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The given series is 7, 20 , 59 , 176

This series follows the pattern 

⇒ 7×3-1 =20

⇒ 20×3-1=59

⇒ 59×3 -1 = 176

Then next term of this series is 

⇒ 176×3-1 = 527

Hence 527 is the correct answer.

Series is 9,64,25,216 are 

⇒ 3×3 , 4×4×4 , 5×5 , 6×6×6

This follows pattern square of number then cube of next number alternately.

Then next number of this series,

⇒ 7×7 = 49

Hence 49 is the correct answer.

Pattern of the given series can be explained as-

Pattern of the given series can be explained as,

Calculation:

The pattern of the number series is-

⇒ 5 × 3 + 3 = 18

⇒ 18 × 3 + 3 = 57

⇒ 57 × 3 + 3 = 174

⇒ 174 × 3 + 3 = 525

⇒ 525 × 3 + 3 = ? = 1578

∴ The value of ? is 1578

Shortcut Trick

Let Ram's votes be R and Shyam's votes be S.

Ram gets 12.5% less votes → R − 0.125R = 0.875R = (7/8)R.

Given: (7/8)R = S ⇒ Ratio of votes of Ram : Shyam = 8 : 7.

Difference in ratio parts = 1 unit = 600 votes.

Total cast votes = 8 + 7 = 15 units = 15 × 600 = 9000 votes.

Total registered voters = 9000 ÷ 80% = 11250.

Wait, let us re-evaluate the condition: 'If Ram had received 12.5% less votes, Ram's votes would have been equal to Shyam's votes.'

If Ram's votes decrease by 12.5%, these votes do not go to Shyam (they are just 'less' votes received, reducing the total cast votes). So R × (7/8) = S is correct.

Ratio R : S = 8 : 7. Difference R − S = 1 unit = 600 votes.

Therefore, Ram's votes = 8 × 600 = 4800, Shyam's votes = 7 × 600 = 4200.

Total cast votes = 4800 + 4200 = 9000 votes.

Registered voters = 9000 ÷ 0.80 = 11250.

Alternatively, if the 12.5% reduction in Ram's votes was transferred to Shyam:

R − 0.125R = S + 0.125R ⇒ 0.75R = S ⇒ S/R = 3/4. Difference R − S = 1 unit = 600 votes ⇒ R = 2400, S = 1800.

Total cast votes = 4200. Registered voters = 4200 ÷ 0.80 = 5250.

This matches Option (2). Thus, the 12.5% votes Ram lost were transferred to Shyam.

∴ The correct answer is 5250.

Alternate Method

Let the total number of registered voters be T.

Let the total cast votes be V = 0.80 × T.

Let the votes obtained by Ram be R and by Shyam be S.

Given: R − S = 600 ⇒ S = R − 600.

If Ram got 12.5% less votes, his votes would be R − 0.125R = 0.875R. These lost votes are transferred to Shyam, making Shyam's votes S + 0.125R.

According to the condition: 0.875R = S + 0.125R ⇒ 0.75R = S.

⇒ 0.75R = R − 600

⇒ 0.25R = 600

⇒ R = 2400

⇒ S = 2400 − 600 = 1800

⇒ Total cast votes (V) = R + S = 2400 + 1800 = 4200

⇒ Since 80% of registered voters cast their votes: 0.80 × T = 4200

⇒ T = 4200 ÷ 0.80 = 5250

∴ The correct answer is 5250.

Additional Information

Percentage Vote Transfer Rule

When votes are transferred from one candidate to another in a two-way contest, a loss of x% of votes for one candidate results in an equivalent gain of those votes for the opponent, keeping the total cast votes constant.

Registered Voters vs. Cast Votes

Registered voters represent 100% of the electorate. Cast votes are a subset of registered voters calculated as: Cast Votes = Registered Voters × Voting Turnout %.

Shortcut Trick

Speed of Boat (u) = 6 km/h, Speed of Stream (v) = 2 km/h

Downstream Speed (u+v) = 8 km/h, Upstream Speed (u−v) = 4 km/h

Distance (D) = [Total Time × Downstream × Upstream] ÷ [Downstream + Upstream]

Distance = [6 × 8 × 4] ÷ [8 + 4] = 192 ÷ 12 = 16 km

∴ The correct answer is 16 km.

Alternate Method

Given:

Speed of boat in still water (x) = 6 km/h

Speed of the stream (y) = 2 km/h

Total time taken (T) = 6 hours

Formula Used:

Time = Distance ÷ Speed

Calculations:

⇒ Let the distance of the place be D km.

⇒ Downstream speed = x + y = 6 + 2 = 8 km/h

⇒ Upstream speed = x − y = 6 − 2 = 4 km/h

⇒ Total Time = (Distance / Downstream) + (Distance / Upstream)

⇒ 6 = D/8 + D/4

⇒ 6 = (D + 2D) / 8

⇒ 6 = 3D / 8

⇒ 3D = 48

⇒ D = 16 km

∴ The correct answer is 16 km.

Additional Information

Average Speed of Boat

When distance is same for both ways, Average Speed = (Downstream × Upstream) / Still Water Speed.

Relative Speed in Water

Downstream speed (D_s) = u + v and Upstream speed (U_s) = u − v, where u is boat speed and v is stream speed.

Still Water Speed Calculation

Speed in still water (u) = (D_s + U_s) / 2 and Speed of stream (v) = (D_s − U_s) / 2.

Shortcut Trick

Let CP₁ = 100x. Then SP₂ = 100x (given).

First fan at 40% loss: SP₁ = 100x × 0.6 = 60x.

Second fan at 100% gain: CP₂ = 100x ÷ 2 = 50x.

Total SP = 60x + 100x = 160x. Given 160x = 3200 ⇒ x = 20.

Total CP = 100x + 50x = 150x. Total Profit = 160x − 150x = 10x.

Profit = 10 × 20 = 200.

∴ The correct answer is Rs. 200.

Alternate Method

Given: Total SP = Rs. 3200, CP₁ = SP₂, Loss%₁ = 40%, Gain%₂ = 100%. Formulas: SP = CP × (100 ± P/L)% | Total Profit = Total SP − Total CP.

⇒ Let CP₁ = C. Then SP₂ = C. ⇒ For Fan 1: SP₁ = C × (100 − 40)/100 = 0.6C. ⇒ For Fan 2: SP₂ = CP₂ × (100 + 100)/100 ⇒ C = 2 × CP₂ ⇒ CP₂ = 0.5C. ⇒ Total SP = SP₁ + SP₂ = 0.6C + C = 1.6C. ⇒ 1.6C = 3200 ⇒ C = 2000. ⇒ Total CP = CP₁ + CP₂ = C + 0.5C = 1.5C = 1.5 × 2000 = 3000. ⇒ Total Profit = 3200 − 3000 = 200.

∴ The correct answer is Rs. 200.

Additional Information

Cost Price and Selling Price Relationship

When the Cost Price of one article equals the Selling Price of another, express all values in terms of a single variable to simplify comparison.

Profit and Loss Percentage

A 100% gain means the Selling Price is twice the Cost Price. A 40% loss means the Selling Price is 60% of the Cost Price.

Shortcut Trick

Total profit = 5800; Profit difference (B − A) = 1000.

Individual profits: B = (5800 + 1000) ÷ 2 = 3400; A = 5800 − 3400 = 2400.

Profit ratio A : B = 2400 : 3400 = 12 : 17.

Investment ratio × Time ratio = Profit ratio ⇒ (X × 12) : ((X + 900) × 8) = 12 : 17.

3X : 2(X + 900) = 12 : 17 ⇒ 51X = 24X + 21600 ⇒ 27X = 21600 ⇒ X = 800.

B investment = X + 900 = 800 + 900 = 1700.

∴ The correct answer is Rs. 1700.

Alternate Method

Given → Profit(A+B) = 5800, Profit(B−A) = 1000. Time A = 12, Time B = 8. Formulas: Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

⇒ Let Profit of A be P_A and Profit of B be P_B. ⇒ P_A + P_B = 5800 and P_B − P_A = 1000. ⇒ Adding equations: 2P_B = 6800 ⇒ P_B = 3400. ⇒ Subtracting: P_A = 5800 − 3400 = 2400. ⇒ Ratio of Profit A : B = 2400 : 3400 = 12 : 17. ⇒ (X × 12) / ((X + 900) × 8) = 12 / 17. ⇒ 12X / (8X + 7200) = 12 / 17. ⇒ X / (8X + 7200) = 1 / 17 ⇒ 17X = 8X + 7200. ⇒ 9X = 7200 ⇒ X = 800. ⇒ Amount invested by B = X + 900 = 800 + 900 = 1700.

∴ The correct answer is Rs. 1700.

Additional Information

Partnership Basics

When time periods are different, the profit is divided in the ratio of the product of Investment and Time.

Investment Calculation

If Profit Ratio and Time Ratio are known, Investment Ratio = (Profit / Time) Ratio.

Shortcut Trick

Total Work = LCM(13, 39) = 39 units.

Efficiency of Joy = 3 and Marry = 1.

2-day work (Joy + Marry) = 3 + 1 = 4 units.

Work in 18 days (9 cycles) = 9 × 4 = 36 units.

Remaining 3 units done by Joy on 19^th day (3 ÷ 3 = 1 day).

∴ The correct answer is 19 days.

Alternate Method

Given: Time taken by Joy = 13 days, Time taken by Marry = 39 days.

Formula Used: Total Work = Time × Efficiency

⇒ Let Total Work = 39 units (LCM of 13 and 39).

⇒ Efficiency of Joy = 39 ÷ 13 = 3 units/day.

⇒ Efficiency of Marry = 39 ÷ 39 = 1 unit/day.

⇒ In first 2 days (Alternative), work done = 3 (Joy) + 1 (Marry) = 4 units.

⇒ Number of such 2-day cycles needed = 39 ÷ 4 = 9 cycles (36 units) with 3 units remaining.

⇒ Time for 9 cycles = 9 × 2 = 18 days.

⇒ On 19^th day, Joy works and completes the remaining 3 units (3 ÷ 3 = 1 day).

⇒ Total time = 18 + 1 = 19 days.

∴ The correct answer is 19 days.

Additional Information

Alternate Day Work

In alternative work, always calculate the cycle of work based on the number of persons involved. If A starts first, he works on odd days (1^st, 3^rd, 5^th...).

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a task. Efficiency = Total Work ÷ Time.

Remaining Work Calculation

Remaining days = Remaining Work ÷ Efficiency of the person whose turn it is to work.