Given:

A candidate who gets 38% of the votes is rejected by 28,800 votes.

Calculation:

Let the total votes polled be x.

The number of votes the winner gets = 62% of x

⇒ $62x\over100$ 

Number of votes received by the lost candidates = 38% of x

⇒ $38x\over100$ 

Difference,

$62x\over100$ - $38x\over100$ = 2,88,00

⇒ $24x\over100$ = 2,88,00

⇒ x = ${2,88,00 \times 100}\over24$ = 1,20,000

∴ The total number of votes polled is 1,20,000.

Shortcut Trick

Rejected candidates gets vote 38%

Selected candidate gets vote (100 - 38)% = 62%

Difference between both candidates vote = 62% - 38% = 24%

Value of 24 % = 28,800

⇒ 1% = 28,800 / 24 =1200

⇒ 100% = 1200 × 100 = 1,20,000 (Total vote)

∴ Option 4 is correct.

Shortcut Trick

Let, the cost price = 100 units

∴ Market price = 100 + 100  of 35% = 135 units

The selling price = 135 - 135 of 20% = 108 units

So, 108 units = Rs. 432

⇒ 100 units = (432/108) × 100 = Rs. 400

∴ The cost price of the article is Rs. 400

Alternate Method

Given:

The marked price of an article is 35% above its cost price

The selling price = Rs. 432

Formula used:

S.P. = (100 - Discount%)/100 × M.P.

Calculation:

Let, the cost price of the article = x

Then, the market price = 135x/100

So, (135x/100) × [(100 - 20)/100] = 432

⇒ (135x/100) × (80/100) = 432

⇒ x = 432 × (100/135) × (100/80)

⇒ x = 400

∴ The cost price of the article is Rs. 400

Shortcut Trick

Calculate the Total Interest Percentage: (Interest ÷ Principal) × 100

Total Interest % = (1260 ÷ 18000) × 100 = 7%

Given the Annual Rate of Interest is 7% per annum.

Since the total interest percentage (7%) matches the annual rate (7%), the Time Period must be exactly 1 year.

∴ The correct answer is 1 year.

Alternate Method

Given:

Principal (P) = ₹18,000

Compound Interest (CI) = ₹1,260

Rate (R) = 7% per annum

Formula: Amount (A) = Principal (P) + Compound Interest (CI)

Calculation:

⇒ Amount (A) = 18000 + 1260 = ₹19,260

⇒ Formula for Compound Interest: A = P(1 + R/100)ⁿ

⇒ 19260 = 18000(1 + 7/100)ⁿ

⇒ 19260 ÷ 18000 = (107/100)ⁿ

⇒ 1.07 = (1.07)ⁿ

⇒ (1.07)¹ = (1.07)ⁿ

⇒ n = 1

∴ The correct answer is 1 year.

Additional Information

Compound Interest Formula

The formula for the total amount is A = P(1 + r/n)^nt, where n is the number of times interest is compounded per year.

CI vs SI for the First Year

For the first year (or the first compounding period), Simple Interest and Compound Interest are always equal on the same principal and rate.

Effective Interest Rate

When interest is compounded more than once a year, the Effective Annual Rate (EAR) is higher than the nominal rate, calculated as [(1 + r/n)ⁿ − 1] × 100.

Given:

Principal Amount (P) = Rs.1000

Rate of Interest (R) = 5% per annum

Time Period (T) = 2 years

Formula Used:

Simple Interest (SI) = (P × R × T) / 100

Calculation:

SI = (1000 × 5 × 2) / 100

⇒ SI = (10000) / 100

⇒ SI = 100

The simple interest earned is Rs.100.

Given:

The wallet contains Rs. 900 

A wallet containing 100 rupee notes, Rs. 50 rupee notes and 20 rupee notes in the ratio of 2 : 3 : 5

Calculation:

A wallet containing 100 rupee notes, Rs. 50 rupee notes and 20 rupee notes in the ratio of 2 : 3 : 5

So, total 100 rupees notes are 2x, total 50 rupees notes are 3x, total 20 rupees notes are 5x

Now, A/Q

(2x) × 100 + (3x) × 50 + (5x) × 20 = 900

⇒ 200x + 150x + 100x = 900

⇒ 450x = 900

⇒ x = 2

So, Total Rs 100 notes is 2 × 2 = 4

Total Rs 50 notes = 3 × 2 = 6

Total Rs 20 notes = 5 × 2 = 10

Formula Used:

Average Speed = 2ab/(a + b)

Applicable when one travels at speed a for half the distance and speed b for the other half of the distance.

In this case, the average speed is the harmonic mean of the two speeds. On similar lines, you can modify this formula for one-third distance.

Calculation:

Using the above formula - 

⇒ a = 10 km/hr, b = 15 km/hr

⇒ Average speed = $\frac{2\times 10\times 15}{10+15}$ = 12 km/hr

∴ The correct answer is 12km/hr 

Additional Information

1. Average Speed = (a + b)/2

Applicable when one travels at speed a for half the time and speed b for another half of the time.

In this case, the average speed is the arithmetic mean of the two speeds.

2. Average Speed = 3abc/(ab + bc + ca)

Applicable when one travels at speed a for one-third of the distance, at speed b for another one-third of the distance, and at speed c for the rest of the one-third of the distance.

Shortcut Trick

Profit Ratio of A, B and C = (10 × 5) : (12 × 6) : (15 × 9) = 50 : 72 : 135

Total profit in ratio parts = 50 + 72 + 135 = 257 parts

Given A's profit = 50 parts = Rs. 2500 ⇒ 1 part = Rs. 50

Total Profit = 257 parts = 257 × 50 = Rs. 12,850

∴ The correct answer is Rs. 12,850.

Alternate Method

Given:

Investment of A = Rs. 10,000, Time = 5 months

Investment of B = Rs. 12,000, Time = 6 months

Investment of C = Rs. 15,000, Time = 9 months

A's profit share = Rs. 2,500

Formula Used:

Profit Ratio = Investment × Time

Calculations:

⇒ Profit ratio (A : B : C) = (10000 × 5) : (12000 × 6) : (15000 × 9)

⇒ Profit ratio = 50000 : 72000 : 135000 = 50 : 72 : 135

⇒ Let the profits of A, B, and C be 50x, 72x, and 135x respectively.

⇒ Total profit = 50x + 72x + 135x = 257x

⇒ Given A's profit = 50x = 2500

⇒ x = 2500 ÷ 50 = 50

⇒ Total Profit = 257x = 257 × 50 = 12850

∴ The correct answer is Rs. 12,850.

Additional Information

Compound Partnership

When capitals of partners are invested for different time periods, the profit or loss is distributed in the ratio of the product of their capital and time.

Simple Partnership

When capitals of partners are invested for the same period of time, the profit or loss is distributed in the exact ratio of their original investments.

Working and Sleeping Partners

A working partner manages the business and receives a specific salary or percentage. A sleeping partner only invests capital. The remaining profit after salary deduction is shared in the investment ratio.

Given:

He sells 60% apples.

Apples left = 420

Calculation:

Let originally he had x apples.

Then, according to the question

⇒ (100 - 60)% of x = 420

⇒ ($\frac{40}{{100}}$ × x) = 420

⇒ $\frac{2x}{{5}}$ = 420

⇒ x = 1050 

∴ The required apples he had is 1050.

Shortcut TrickHe sells 60% apples i.e., 40% is still left

⇒ 40% = 420 apples

⇒ 100% = 1050 apples

Given:

A train crosses a station platform in 36 seconds

And crosses a man standing on the platform in 20 seconds

The speed of the train is 54 km/h

Formula Used:

Speed = Distance/Time

Calculation:

Let the length of the train be x m and the length of the platform be y m 

According to the question

54 × (5/18) = x/20

⇒ 15 × 20 = x

⇒ x = 300 m

Again, According to the question

⇒ 54 × (5/18) = (300 + y) /36

⇒ 15 × 36 = 300 + x

⇒ y = 540 – 300

⇒ y = 240

∴ The length of the platform is 240.

Shortcut Trick As given train cross its length in 20 seconds.

So the train cross the platform in = 36 – 20 = 16 seconds

The distance covered by train in 16 seconds = 54 × (5/18) × 16 = 240

The logic followed here is:

Option 1) # = MLI, % = KJK ⇒ MLI : HJE : : OQL : KJK

Option 2) # = HGY, % = LHU ⇒ HGY : HJE : : OQL : LHU 

Option 3) # = LNI, % = KMH ⇒ LNI : HJE : : OQL : KMH 

Option 4) # = PKO, % = NHU ⇒ PKO : HJE : : OQL : NHU 

Hence, the correct answer is "Option 3".