their present ages.

Solution: Let ages be 5x and 3x. After 6 years: (5x + 6)/(3x + 6) = 3/2 Cross multiply: 2(5x + 6) = 3(3x + 6) 10x + 12 = 9x + 18 x = 6 Ages: A = 30 years, B = 18 years

Example 2: A father is 30 years older than his son. 5 years ago, the father was 7 times the son’s age. Find the son’s present age.

Solution: Let son’s present age = x. Father = x + 30. 5 years ago: (x + 30 - 5) = 7(x - 5) x + 25 = 7x - 35 60 = 6x, so x = 10 years

3. Profit and Loss

Key Formulas

Term	Formula
Profit	SP - CP
Loss	CP - SP
Profit %	(Profit / CP) x 100
Loss %	(Loss / CP) x 100
SP (when profit)	CP x (100 + P%) / 100
SP (when loss)	CP x (100 - L%) / 100
If two items sold at same SP, one at x% profit and other at x% loss	There is always a net loss = (x/10)² %

Shortcuts

Shortcut	When to Use
Successive discounts of a% and b% = a + b - ab/100	Equivalent single discount
Marked Price x (100 - Discount%)/100 = SP	Finding SP from MP
If CP of x articles = SP of y articles, Profit% = [(x-y)/y] x 100	Quick profit calculation

Solved Examples

Example 1: A shopkeeper buys an item for Rs 500 and sells it for Rs 600. Find the profit percentage.

Solution: Profit = 600 - 500 = 100. Profit% = (100/500) x 100 = 20%

Example 2: The CP of 20 articles equals the SP of 16 articles. Find the profit %.

Solution: Profit% = [(20 - 16)/16] x 100 = (4/16) x 100 = 25%

Example 3: Two articles are sold at Rs 1,000 each. On one, there is 20% profit and on the other, 20% loss. Find the overall gain or loss.

Solution: Net loss% = (20/10)² = 4%. This is always a loss. Alternatively: CP1 = 1000 x 100/120 = 833.33. CP2 = 1000 x 100/80 = 1250. Total CP = 2083.33. Total SP = 2000. Loss = 83.33. Loss% = (83.33/2083.33) x 100 = 4% loss

4. Partnership

Key Concept

Profit is shared in the ratio of (Capital x Time) invested by each partner.

Types

Type	Details
Simple partnership	All partners invest for the same duration; profit shared in ratio of capitals
Compound partnership	Partners invest for different durations; profit shared in ratio of (Capital x Time)

Solved Examples

Example 1: A invests Rs 40,000 and B invests Rs 60,000. They earn Rs 50,000 profit. Find each share.

Solution: Ratio = 40,000 : 60,000 = 2:3 A’s share = (2/5) x 50,000 = Rs 20,000 B’s share = (3/5) x 50,000 = Rs 30,000

Example 2: A invests Rs 10,000 for 12 months. B invests Rs 15,000 for 8 months. Profit is Rs 25,000. Find each share.

Solution: A’s investment = 10,000 x 12 = 1,20,000 B’s investment = 15,000 x 8 = 1,20,000 Ratio = 1:1 Each gets Rs 12,500

5. Time and Work

Key Concepts

Concept	Explanation
If A can do work in n days	A’s 1 day work = 1/n
A and B together	1 day work = 1/a + 1/b; Total days = ab/(a+b)
If A is twice as efficient as B	A takes half the time of B

LCM Method (Shortcut)

Instead of fractions, take LCM of given days as total units of work.

Solved Examples

Example 1: A can do a work in 10 days. B can do it in 15 days. In how many days can they do it together?

Fraction method: 1/10 + 1/15 = 5/30 = 1/6. Together = 6 days

LCM method: LCM(10, 15) = 30 units. A’s rate = 30/10 = 3 units/day. B’s rate = 30/15 = 2 units/day. Together = 5 units/day. Time = 30/5 = 6 days

Example 2: A and B can do a work in 12 days. B and C in 15 days. A and C in 20 days. In how many days can all three do it together?

Solution: 1/A+B + 1/B+C + 1/A+C = 1/12 + 1/15 + 1/20 = (5 + 4 + 3)/60 = 12/60 = 1/5 This equals 2(A+B+C)‘s 1 day work. So A+B+C’s 1 day work = 1/10. Together = 10 days

Example 3: A can do a work in 6 days. B can destroy (undo) the same work in 12 days. If both work together, when will the work be completed?

Solution: Net work per day = 1/6 - 1/12 = 1/12. Time = 12 days

6. Pipes and Cisterns

Key Concepts

Concept	Explanation
Filling pipe	Positive work (adds water)
Emptying pipe (leak)	Negative work (removes water)
If a pipe fills in n hours	Rate = 1/n per hour
Net rate with fill and drain	1/fill - 1/drain

Solved Example

Example: Pipe A fills a tank in 20 minutes. Pipe B empties it in 30 minutes. If both are open, how long to fill the tank?

Solution: Net rate = 1/20 - 1/30 = (3-2)/60 = 1/60. Time to fill = 60 minutes

7. Quick Reference — Common Fractions to Percentages

Fraction	Percentage
1/2	50%
1/3	33.33%
1/4	25%
1/5	20%
1/6	16.67%
1/7	14.28%
1/8	12.5%
1/9	11.11%
1/10	10%
1/11	9.09%
1/12	8.33%

PSC-Focused Quick Recall

Question Pattern	Answer/Shortcut
Total in a row formula	Left position + Right position - 1
Age difference over time	Stays constant
Two items sold at same SP with x% profit and x% loss	Always net loss of (x/10)² %
Partnership profit sharing	Ratio of (Capital x Time)
A + B work together formula	ab/(a+b) days
Pipe A fills in m min, B empties in n min, together?	mn/(n-m) minutes (if n is greater than m)
CP of x items = SP of y items, Profit%	[(x-y)/y] x 100
Successive discounts a% and b%	Equivalent = a + b - ab/100
LCM method advantage	Avoids fractions; faster calculation