Quantitative Aptitude PYQs

Common Calculation

For A,

Total user = 1600

Number of active users = 1600 × 75 /100 = 1200

Number of Inactive users = 1600 – 1200 = 400

Number of users who deleted the app = 400 × 3 / 5 = 240

Others state values is given in the table.

Calculation

Sum deleted [A + B + C] = [240 + 270 + 50] = 560.

Given sum = 5n.

So 5n=560

⇒ n = [560 ÷ 5] = 112

Common Calculation

For A,

Total user = 1600

Number of active users = 1600 × 75 /100 = 1200

Number of Inactive users = 1600 – 1200 = 400

Number of users who deleted the app = 400 × 3 / 5 = 240

Others state values is given in the table.

Calculation

Active in D = 20% more than C

⇒ [1350 × 1.20] = 1350 + 270 = 1620

Sum of active users in D and A = 1620 + 1200 = 2820

Common Calculation

For A,

Total user = 1600

Number of active users = 1600 × 75 /100 = 1200

Number of Inactive users = 1600 – 1200 = 400

Number of users who deleted the app = 400 × 3 / 5 = 240

Others state values is given in the table.

Calculation

Active users in C = 1350.

Ratio male: female = 3:2

⇒ Total parts = 5.

Difference between Male and Female

= [1/5] × 1350 = (1350 ÷ 5) = 270

Common Calculation

For A,

Total user = 1600

Number of active users = 1600 × 75 /100 = 1200

Number of Inactive users = 1600 – 1200 = 400

Number of users who deleted the app = 400 × 3 / 5 = 240

Others state values is given in the table.

Calculation

Deleted in A = 5t.

So 5t = 240.

So, t = [240 ÷ 5] = 48

So, [t + 5] = [48 + 5] = 53

Common Calculation

For A,

Total user = 1600

Number of active users = 1600 × 75 /100 = 1200

Number of Inactive users = 1600 – 1200 = 400

Number of users who deleted the app = 400 × 3 / 5 = 240

Others state values is given in the table.

Calculation

Inactive in B = 432.

Ratio male: female = 5:3

⇒ total parts = 8.

Female inactive in B = [3/8] × 432 = 54 × 3 = 162

Shortcut Trick

Let the total number of voters in the electoral roll be 100x.

Winner votes = 60% of 100x = 60x.

Total votes cast = 90x. Invalid votes = 600. Valid votes = 90x − 600.

Loser votes = (90x − 600) − 60x = 30x − 600.

Margin = Winner − Loser ⇒ 60x − (30x − 600) = 4200 ⇒ 30x + 600 = 4200 ⇒ 30x = 3600 ⇒ x = 120.

Voters who cast their votes = 90x = 90 × 120 = 10800.

∴ The correct answer is 10800.

Alternate Method

Given: Winner secured 60% of total list. Winning margin = 4200. Invalid votes = 600. Not cast = 10%.

Formula: Total Cast = (Winner Votes + Loser Votes + Invalid Votes)

⇒ Let Total Voters = 100x

⇒ Votes Cast = 90% of 100x = 90x

⇒ Winner Votes = 60% of 100x = 60x

⇒ Valid Votes = 90x − 600

⇒ Loser Votes = Valid Votes − Winner Votes = (90x − 600) − 60x = 30x − 600

⇒ Difference = Winner − Loser = 60x − (30x − 600) = 4200

⇒ 30x + 600 = 4200

⇒ 30x = 3600 ⇒ x = 120

⇒ Total Votes Cast = 90x = 90 × 120 = 10800

∴ The correct answer is 10800.

Additional Information

Election Terms

Electoral Roll refers to the total registered voters. Valid votes are the remaining votes after subtracting invalid ones from the total votes cast.

Base Change in Percentage

Always check if the percentage is based on Total Votes or Valid Votes. In this question, 60% is calculated on the total voter list.

Shortcut Trick

Ratio of Downstream time (T_d) to Upstream time (T_u) for same distance = 7 : 10

Since Speed ∝ 1/Time for a constant distance, Speed ratio (S_d : S_u) = 10 : 7

Upstream speed (S_u) = 70 km ÷ 5 hours = 14 kmph

Using ratio: 7 units → 14 kmph, then 10 units → 20 kmph (Downstream Speed)

Time for 95 km downstream = 95 ÷ 20 = 4.75 hours

∴ The correct answer is 4.75 hours.

Alternate Method

Given: Upstream distance = 70 km, Time = 5 hours. Ratio T_d : T_u = 7 : 10.

Formula Used: Speed = Distance ÷ Time; Downstream Speed (S_d) = u + v; Upstream Speed (S_u) = u − v.

⇒ Upstream speed (S_u) = 70 / 5 = 14 kmph

⇒ For distance x, Time ∝ 1/Speed

⇒ T_d / T_u = S_u / S_d

⇒ 7 / 10 = 14 / S_d

⇒ S_d = (14 × 10) / 7 = 20 kmph

⇒ Required Time = Downstream Distance / S_d

⇒ Required Time = 95 / 20 = 4.75 hours

∴ The correct answer is 4.75 hours.

Additional Information

Downstream and Upstream Speed

Downstream Speed (S_d) = Speed of boat in still water (u) + Speed of stream (v). Upstream Speed (S_u) = u − v.

Boat and Stream Relation

Speed of boat in still water (u) = (S_d + S_u) / 2. Speed of stream (v) = (S_d − S_u) / 2.

Ratio of Speeds

For a constant distance, the ratio of speeds is the inverse of the ratio of time taken: S₁/S₂ = T₂/T₁.

Shortcut Trick

Relative Speed when moving towards each other = Sum of individual speeds.

Relative Speed = 10 + 5 = 15 m/sec.

Meeting Time = Total Distance ÷ Relative Speed.

Meeting Time = 1050 ÷ 15 = 70 sec.

∴ The correct answer is 70 sec.

Alternate Method

Given:

Distance (D) = 1050 m

Speed of Kapil (S₁) = 10 m/sec

Speed of Anand (S₂) = 5 m/sec

Formula Used:

Time = Distance ÷ Relative Speed

Relative Speed (Opposite direction) = S₁ + S₂

Calculations:

⇒ Relative Speed = 10 + 5 = 15 m/sec

⇒ Time = 1050 ÷ 15

⇒ Time = 70 sec

∴ The correct answer is 70 sec.

Additional Information

Relative Speed Concept

When two objects move in opposite directions (towards each other), their relative speed is the sum of their individual speeds (S₁ + S₂).

Same Direction Speed

When two objects move in the same direction, their relative speed is the difference of their individual speeds (S₁ − S₂).

Unit Conversion

To convert km/hr to m/sec, multiply by 5/18. To convert m/sec to km/hr, multiply by 18/5.

Shortcut Trick

Make the total capacity of both vessels equal: Vessel A (5+3 = 8 units) and Vessel B (4+1 = 5 units).

LCM of 8 and 5 is 40. Multiply Vessel A by 5 and Vessel B by 8.

New ratios → A: (5×5) : (3×5) = 25 : 15 | B: (4×8) : (1×8) = 32 : 8.

Total Milk = 25 + 32 = 57 | Total Water = 15 + 8 = 23.

∴ The correct answer is 57 : 23.

Alternate Method

Given: Ratio in A = 5 : 3, Ratio in B = 4 :

1. Capacity A = Capacity B.

Formula: Total Quantity = Sum of components from each vessel.

⇒ Quantity of Milk in C = (5/8) + (4/5)

⇒ Quantity of Milk in C = (25 + 32) / 40 = 57/40

⇒ Quantity of Water in C = (3/8) + (1/5)

⇒ Quantity of Water in C = (15 + 8) / 40 = 23/40

⇒ Required Ratio = (57/40) : (23/40) = 57 : 23

∴ The correct answer is 57 : 23.

Additional Information

Mixtures of Different Capacities

If vessels have different capacities in ratio x : y, multiply the component fractions of each vessel by x and y respectively before adding.

Alligation Rule

Used to find the ratio in which two mixtures are mixed to get a final concentration: (Quantity 1 / Quantity 2) = (C₂ − C_m) / (C_m − C₁).

I. 2x² – 22x + 60 = 0

⇒ 2x² – 12x – 10x + 60 = 0

⇒ (x – 5) (2x – 12) = 0

⇒ x =5, 6

II. y² – 9y + 20 = 0

⇒ y² – 4y – 5y + 20 = 0

⇒ (y – 5) (y – 4) = 0

⇒ y = 5, 4  

 

 

Hence, x ≥ y.

I. 9x² – 27x + 20 = 0

⇒ 9x² – 15x – 12x + 20 = 0

⇒ 3x (3x – 5) – 4 (3x – 5) = 0

⇒ (3x – 5) (3x – 4) = 0

⇒ x = 4/3, 5/3

 

II. 3y² – 10y + 8 = 0

⇒ 3y² – 4y – 6y + 8 = 0

⇒ y (3y – 4) – 2 (3y – 4) = 0

⇒ (3y – 4) (y – 2) = 0

⇒ y = 4/3, 2

 

 

Hence, relationship between x and y cannot be established.

I. 2x² –18= 0   

⇒ x² –9= 0

⇒ x = ± 3

⇒ x = –3, 3

II. y²– 12y + 32 = 0

⇒ y² – 4y – 8y + 32 = 0

⇒ (y – 4) (y – 8) = 0

⇒ y = 4, 8

 

 

Hence, x < y.

I. 3x² – 1x – 24 = 0

⇒ 3x² – 9x + 8x – 24 = 0

⇒ 3x (x – 3) + 8 (x – 3) = 0

⇒ (x – 3) (3x + 8) = 0

⇒ x = -8/3, 3

 

II. 3y² + 4y – 20 = 0

⇒ 3y² + 10y – 6y – 20 = 0

⇒ y (3y + 10) – 2 (3y + 10) = 0

⇒ (y – 2) (3y + 10) = 0

⇒ y = -10/3, 2

 

 

Hence, relationship between x and y cannot be established.

I. 7x² – 37x + 10 = 0

⇒ 7x² – 35x – 2x + 10 = 0

⇒ 7x (x – 5) – 2 (x – 5) = 0

⇒ (7x – 2) (x – 5) = 0

⇒ x = 2/7, 5

 

II. 4y² – 47y + 138 = 0

⇒ 4y² – 24y – 23y + 138 = 0

⇒ 4y (y – 6) – 23(y – 6) = 0

⇒ (4y – 23) (y – 6) = 0

⇒ y = 23/4, 6

Comparison between x and y (via Tabulation):

 

Hence x < y.

Given:

Let number of soaps sold by Raju, Ramu and Rasu be a, b and c respectively while number of detergents packs sold by Raju, Ramu and Rasu be d, e and f respectively.

⇒ a + d + b + e + c + f = 225

⇒ d - 20 = a

⇒ a + d = 80

Solving,

a = 30 and d = 50

f = 50 - 20 = 30

⇒ c - f = 10

⇒ c = 30 + 10 = 40

Then,

⇒ b + e = 225 - 30 - 50 - 40 - 30

⇒ b + e = 75

b : e = 11 : 4

Solving,

b = 55 and e = 20

 

Required difference = b - e

= 55 - 20 = 35

Given:

Let the number of soaps sold by Raju, Ramu and Rasu be a, b and c respectively while number of detergents packs sold by Raju, Ramu and Rasu be d, e and f respectively.

⇒ a + d + b + e + c + f = 225

⇒ d - 20 = a

⇒ a + d = 80

Solving,

a = 30 and d = 50

⇒ f = 50 - 20 = 30

⇒ c - f = 10

⇒ c = 30 + 10 = 40

Then,

⇒ b + e = 225 - 30 - 50 - 40 - 30

⇒ b + e = 75

b : e = 11 : 4

Solving,

b = 55 and e = 20

 

Total number of soaps sold by three = a + b + c = 30 + 55 + 40 = 125

Required percentage =

= 30 / 125 × 100

= 24%

Given:

Let number of soaps sold by Raju, Ramu and Rasu be a, b and c respectively while number of detergents packs sold by Raju, Ramu and Rasu be d, e and f respectively.

⇒ a + d + b + e + c + f = 225

⇒ d - 20 = a

⇒ a + d = 80

Solving,

a = 30 and d = 50

⇒ f = 50 - 20 = 30

⇒ c - f = 10

⇒ c = 30 + 10 = 40

Then,

⇒ b + e = 225 - 30 - 50 - 40 - 30

⇒ b + e = 75

b : e = 11 : 4

Solving,

b = 55 and e = 20

 

Total number of soaps sold by three = a + b + c = 30 + 55 + 40 = 125

Total number od detergents by three = d + e + f = 50 + 20 + 30 = 100

Required percentage = 125 = 100 + 100 × ? / 100

⇒ ? = 25%

Shortcut Trick

Interest for 1 month = ₹400 → Interest for 1 year = ₹400 × 12 = ₹4800.

Principal X (Simple Interest at 10%) = 4800 ÷ 0.10 = ₹48000.

Principal Y (CI semi-annually): Rate = 5% per half-year, n = 2 periods. Effective Rate = 5 + 5 + (5×5)/100 = 10.25%.

Value of Y = 4800 ÷ 0.1025 ≈ ₹46829.

Difference X − Y = 48000 − 46829 = ₹1171.

∴ The correct answer is ₹1171.

Alternate Method

Given: Monthly interest = ₹400, Rate (R) = 10% p.a., Time (T) = 1 year.

Formula Used: SI = (P × R × T) ÷ 100 | CI = P[(1 + r/100)ⁿ − 1]

Step 1: Calculate X

⇒ Annual interest = 400 × 12 = ₹4800

⇒ X = (SI × 100) ÷ (R × T) = (4800 × 100) ÷ (10 × 1) = ₹48000

Step 2: Calculate Y

⇒ For compounding semi-annually: New Rate = 10% ÷ 2 = 5%; Time = 1 year × 2 = 2 periods

⇒ Interest = Y × [(1 + 5/100)² − 1] = 4800

⇒ Y × [1.1025 − 1] = 4800

⇒ Y = 4800 ÷ 0.1025 = ₹46829.26

Step 3: Find Difference

⇒ Difference = X − Y = 48000 − 46829.26 = ₹1170.74 ≈ ₹1171

∴ The correct answer is ₹1171.

Additional Information

Effective Rate of Interest

When interest is compounded semi-annually, the effective annual rate is given by R_eff = (1 + r/200)² − 1, where r is the nominal annual rate.

Simple Interest vs Compound Interest

For the same interest earned, a compounding account requires a smaller principal (P) compared to a simple interest account because interest also earns interest.

Compounding Periods

Quarterly: Rate = R/4, Time = 4T. Monthly: Rate = R/12, Time = 12T.

Shortcut Trick

Let the Marked Price be 100%.

Discounted Price = 100% − 20% = 80%.

Price with GST = 80% × (100 + 8)% ÷ 100 = 80% × 1.08 = 86.4%.

According to the question, 86.4% = 4104.

Marked Price (100%) = (4104 ÷ 86.4) × 100 = 4750.

∴ The correct answer is Rs.4750.

Alternate Method

Given → Discount = 20%, GST = 8%, Final Price = Rs.4104.
Formulas → SP = MP × (100 − D%) ÷ 100 | Final Price = SP × (100 + GST%) ÷ 100.

⇒ Let Marked Price be x.

⇒ Price after 20% discount = x × (80 ÷ 100) = 0.8x.

⇒ Final price after 8% GST = 0.8x × (108 ÷ 100) = 0.864x.

⇒ 0.864x = 4104.

⇒ x = 4104 ÷ 0.864 = 4750.

∴ The correct answer is Rs.4750.

Additional Information

Marked Price & Discount

The price listed on the product is the Marked Price. Discount is always calculated on the Marked Price to arrive at the Selling Price.

Sales Tax/GST

GST is an indirect tax applied on the final selling price. It increases the amount the customer has to pay: Final Price = SP × (100 + Tax%) ÷ 100.

Shortcut Trick

Let notes of ₹10, ₹50, ₹100, ₹200 be N₁₀, N₅₀, N₁₀₀, N₂₀₀.

Ratio N₁₀ : N₅₀ = 30 : 7 and N₁₀₀ : N₂₀₀ = 5 : 3.

Amount equality: 10 × N₁₀ = 200 × N₂₀₀ ⇒ N₁₀ = 20 × N₂₀₀.

Substitute ratio values: Let N₂₀₀ = 3k ⇒ N₁₀ = 60k. Since N₁₀ : N₅₀ is 30 : 7, then N₅₀ = 14k. Given N₁₀₀ : N₂₀₀ is 5 : 3, N₁₀₀ = 5k.

Sum of notes: 60k + 5k = 325 ⇒ 65k = 325 ⇒ k = 5. Total amount = (10×300) + (50×70) + (100×25) + (200×15) = 3000 + 3500 + 2500 + 3000 = ₹12,000.

∴ The correct answer is ₹12,000.

Alternate Method

Given: Ratio N₁₀:N₅₀ = 30:7, Ratio N₁₀₀:N₂₀₀ = 5:3, Amount ₹10 = Amount ₹200, N₁₀ + N₁₀₀ = 325.

⇒ Let N₁₀ = 30x, N₅₀ = 7x and N₁₀₀ = 5y, N₂₀₀ = 3y.

⇒ Condition 1: 10 × 30x = 200 × 3y

⇒ 300x = 600y ⇒ x = 2y

⇒ Condition 2: N₁₀ + N₁₀₀ = 325

⇒ 30x + 5y = 325

⇒ Substitute x = 2y: 30(2y) + 5y = 325

⇒ 60y + 5y = 325 ⇒ 65y = 325 ⇒ y = 5

⇒ Then x = 2 × 5 = 10

⇒ Notes: N₁₀ = 300, N₅₀ = 70, N₁₀₀ = 25, N₂₀₀ = 15

⇒ Total Amount = (300×10) + (70×50) + (25×100) + (15×200)

⇒ Total Amount = 3000 + 3500 + 2500 + 3000 = ₹12,000

∴ The correct answer is ₹12,000.

Additional Information

Ratio of Value and Number

The total value of a specific denomination is calculated as (Face Value × Number of Notes). Always convert ratios to a single variable when conditions link different groups.

Proportionality in Denominations

If the total amount of two different denominations is equal, the ratio of their count is the inverse of the ratio of their face values (N₁ : N₂ = V₂ : V₁).