Shortcut Trick

Convert speed to kmph: 15 × (18/5) = 54 kmph.

Convert time to hours: 3 hours 20 mins = 3 + (1/3) = 10/3 hours.

Distance = 54 × (10/3) = 18 × 10 = 180 km.

∴ The correct answer is 180 km.

Alternate Method

Given: Speed = 15 m/s, Time = 3 hours 20 minutes.

Formula Used: Distance = Speed × Time

⇒ Speed in kmph = 15 × (18/5) = 54 kmph

⇒ Time in hours = 3 + (20/60) = 3 + 1/3 = 10/3 hours

⇒ Distance = 54 × (10/3)

⇒ Distance = 18 × 10 = 180 km

∴ The correct answer is 180 km.

Additional Information

Unit Conversion

To convert m/s to km/h, multiply by 18/5. To convert km/h to m/s, multiply by 5/18.

Average Speed

When a distance is covered at speed x and the same distance at speed y, Average Speed = (2 × x × y) ÷ (x + y).

Relative Speed

For two objects moving in the same direction, speed = S₁ − S₂. In opposite directions, speed = S₁ + S₂.

Given:

Sum of two numbers  = 72

Ratio of two numbers = 4 : 5

Calculation:

Let the two numbers be 4x and 5x

According to the question,

4x + 5x = 72

⇒ 9x = 72

⇒ x = 72/9 = 8

Now, If each number is increased by 8.

(4x + 8) : (5x + 8)

⇒ (4 × 8 + 8) : (5 × 8 + 8)

⇒ (32+ 8) : (40 + 8)

⇒ 40 : 48 = 5 : 6

∴ The ratio of the new numbers is 5 : 6.

If 8 numbers are 1, 2, 3, 4, 5, 6, 7, 8

The average of these numbers is

⇒ (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)/8 = 4.5

If next four numbers are added, then,

⇒ (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12)/12 = 6.5

∴ The average is increased by 2

[Note: No matter which consecutive numbers we choose, the increase or decrease in average is same]

Shortcut Trick

Total Average of 9 numbers = 32

Net deviation for overlap = 5 × (22 − 32) + 5 × (38 − 32)

Net deviation = 5 × (−10) + 5 × (6) = −50 + 30 = −20

Fifth number = 32 − 20 = 12

∴ The correct answer is 12.

Alternate Method

Given:

Average of 9 numbers = 32

Average of first 5 numbers = 22

Average of last 5 numbers = 38

Formula Used:

Sum = Average × Number of observations

Fifth number (overlap) = (Sum of first 5 + Sum of last 5) − Total Sum of 9 numbers

Calculations:

⇒ Sum of 9 numbers = 9 × 32 = 288

⇒ Sum of first 5 numbers = 5 × 22 = 110

⇒ Sum of last 5 numbers = 5 × 38 = 190

⇒ Sum of (first 5 + last 5) = 110 + 190 = 300

⇒ Fifth number = 300 − 288 = 12

∴ The correct answer is 12.

Additional Information

Average of Overlapping Terms

When the sum of the subsets (m + n) exceeds the total number of terms (N), the overlapping term is calculated as: Overlap = (Sum_m + Sum_n) − Total Sum.

Average of Missing Terms

If the sum of subsets (m + n) is less than the total terms (N), the missing term is: Missing = Total Sum − (Sum_m + Sum_n).

Arithmetic Mean Formula

The basic formula for average is Sum of all terms ÷ Total number of terms.

Shortcut Trick

For consecutive years, the rate of interest is calculated on the previous year's amount.

Rate % = [(Difference in Amounts) ÷ (Initial Amount)] × 100

Rate % = [(2,977.54 − 2,809) ÷ 2,809] × 100

Rate % = [168.54 ÷ 2,809] × 100 = 6%

∴ The correct answer is 6%.

Alternate Method

Given:

Amount after 2 years (A₂) = ₹2,809

Amount after 3 years (A₃) = ₹2,977.54

Formula Used:

In compound interest, A = P(1 + R/100)ⁿ. For consecutive years, A_n+1 = A_n(1 + R/100).

Calculations:

⇒ Interest for the 3^rd year = A₃ − A₂

⇒ Interest = 2,977.54 − 2,809 = 168.54

⇒ This interest is earned on the principal of ₹2,809 (which is the amount at the end of year 2).

⇒ Rate (R) = (Interest ÷ Principal) × 100

⇒ R = (168.54 ÷ 2,809) × 100

⇒ R = 16854 ÷ 2809

⇒ R = 6%

∴ The correct answer is 6%.

Additional Information

Compound Interest Formula

The amount A after n years is given by A = P(1 + R/100)ⁿ, where P is the principal and R is the annual rate.

Successive Percentage Change

Compound interest is essentially a successive percentage increase. For two years, the effective rate is x + y + (xy/100).

Difference between CI and SI

For 2 years, the difference between CI and SI is P(R/100)².

Shortcut Trick

The extra time of 4 seconds (20 − 16) is taken to cover the 60-metre platform.

Speed = Platform Length ÷ Time Difference

Speed = 60 ÷ 4 = 15 m/s

Speed in kmph = 15 × (18/5) = 54 kmph

∴ The correct answer is 54 kmph.

Alternate Method

Given:

Length of platform = 60 m

Time to cross platform = 20 sec

Time to cross man = 16 sec

Formula Used:

Speed = Distance ÷ Time

Calculations:

Let the length of the train be L and speed be V.

When the train crosses a man, it covers its own length:

⇒ V = L ÷ 16 → (1)

When the train crosses the platform, it covers its own length plus platform length:

⇒ V = (L + 60) ÷ 20 → (2)

Equating (1) and (2):

⇒ L ÷ 16 = (L + 60) ÷ 20

⇒ 20L = 16(L + 60)

⇒ 20L = 16L + 960

⇒ 4L = 960

⇒ L = 240 m

Substitute L in equation (1):

⇒ Speed (V) = 240 ÷ 16 = 15 m/s

Converting Speed to kmph:

⇒ Speed = 15 × (18 ÷ 5) = 54 kmph

∴ The correct answer is 54 kmph.

Additional Information

Unit Conversion

To convert speed from m/s to kmph, multiply by 18/5. To convert kmph to m/s, multiply by 5/18.

Train Crossing Concepts

When a train crosses a pole/man, distance = train length. When it crosses a bridge/platform, distance = train length + bridge/platform length.

Relative Speed

Same direction: Subtract speeds (S₁ − S₂). Opposite direction: Add speeds (S₁ + S₂).

Given:

Present population of village = 28000

Rate at which population increases = 5%

Time period = 2 years

Formula used:

\({P_n} = P{\left( {1 + \frac{R}{{100}}} \right)^n}\)

Where, P_n = Population after n years

P = Present population

R = Rate of growth of population

n = time period (in years)

Calculations:

Population after 2 years,

\( \Rightarrow {P_2} = 28000{\left( {1 + \frac{5}{{100}}} \right)^2}\)

\( \Rightarrow {P_2} = 28000{\left( {\frac{{21}}{{20}}} \right)^2}\)

\( \Rightarrow {P_2} = 28000\left( {\frac{{441}}{{400}}} \right)\)

⇒ P₂ = 70 × 441

⇒ P₂ = 30870

∴ The population after 2 years will be 30870

Shortcut Trick

A has 55% and B has 45% votes (Difference = 10%).

For a tie, A must give half of the difference (5% of total) to B.

Given: 5% of Total = 10,000 votes.

Total votes = (10,000 ÷ 5) × 100 = 200,000 votes.

∴ The correct answer is 200000 votes.

Alternate Method

Given: A = 55% of total, B = 45% of total.

Formula: Total Votes = 100%.

A: 55% B: 45% 10,000 votes transferred (5% shift) Tie Point = 50%

⇒ Let the total number of votes be T.

⇒ Initial votes for A = 0.55T.

⇒ Initial votes for B = 100% − 55% = 45% = 0.45T.

⇒ According to the condition: 0.55T − 10,000 = 0.45T + 10,000.

⇒ 0.55T − 0.45T = 10,000 + 10,000.

⇒ 0.10T = 20,000.

⇒ T = 20,000 ÷ 0.10 = 200,000.

∴ The correct answer is 200000 votes.

Additional Information

Voting Terminology

Total Votes = Valid Votes + Invalid Votes. Majority is the difference between the winner and the runner-up.

Percentage Transfer Concept

If quantity X gives value 'v' to quantity Y and they become equal, then the original difference is 2 × v.

Successive Percentage

For two successive percentage changes a% and b%, the effective change is [a + b + (a × b)/100]%.

∴ The correct answer is 200000 votes.

Given∶

Ratio of investment = 3 ∶ 1

Profit goes to charity = 4% of total profit

Manmohan's share = Rs. 1080

Formula Used∶ 

Basics of Ratio and Unitary Method.

Calculation∶

Let the total profit be of Rs. 100

Given ratio = 3 ∶ 1

Sum of the ratio = 3 + 1 = 4

After paying to charity, Manmohan's share = (96 × 3/4) = Rs. 72

If share is Rs. 72, then total profit = Rs. 100

If Manmohan's share is Rs.1, then total profit = Rs.100/72

If Manmohan's share is Rs. 1080, then total profit = (100 /72) × 1080 = 1500

∴ The total profit is of Rs. 1500.

The logic followed here is:

Given: # : LNI :: QSN : %

So, checking each option one by one:

Option 1) # = LKD, % = VML → LKD : LNI :: QSN : VML

Option 2) # = GID, % = VXS → GID : LNI :: QSN : VXS

Option 3) # = JDD, % = VNH  → JDD : LNI :: QSN : VNH

Option 4) # = JHD, % = MJP → JHD : LNI :: QSN : MJP

Hence, "Option 2" is the correct answer.