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When time is constant, the distance ratio is equal to the speed ratio.

Speed reduced by 3/5 → New Speed = 1 − 3/5 = 2/5 of Original Speed.

Ratio of Speeds (Original : New) = 5 : 2 ∴ Ratio of Distance = 5 : 2.

Difference in distance = 5 − 2 = 3 units → 54 km. So, 1 unit = 18 km.

Original Distance = 5 units × 18 = 90 km. Speed = 90 km ÷ 45 hrs = 2 km/hr.

∴ The correct answer is 2 km/hr.

Alternate Method

Given: Time (T) = 45 hrs, Speed reduction = 3/5, Distance reduction = 54 km.

Formula: Distance = Speed × Time

⇒ Let the original speed be 's' km/hr.

⇒ Reduced speed = s − (3/5)s = (2/5)s km/hr.

⇒ Original Distance (D₁) = s × 45

⇒ New Distance (D₂) = (2/5)s × 45 = 18s

⇒ Difference in distance (D₁ − D₂) = 45s − 18s = 54 km

⇒ 27s = 54

⇒ s = 54 ÷ 27 = 2 km/hr

∴ The correct answer is 2 km/hr.

Additional Information

Constant Time Property

When time is kept constant, the ratio of the speeds of two objects is directly proportional to the ratio of the distances covered by them (S₁/S₂ = D₁/D₂).

Average Speed

If a person covers a certain distance at speed x and the same distance at speed y, the average speed for the whole journey is (2xy)/(x + y).

Relative Speed

Speed of two bodies moving in the same direction is the difference of their speeds (u − v), and in the opposite direction is the sum of their speeds (u + v).

Shortcut Trick

Ratio of numbers = 5 : 4 : 8

Sum of (1^st + 2^nd) ratio units = 5 + 4 = 9 units

Difference between this sum and 3^rd ratio unit = 9 − 8 = 1 unit

Given 1 unit = 18

Difference between 3^rd (8 units) and 2^nd (4 units) = 8 − 4 = 4 units

4 units = 4 × 18 = 72

∴ The correct answer is 72.

Alternate Method

Given: Ratio of three numbers = 5 : 4 : 8

Let the three numbers be 5x, 4x, and 8x.

As per the question:

⇒ (5x + 4x) = 8x + 18

⇒ 9x = 8x + 18

⇒ 9x − 8x = 18

⇒ x = 18

To find the difference between the third number and second number:

⇒ Difference = 8x − 4x

⇒ Difference = 4x

⇒ Difference = 4 × 18

⇒ Difference = 72

∴ The correct answer is 72.

Additional Information

Ratio and Proportion

A ratio compares values by division. If a : b : c is given, numbers are assumed as ax, bx, and cx where x is the common multiplier.

Property of Equations

When translating word problems to linear equations, carefully group terms like 'sum of' and 'difference between' to maintain equality.

Shortcut Trick

Total ratio parts = 11 + 9 = 20 units.

Total profit = Rs. 5,000 ⇒ 1 unit = 5,000 ÷ 20 = Rs. 250.

Goutam profit = 9 units = 9 × 250 = Rs. 2,250.

Required value = 120% of Goutam profit = 1.2 × 2,250 = Rs. 2,700.

∴ The correct answer is Rs. 2,700.

Alternate Method

Given → Investment ratio Suresh : Goutam = 11 : 9, Time = 1 year, Total Profit = Rs. 5,000. Formulas: Profit Ratio = Investment × Time | Individual Profit = (Own Ratio ÷ Total Ratio) × Total Profit.

⇒ Ratio of profits = 11 × 1 : 9 × 1 = 11 : 9. ⇒ Goutam share of profit = (9 ÷ (11 + 9)) × 5,000. ⇒ Goutam profit = (9 ÷ 20) × 5,000 = 9 × 250 = Rs. 2,250. ⇒ Amount 20% more than Goutam profit = 2,250 + (20% of 2,250). ⇒ Result = 2,250 + 450 = Rs. 2,700.

∴ The correct answer is Rs. 2,700.

Additional Information

Partnership Concept

Profit is distributed in the ratio of the product of Investment and Time Period.

Percentage Increase

To find a value x% more than A, use the formula: Result = A × (100 + x) ÷ 100.

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Successive percentage formula: Net Change = a + b + (a × b) ÷ 100

Here, profit markup (a) = +25% and discount (b) = −17%.

Net Profit% = 25 − 17 + (25 × −17) ÷ 100

Net Profit% = 8 − 425 ÷ 100 = 8 − 4.25 = 3.75%.

The approximate value is 3.7%.

∴ The correct answer is 3.7%.

Alternate Method

Given: Cost Price (CP) = Rs. 964, Markup Profit% = 25%, Discount% = 17%. Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Profit% = (SP − CP) ÷ CP × 100.

⇒ Assume CP = 100 units (Note: Actual CP of 964 is redundant for percentage calculation). ⇒ Marked Price (MP) = 100 + 25% of 100 = 125 units. ⇒ Selling Price (SP) = 125 − 17% of 125 = 125 − 21.25 = 103.75 units. ⇒ Profit = 103.75 − 100 = 3.75 units. ⇒ Profit% = (3.75 ÷ 100) × 100 = 3.75%. ⇒ Approx profit percentage = 3.7%.

∴ The correct answer is 3.7%.

Additional Information

Marked Price & Discount

Marked price is the price tagged on an article. Discount is always calculated on the Marked Price unless stated otherwise.

Redundant Data

When the final answer is required in percentage and the initial conditions are in percentages, the absolute value (Rs. 964) does not affect the result.

Shortcut Trick

Total Votes = 8000

Valid Votes = 70% of 8000 = 5600

Other candidate percentage = 100% − 60% = 40%

Required Votes = 40% of 5600 = 2240

∴ The correct answer is 2240.

Alternate Method

Given:

Total votes = 8000

Invalid votes = 30%

First candidate share = 60% of valid votes

Calculations:

⇒ Valid votes % = 100% − 30% = 70%

⇒ Number of valid votes = 8000 × (70/100) = 5600

⇒ Share of the other candidate = 100% − 60% = 40%

⇒ Votes of the other candidate = 5600 × (40/100)

⇒ Votes of the other candidate = 56 × 40 = 2240

∴ The correct answer is 2240.

Additional Information

Percentage Change Base

Always ensure the percentage is applied to the correct base. Here, the candidate share is 60% of valid votes, not total votes.

Valid vs Invalid Votes

Valid Votes = Total Votes − Invalid Votes. Only valid votes are distributed among candidates.

Successive Multiplication

The result can be found in one step: Result = Total × (Valid% / 100) × (Other Candidate% / 100).

Here, persons K, L, M, N, O, P, Q and R.

1. Three persons live between O and P.

2. O lives on an odd number floor below the fifth floor.

 

3. One person lives between M and L who lives above M.

4. K lives immediately above R.

5. R lives above Q but below M.

 

6. N and Q lives on an adjacent floor. Therefore, case 2 is eliminated.

7. N does not live just below O. Hence, the final arrangement is as follows

 

Hence, O lives three floors below the floor of M.

Here, persons K, L, M, N, O, P, Q and R.

1. Three persons live between O and P.

2. O lives on an odd number floor below the fifth floor.

 

3. One person lives between M and L who lives above M.

4. K lives immediately above R.

5. R lives above Q but below M.

 

6. N and Q lives on an adjacent floor. Therefore, case 2 is eliminated.

7. N does not live just below O. Hence, the final arrangement is as follows

 

Since, one person lives below the floor of Q and one person lives above the floor of P.

Hence, the correct answer is P.

Here, persons K, L, M, N, O, P, Q and R.

1. Three persons live between O and P.

2. O lives on an odd number floor below the fifth floor.

 

3. One person lives between M and L who lives above M.

4. K lives immediately above R.

5. R lives above Q but below M.

 

6. N and Q lives on an adjacent floor. Therefore, case 2 is eliminated.

7. N does not live just below O. Hence, the final arrangement is as follows

 

Hence, M lives on sixth floor.

Here, persons K, L, M, N, O, P, Q and R.

1. Three persons live between O and P.

2. O lives on an odd number floor below the fifth floor.

 

3. One person lives between M and L who lives above M.

4. K lives immediately above R.

5. R lives above Q but below M.

 

6. N and Q lives on an adjacent floor. Therefore, case 2 is eliminated.

7. N does not live just below O. Hence, the final arrangement is as follows

 

Since, M lives just above the floor of K. Therefore, statement first is correct.

N lives on an odd number floor. Therefore, statement second is correct.

O lives below R. Therefore, statement third is correct.

Hence, the correct answer is all of these.

Here, persons K, L, M, N, O, P, Q and R.

1. Three persons live between O and P.

2. O lives on an odd number floor below the fifth floor.

 

3. One person lives between M and L who lives above M.

4. K lives immediately above R.

5. R lives above Q but below M.

 

6. N and Q lives on an adjacent floor. Therefore, case 2 is eliminated.

7. N does not live just below O. Hence, the final arrangement is as follows

 

Since, the position of M and Q remain the same.

Hence, the correct answer is two persons.