Quantitative Aptitude PYQs

Shortcut Trick

Upstream speed (U) = (x + 4) − (x − 5) = 9 km/hr.

Downstream speed (D) = (x + 4) + (x − 5) = 2x − 1 km/hr.

Upstream time = 108 ÷ 9 = 12 hours.

Downstream time = 18 − 12 = 6 hours.

Downstream speed (D) = 114 ÷ 6 = 19 km/hr ⇒ 2x − 1 = 19 ⇒ x = 10.

Speed of stream = x − 5 = 10 − 5 = 5 km/hr.

∴ The correct answer is 5 km/hr.

Alternate Method

Given:

Speed of boat in still water (u) = x + 4 km/hr

Speed of stream (v) = x − 5 km/hr

Formula Used:

Upstream Speed = u − v | Downstream Speed = u + v | Time = Distance ÷ Speed

Calculations:

⇒ Upstream Speed = (x + 4) − (x − 5) = 9 km/hr

⇒ Downstream Speed = (x + 4) + (x − 5) = (2x − 1) km/hr

⇒ Total Time = Upstream Time + Downstream Time

⇒ 18 = 108 ÷ 9 + 114 ÷ (2x − 1)

⇒ 18 = 12 + 114 ÷ (2x − 1)

⇒ 6 = 114 ÷ (2x − 1)

⇒ 2x − 1 = 114 ÷ 6 = 19

⇒ 2x = 20 ⇒ x = 10

⇒ Speed of stream = x − 5 = 10 − 5 = 5 km/hr

∴ The correct answer is 5 km/hr.

Additional Information

Upstream and Downstream Basics

When a boat moves against the current, it is upstream (u − v). When it moves with the current, it is downstream (u + v).

Average Speed of Boat

Average Speed = (Upstream Speed × Downstream Speed) ÷ Speed of Boat in Still Water (when distances are equal).

Speed Conversion

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

I. x2 + 11x − 80 = 0

⇒ x2 + 16x − 5x − 80 = 0

⇒ (x + 16), (x − 5) = 0

x = − 16, 5

II. y2 − 27y + 110 = 0

⇒ (y − 22)(y − 5) = 0

 y = 22, 5

∴ x ≤ y

Calculation:

2x² + 15x + 28 = 0

⇒ 2x² + 8x + 7x + 28 = 0

⇒ 2x(x + 4) + 7(x + 4)

⇒ x = -4, -7/2

3y² + 11y + 6 = 0

⇒ 3y2 + 9y + 2y + 6 = 0

⇒ 3y(y + 3) + 2(y + 3)

⇒ Y = -3, -2/3

Comparison between X and Y (via tabulation):

∴ x < y

According to given equations

I. x² + 28x + 147 = 0

⇒ x² + 21x + 7x + 147 = 0

⇒ x(x + 21) + 7(x + 2) = 0

⇒ (x + 21) (x + 7) = 0

⇒ x = -21, -7

II. y² – 15y – 154 = 0 

⇒ y² – 22y + 7y – 154 = 0

⇒ y(y – 22) + 7(y – 22) = 0

⇒ (y – 22) (y + 7) = 0

⇒ y = 22, -7

Comparison between x and y (via Tabulation):

 ∴ y ≥ x

According to given equations

I. x² – 15x + 44 = 0

⇒ x² – 11x – 4x + 44 = 0

⇒ x(x – 11) – 4(x – 11) = 0

⇒ (x – 4) (x – 11) = 0

⇒ x = 11, 4

II. 3y² – 22y + 35 = 0

⇒ 3y² – 15y – 7y + 35 = 0

⇒ 3y(y – 5) – 7(y – 5) = 0

⇒ (y – 5) (3y – 7) = 0

⇒ y = 7/3, 5

Comparison between x and y (via Tabulation):

∴ The relationship between x and y cannot be established.

Shortcut Trick

Net change in weight = (10 × (45 − 48)) − (5 × (50 − 48))

Net change = −30 − 10 = −40 kg

New total people = 50 + 10 − 5 = 55

New Average = 48 + (−40 ÷ 55) ≈ 48 − 0.72 = 47.28 kg

∴ The correct answer is 47 kg.

Alternate Method

Given: Initial 50 people (avg 48kg), Added 10 people (avg 45kg), Removed 5 people (avg 50kg).

Formula: Average = Total Weight ÷ Number of People

⇒ Total Weight = (50 × 48) + (10 × 45) − (5 × 50)

⇒ Total Weight = 2400 + 450 − 250 = 2600 kg

⇒ Total Number of People = 50 + 10 − 5 = 55

⇒ New Average = 2600 ÷ 55 = 47.2727... kg

⇒ Approximate Average ≈ 47 kg

∴ The correct answer is 47 kg.

Additional Information

Weighted Average Formula

Average = (n₁x₁ + n₂x₂ − n₃x₃) ÷ (n₁ + n₂ − n₃) where x represents the average and n represents number of observations.

Deviation Method for Averages

Net Deviation = ∑(Individual Values − Assumed Average). New Average = Assumed Average + (Net Deviation ÷ Total Number).

Shortcut Trick

Let original speed be S kmph. Using the product of speed formula: Distance = [S1 × S2 / (S2 − S1)] × Time Difference.

450 = [S × (S + 10) / 10] × (30/60) ⇒ 450 = [S(S + 10) / 10] × 1/2.

S(S + 10) = 9000 ⇒ S = 90 kmph (as 90 × 100 = 9000).

Time for 3/5^th journey = (3/5 × 450) ÷ 90 = 270 ÷ 90 = 3 hr.

∴ The correct answer is 3 hr.

Alternate Method

Given: Total distance = 450 km, Speed increase = 10 kmph, Time saved = 30 mins = 0.5 hr.

Formula Used: Time = Distance / Speed

⇒ (450 / S) − [450 / (S + 10)] = 1/2

⇒ 450 [ (S + 10 − S) / (S(S + 10)) ] = 1/2

⇒ 450 [ 10 / (S² + 10S) ] = 1/2

⇒ 4500 × 2 = S² + 10S

⇒ S² + 10S − 9000 = 0

⇒ S² + 100S − 90S − 9000 = 0 ⇒ (S + 100)(S − 90) = 0

⇒ S = 90 kmph (Speed cannot be negative)

⇒ 3/5^th Distance = 3/5 × 450 = 270 km

⇒ Time required = 270 / 90 = 3 hr

∴ The correct answer is 3 hr.

Additional Information

Speed-Time Inversely Proportional

For a constant distance, speed is inversely proportional to time (S₁/S₂ = T₂/T₁).

Distance Formula for Speed Change

Distance = [v₁v₂ / |v₁−v₂|] × Δt, where Δt is the difference in time taken at speeds v₁ and v₂.

Average Speed

Average Speed = Total Distance ÷ Total Time. For two equal halves of a journey with speeds x and y, it is 2xy/(x+y).

Common Calculation

Number of TV sold in shop A is 120.

Difference between number of TV and cell phone sold in shop A is 40.

Number of TV sold in each shop is more than number of cell phone sold.

So, Number of cell phone sold in shop A is 120 – 40 = 80.

Similarly, we can calculate others shops values also.

Calculation

Shop C revenue = (240 × 15000) + (160 × 8000) = 48,80,000

Shop B revenue = (160 × 15000) + (100 × 8000) = 32,00,000

Difference = 4880000 - 3200000 = 16,80,000

Common Calculation

Number of TV sold in shop A is 120.

Difference between number of TV and cell phone sold in shop A is 40.

Number of TV sold in each shop is more than number of cell phone sold.

So, Number of cell phone sold in shop A is 120 – 40 = 80.

Similarly, we can calculate others shops values also.

Calculation

Total TVs = 800 → Avg = 160

Total Cell phones = 540 → Avg = 108

% more = [(160 - 108)/108] × 100 ≈ 48.15%

Common Calculation

Number of TV sold in shop A is 120.

Difference between number of TV and cell phone sold in shop A is 40.

Number of TV sold in each shop is more than number of cell phone sold.

So, Number of cell phone sold in shop A is 120 – 40 = 80.

Similarly, we can calculate others shops values also.

Calculation

TVs in A + C = 120 + 240 = 360

Cell phones in B + D = 100 + 80 = 180

Ratio = 360 : 180 = 2 : 1

Common Calculation

Number of TV sold in shop A is 120.

Difference between number of TV and cell phone sold in shop A is 40.

Number of TV sold in each shop is more than number of cell phone sold.

So, Number of cell phone sold in shop A is 120 – 40 = 80.

Similarly, we can calculate others shops values also.

Calculation

Shop A total = 120 + 80 = 200

Shop E total = 180 + 120 = 300

Increase = 100

% Increase = (100/200) × 100 = 50%

Common Calculation

Number of TV sold in shop A is 120.

Difference between number of TV and cell phone sold in shop A is 40.

Number of TV sold in each shop is more than number of cell phone sold.

So, Number of cell phone sold in shop A is 120 – 40 = 80.

Similarly, we can calculate others shops values also.

Calculation

Shop D: TVs good = [100 × 75/100] =75, Cell phones = 80 → Total = 155

Shop A: TVs = 120, Cell phones good = [80 × 80/100] = 64 → Total = 184

Ratio = 155 : 184

Shortcut Trick

Compound Interest (CI) on ₹4000 at 10% for 2 years: Effective rate = 10 + 10 + (10×10)/100 = 21%.

CI = 21% of 4000 = ₹840.

Simple Interest (SI) = 1/2 × 840 = ₹420.

SI % for 3 years at 8% = 3 × 8 = 24%.

Principal = 420 ÷ 24% = 420 ÷ 0.24 = ₹1750.

∴ The correct answer is ₹1750.

Alternate Method

Given:

Compound Interest: P = ₹4000, R = 10%, T = 2 years.

Simple Interest: R = 8%, T = 3 years.

Formula Used:

CI = P[(1 + R/100)^T − 1]

SI = (P × R × T) / 100

Calculations:

⇒ CI = 4000 × [(1 + 10/100)² − 1]

⇒ CI = 4000 × [1.21 − 1] = 4000 × 0.21 = ₹840.

⇒ Given: SI = 1/2 × 840 = ₹420.

⇒ 420 = (Sum × 8 × 3) / 100

⇒ 420 = (Sum × 24) / 100

⇒ Sum = (420 × 100) / 24

⇒ Sum = ₹1750.

∴ The correct answer is ₹1750.

Additional Information

Effective Rate for Compound Interest

For 2 years, the effective rate is calculated as (x + y + xy/100)%. This helps avoid long power calculations.

Difference between CI and SI

The difference for 2 years is P(R/100)². For 3 years, it is P(R/100)² × (3 + R/100).

Simple Interest Property

Simple interest remains constant every year on the same principal, unlike compound interest where interest is earned on interest.

Shortcut Trick

C present age = 34 years → C (2 yrs later) = 36 years.

A (2 yrs later) = 2/3 × 36 = 24 years → A present age = 22 years.

A (4 yrs ago) = 18 years, which is 50% of D (4 yrs ago) → D (4 yrs ago) = 36 years.

D present = 40 years. Required Difference = |(22 − 3) − (40 + 5)| = |19 − 45| = 26 years.

∴ The correct answer is 26 years.

Alternate Method

Given: Ratio A:B = 11:9 | C present = 34 | A (present + 2) = 2/3 × C (present + 2)

⇒ C (2 years hence) = 34 + 2 = 36 years

⇒ A (2 years hence) = (2/3) × 36 = 24 years

⇒ A (Present age) = 24 − 2 = 22 years

⇒ Since A:B is 11:9, 11 units = 22 → 1 unit = 2 (B = 18 years, though not needed)

⇒ A (4 years ago) = 22 − 4 = 18 years

⇒ A (4 years ago) = 50% of D (4 years ago) ⇒ 18 = 0.5 × D_{4_ago} ⇒ D (4 years ago) = 36 years

⇒ D (Present age) = 36 + 4 = 40 years

⇒ Age of A 3 years ago = 22 − 3 = 19 years

⇒ Age of D 5 years hence = 40 + 5 = 45 years

⇒ Difference = 45 − 19 = 26 years

∴ The correct answer is 26 years.

Additional Information

Ratio and Proportions in Ages

When the ratio of ages at two different time points is given, the difference between the ages remains constant. Use 1 unit value to find actual ages.

Percentage and Fraction conversion

50% of a value is equivalent to 1/2 of the value. If X is 50% of Y, then Y = 2 × X.

Age Difference Consistency

The difference between the ages of two people always remains the same regardless of how many years pass or have passed.

Shortcut Trick

Assume total population = 100 units.

Males = 60; Females = 40.

Literate Males = 40% of 60 = 24; Literate Females = 25% of 40 = 10.

Total Literate = 24 + 10 = 34 units → 102000.

1 unit = 3000 ∴ 100 units = 300000.

∴ The correct answer is 300000.

Alternate Method

Given:

Total literate population = 102000

Calculations:

⇒ Let total population be x.

⇒ Males = 60% of x = 0.60x

⇒ Females = 40% of x = 0.40x

⇒ Literate Males = 40% of 0.60x = 0.24x

⇒ Literate Females = 25% of 0.40x = 0.10x

⇒ Total Literate = 0.24x + 0.10x = 0.34x

⇒ 0.34x = 102000

⇒ x = 102000 ÷ 0.34

⇒ x = 300000

∴ The correct answer is 300000.

Additional Information

Percentage Base Change

When calculating percentages of sub-groups, always multiply the sub-group percentage by its parent group percentage (e.g., % of Literate Males = % Literate × % Males).

Successive Percentage

If a value changes by a% and then b%, the effective change is [a + b + (ab/100)]%.

Unitary Method in Percentage

If p% of a total is known (V), then the total 100% is calculated as (V ÷ p) × 100.

General Solution:

Let’s find how many Wooden and Mechanical pencils each manufacturer sold.

Equation (per manufacturer):

Let W = wooden count, M = mechanical count.

W + M = Total

12W + 18M = Revenue

Reduce: divide the revenue equation by 6 → 2W + 3M = Revenue/6.

Substitute W = Total − M → 2(Total − M) + 3M = Revenue/6 → 2 × Total + M = Revenue/6

So

⇒ M = Revenue/6 − 2 × Total

⇒ W = Total − M

Now let's compute for each manufacturer:

P:

Revenue/6 = 18,000 ÷ 6 = 3,000

M = 3,000 − 2×1,200 = 3,000 − 2,400 = 600

W = 1,200 − 600 = 600

Q:

Revenue/6 = 25,500 ÷ 6 = 4,250

M = 4,250 − 2×1,500 = 4,250 − 3,000 = 1,250

W = 1,500 − 1,250 = 250

R:

Revenue/6 = 11,700 ÷ 6 = 1,950

M = 1,950 − 2×900 = 1,950 − 1,800 = 150

W = 900 − 150 = 750

S:

Revenue/6 = 29,700 ÷ 6 = 4,950

M = 4,950 − 2×1,800 = 4,950 − 3,600 = 1,350

W = 1,800 − 1,350 = 450

T:

Revenue/6 = 36,000 ÷ 6 = 6,000

M = 6,000 − 2×2,000 = 6,000 − 4,000 = 2,000

W = 2,000 − 2,000 = 0

Thus,

 

Calculations:

Wooden (P + Q) = 600 + 250 = 850

Mechanical (R + S) = 150 + 1350 = 1500

Required ratio = Wooden (P + Q) : Mechanical (R + S)

= 850 : 1500

= 17 : 30

Hence, the correct answer is 17:30.

General Solution:

Let’s find how many Wooden and Mechanical pencils each manufacturer sold.

Equation (per manufacturer):

Let W = wooden count, M = mechanical count.

W + M = Total

12W + 18M = Revenue

Reduce: divide the revenue equation by 6 → 2W + 3M = Revenue/6.

Substitute W = Total − M → 2(Total − M) + 3M = Revenue/6 → 2 × Total + M = Revenue/6

So

⇒ M = Revenue/6 − 2 × Total

⇒ W = Total − M

Now let's compute for each manufacturer:

P:

Revenue/6 = 18,000 ÷ 6 = 3,000

M = 3,000 − 2×1,200 = 3,000 − 2,400 = 600

W = 1,200 − 600 = 600

Q:

Revenue/6 = 25,500 ÷ 6 = 4,250

M = 4,250 − 2×1,500 = 4,250 − 3,000 = 1,250

W = 1,500 − 1,250 = 250

R:

Revenue/6 = 11,700 ÷ 6 = 1,950

M = 1,950 − 2×900 = 1,950 − 1,800 = 150

W = 900 − 150 = 750

S:

Revenue/6 = 29,700 ÷ 6 = 4,950

M = 4,950 − 2×1,800 = 4,950 − 3,600 = 1,350

W = 1,800 − 1,350 = 450

T:

Revenue/6 = 36,000 ÷ 6 = 6,000

M = 6,000 − 2×2,000 = 6,000 − 4,000 = 2,000

W = 2,000 − 2,000 = 0

Thus,

 

Calculations:

Manufacturer Q = 1500

Manufacturer T = 2000

Percentage Increase = ((New Value - Old Value) / Old Value) × 100

Percentage Increase = ((2000 - 1500) / 1500) × 100

= (500 / 1500) × 100

= (1/3) × 100

= 33.33%

The total number of pencils produced by Manufacturer T increased by 33.33% compared to Manufacturer Q.

Hence, the correct answer is 33.33%.

General Solution:

Let’s find how many Wooden and Mechanical pencils each manufacturer sold.

Equation (per manufacturer):

Let W = wooden count, M = mechanical count.

W + M = Total

12W + 18M = Revenue

Reduce: divide the revenue equation by 6 → 2W + 3M = Revenue/6.

Substitute W = Total − M → 2(Total − M) + 3M = Revenue/6 → 2 × Total + M = Revenue/6

So

⇒ M = Revenue/6 − 2 × Total

⇒ W = Total − M

Now let's compute for each manufacturer:

P:

Revenue/6 = 18,000 ÷ 6 = 3,000

M = 3,000 − 2×1,200 = 3,000 − 2,400 = 600

W = 1,200 − 600 = 600

Q:

Revenue/6 = 25,500 ÷ 6 = 4,250

M = 4,250 − 2×1,500 = 4,250 − 3,000 = 1,250

W = 1,500 − 1,250 = 250

R:

Revenue/6 = 11,700 ÷ 6 = 1,950

M = 1,950 − 2×900 = 1,950 − 1,800 = 150

W = 900 − 150 = 750

S:

Revenue/6 = 29,700 ÷ 6 = 4,950

M = 4,950 − 2×1,800 = 4,950 − 3,600 = 1,350

W = 1,800 − 1,350 = 450

T:

Revenue/6 = 36,000 ÷ 6 = 6,000

M = 6,000 − 2×2,000 = 6,000 − 4,000 = 2,000

W = 2,000 − 2,000 = 0

Thus,

 

Calculations:

Wooden (P) = 600

Wooden (R) = 750

Wooden (S) = 450

Average = (600 + 750 + 450) ÷ 3

= 1800 ÷ 3

= 600

Thus, the average number of Wooden pencils manufactured by P, R, and S is 600.

General Solution:

Let’s find how many Wooden and Mechanical pencils each manufacturer sold.

Equation (per manufacturer):

Let W = wooden count, M = mechanical count.

W + M = Total

12W + 18M = Revenue

Reduce: divide the revenue equation by 6 → 2W + 3M = Revenue/6.

Substitute W = Total − M → 2(Total − M) + 3M = Revenue/6 → 2 × Total + M = Revenue/6

So

⇒ M = Revenue/6 − 2 × Total

⇒ W = Total − M

Now let's compute for each manufacturer:

P:

Revenue/6 = 18,000 ÷ 6 = 3,000

M = 3,000 − 2×1,200 = 3,000 − 2,400 = 600

W = 1,200 − 600 = 600

Q:

Revenue/6 = 25,500 ÷ 6 = 4,250

M = 4,250 − 2×1,500 = 4,250 − 3,000 = 1,250

W = 1,500 − 1,250 = 250

R:

Revenue/6 = 11,700 ÷ 6 = 1,950

M = 1,950 − 2×900 = 1,950 − 1,800 = 150

W = 900 − 150 = 750

S:

Revenue/6 = 29,700 ÷ 6 = 4,950

M = 4,950 − 2×1,800 = 4,950 − 3,600 = 1,350

W = 1,800 − 1,350 = 450

T:

Revenue/6 = 36,000 ÷ 6 = 6,000

M = 6,000 − 2×2,000 = 6,000 − 4,000 = 2,000

W = 2,000 − 2,000 = 0

Thus,

 

Calculations:

Price of Mechanical pencil increased by 20% → New price = 18 × 1.2 = ₹21.6

 

Revenue from Wooden pencils = 250 × 12 = ₹3,000

Revenue from Mechanical pencils = 1250 × 21.6 = ₹27,000

Total New Revenue = 3,000 + 27,000 = ₹30,000

Hence, the correct answer is ₹30,000.

General Solution:

Let’s find how many Wooden and Mechanical pencils each manufacturer sold.

Equation (per manufacturer):

Let W = wooden count, M = mechanical count.

W + M = Total

12W + 18M = Revenue

Reduce: divide the revenue equation by 6 → 2W + 3M = Revenue/6.

Substitute W = Total − M → 2(Total − M) + 3M = Revenue/6 → 2 × Total + M = Revenue/6

So

⇒ M = Revenue/6 − 2 × Total

⇒ W = Total − M

Now let's compute for each manufacturer:

P:

Revenue/6 = 18,000 ÷ 6 = 3,000

M = 3,000 − 2×1,200 = 3,000 − 2,400 = 600

W = 1,200 − 600 = 600

Q:

Revenue/6 = 25,500 ÷ 6 = 4,250

M = 4,250 − 2×1,500 = 4,250 − 3,000 = 1,250

W = 1,500 − 1,250 = 250

R:

Revenue/6 = 11,700 ÷ 6 = 1,950

M = 1,950 − 2×900 = 1,950 − 1,800 = 150

W = 900 − 150 = 750

S:

Revenue/6 = 29,700 ÷ 6 = 4,950

M = 4,950 − 2×1,800 = 4,950 − 3,600 = 1,350

W = 1,800 − 1,350 = 450

T:

Revenue/6 = 36,000 ÷ 6 = 6,000

M = 6,000 − 2×2,000 = 6,000 − 4,000 = 2,000

W = 2,000 − 2,000 = 0

Thus,

 

Calculations:

Total Wooden Pencils = 600 + 250 + 740 + 450 + 0 = 2050

Total Mechanical Pencils = 600 + 1250 + 150 + 1350 + 2000 = 5350

Difference = 5350 − 2050 = 3300

The difference between the total number of Wooden and Mechanical pencils is 3300.

Hence, the correct answer is 3300.