Quantitative Aptitude PYQs

Shortcut Trick

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.

Shortcut Trick

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).

Total shoes sold by Retailer E = 620 + 580 = 1200

Total shoes sold by Retailer A = 450 + 320 = 770

Percentage = (Total E ÷ Total A) × 100 = (1200 ÷ 770) × 100 ≈ 155.84%

Hence, the correct answer is 155.84%.

Total Non-leather shoes sold by all retailers:

320 + 480 + 500 + 450 + 580 = 2330

Number of retailers = 5

Average = Total ÷ Number of retailers = 2330 ÷ 5 = 466

Hence, the correct answer is 466.

Total Leather shoes sold by A and B:

Leather (A + B) = 450 + 510 = 960

Total Non-leather shoes sold by B and C:

Non-leather (B + C) = 480 + 500 = 980

Leather (A + B) : Non-leather (B + C) = 960 : 980 = 48 : 49

Hence, the correct answer is 48 : 49.

Leather shoes sold:

Retailer B = 510

Retailer D = 550

Difference = 550 − 510 = 40

Percentage less = (Difference ÷ D) × 100 = (40 ÷ 550) × 100 ≈ 7.27%

Hence, the correct answer is ≈ 7.27%.

Retailer C → Leather = 480, Non-leather = 500 → Total = 480 + 500 = 980  

Retailer E → Leather = 620, Non-leather = 580 → Total = 620 + 580 = 1200  

Combined total = 980 + 1200 = 2180

Hence, the correct answer is 2180.

Shortcut Trick

Downstream Speed (u) = 11 ÷ 2 = 5.5 km/h

Upstream Speed (v) = 11 ÷ 4 = 2.75 km/h

Boat Speed = (u + v) ÷ 2 = (5.5 + 2.75) ÷ 2 = 8.25 ÷ 2 = 4.125 km/h

Current Speed = (u − v) ÷ 2 = (5.5 − 2.75) ÷ 2 = 2.75 ÷ 2 = 1.375 km/h

∴ The correct answer is 4.125 km/h and 1.375 km/h.

Alternate Method

Given:

Distance (D) = 11 km

Time taken along the stream (t₁) = 2 hours

Time taken against the stream (t₂) = 4 hours

Formula Used:

Speed = Distance ÷ Time

Downstream Speed (u) = x + y | Upstream Speed (v) = x − y

Speed of boat in still water (x) = (u + v) ÷ 2

Speed of current (y) = (u − v) ÷ 2

Calculations:

⇒ Downstream Speed (u) = 11 ÷ 2 = 5.5 km/h

⇒ Upstream Speed (v) = 11 ÷ 4 = 2.75 km/h

⇒ Speed of boat (x) = (5.5 + 2.75) ÷ 2

⇒ x = 8.25 ÷ 2 = 4.125 km/h

⇒ Speed of current (y) = (5.5 − 2.75) ÷ 2

⇒ y = 2.75 ÷ 2 = 1.375 km/h

∴ The correct answer is 4.125 km/h and 1.375 km/h.

Additional Information

Relative Speed in Water

When moving with the flow, the speeds add up (x + y). When moving against the flow, the stream speed is subtracted from the boat speed (x − y).

Average Speed of Boat

If the boat covers equal distance upstream and downstream, Average Speed = (u × v) / x = (x² − y²) / x km/h.

Time Ratio Property

The ratio of Speed of Boat to Speed of Current is given by (t_up + t_down) : (t_up − t_down).

Shortcut Trick

Use the Alligation Method to find the ratio of quantities.

Profit 1: 11%, Profit 2: 18%, Mean Profit: 14%.

Ratio = (18 − 14) : (14 − 11) = 4 : 3.

Total units = 4 + 3 = 7 units.

7 units = 980 kg ⇒ 1 unit = 140 kg.

Quantity at 18% profit = 3 × 140 = 420 kg.

∴ The correct answer is 420 kg.

Alternate Method

Given: Total quantity = 980 kg, Part 1 profit = 11%, Part 2 profit = 18%, Overall profit = 14%.

Formulas: Profit = SP − CP | Weighted Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂)

⇒ Let quantity at 18% be x kg, then quantity at 11% is (980 − x) kg.

⇒ 11(980 − x) + 18x = 14 × 980

⇒ 10780 − 11x + 18x = 13720

⇒ 7x = 13720 − 10780

⇒ 7x = 2940

⇒ x = 420 kg.

∴ The correct answer is 420 kg.

Additional Information

Alligation Rule

It is used to find the ratio in which two ingredients at given prices/profits are mixed to produce a mixture of a mean price/profit.

Profit and Loss on Quantities

When overall profit % is given for two parts of a total quantity, the ratio of quantities is inversely proportional to the deviation of their profit % from the mean profit %.

The pattern of the given series is a Fibonacci series.

Every number from third number is a sum of the previous two numbers.

⇒ 17 = 5 + 12

⇒ 29 = 17 + 12

⇒ 46 = 29 + 17

⇒ 75 = 46 + 29

⇒ 121 = 75 + 46

⇒ 196 = 121 + 75

The series is formed by the following logic

⇒ 3 + 1² = 4

⇒ 4 + 3² = 13

⇒ 13 + 5² = 38

⇒ 38 + 7² = 87

⇒ 87 + 9² = 168

∴ the answer is 38

The pattern of the number series is as follows:

17 × 2 + 4 = 38

38 × 2 + 4 = 80

80 × 2 + 4 = 164

164 × 2 + 4 = 332

332 × 2 + 4 = 668

668 × 2 + 4 = 1340

∴ The required next number is 1340.

Given:

Population of town is 63400

In the first year 15% and second year decreases by 20%

Calculation:

According to question,

⇒ {63400 × (115/100) × (80/100)}

⇒ 58328

∴ After 2 year later population of town is 58328

 

Formula used:

[X + Y + {(XY)/100}%]

⇒ X = 15%, and Y = - 20%

Applying the net% effect formula

⇒ [(+15 - 20) - {15 × 20}/100]

⇒ - 5 - 3 

⇒ - 8

So, (100 - 8)% of 63400

⇒ 92% of 63400

⇒ 58328

∴ After 2 year later population of town is 58328

 

1^st year 15% increase(3/20) , 2^nd year 20% decrease(1/5)

   year  =           Present pop.                last pop.

1^st year =          20                                     23

2^nd yrs =            5                                      4

After 2 yrs present pop. 100  and last pop. 92 

Given to 100 unit = 63400

⇒ 1 unit = 634

⇒ last population = 634 × 92

⇒ 58328

∴ After 2 year later population of town is 58328

Shortcut Trick

Total Time taken from 9 a.m. to 2 p.m. = 5 hours.

Distance = Speed × Time

Distance = 37 × 5 = 185 km.

∴ The correct answer is 185 km.

Alternate Method

Given:

Starting time = 9 a.m.

Arrival time = 2 p.m.

Speed of Train A = 37 km/h

Formula Used:

Distance = Speed × Time

Calculations:

⇒ Time duration = 2 p.m. − 9 a.m.

⇒ Time duration = (12 − 9) + 2 = 5 hours

⇒ Distance = 37 km/h × 5 hours

⇒ Distance = 185 km

∴ The correct answer is 185 km.

Additional Information

Speed-Distance Relation

The fundamental relation is Distance = Speed × Time. If Speed is in km/h and Time is in hours, Distance will be in km.

Unit Conversion

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

Average Speed

Average Speed = Total Distance ÷ Total Time. For two equal distances with speeds x and y, Average Speed = 2xy ÷ (x + y).

Given expression:

⇒ 48% of 250 + 1050 ÷ 3 – √(?) = 423

⇒ (48/100) × 250 + 1050 ÷ 3 – √(?) = 423

⇒ (48/100) × 250 + 350 – √(?) = 423

⇒ 120 + 350 – √(?) = 423

⇒ 470 – √(?) = 423

⇒ √(?) = 470 – 423

⇒ √(?) = 47

⇒ ? = 2209

Given, 139% of 459 + \(5\frac{1}{2}\) of 384

Approximating terms to the closest values, we get

⇒ 140% of 460 + 11/2 × 384

⇒ (140 × 460)/100 + 11/2 × 384

⇒ 644 + 2112 = 2756 ≈ 2760

∴ Approx value of the expression is 2760

⇒ 150 + 45 × 2(36 – 16) ÷ 18 = ?

⇒ 150 + 45 × 2(20) ÷ 18

⇒ 150 + 100 = 250