Quantitative Aptitude PYQs

Shortcut Trick

Ratio of number of workers = 7 : 4

Ratio of wages per worker = 3 : 5

Ratio of Total Expenditure = (7 × 3) : (4 × 5) = 21 : 20

∴ The correct answer is 21 : 20.

Alternate Method

Given:

Initial number of workers = 7x, New number of workers = 4x

Initial wages = 3y, New wages = 5y

Formula Used:

Total Expenditure = (Number of Workers) × (Wages per Worker)

Calculations:

⇒ Previous Expenditure = 7x × 3y = 21xy

⇒ Current Expenditure = 4x × 5y = 20xy

⇒ Ratio = 21xy : 20xy

⇒ Ratio = 21 : 20

∴ The correct answer is 21 : 20.

Additional Information

Compound Ratio

When two or more ratios are multiplied term by term, the resulting ratio is called a Compound Ratio. Here, (a:b) and (c:d) gives (ac:bd).

Expenditure Relation

In commercial arithmetic, Total Cost or Expenditure is always the product of Quantity (workers/items) and Rate (wages/price).

Inverse Variation

If the total expenditure remains constant, the number of workers would be inversely proportional to the wages per worker.

Shortcut Trick

Apply Alligation between Officers and Rest Staff.

Ratio of (Officers : Rest Staff) = (600 − 560) : (4000 − 600) = 40 : 3400 = 1 : 85.

Since 1 unit (Officers) = 12, then 85 units (Rest Staff) = 85 × 12 = 1020.

∴ The correct answer is 1020.

Alternate Method

Given: Average salary of 12 Officers = ₹4000, Average salary of Rest Staff = ₹560, Average salary of Total Staff = ₹600.

Calculations:

⇒ Let the number of Rest Staff be N.

⇒ Total salary = (12 × 4000) + (N × 560) = (12 + N) × 600

⇒ 48000 + 560N = 7200 + 600N

⇒ 48000 − 7200 = 600N − 560N

⇒ 40800 = 40N

⇒ N = 40800 ÷ 40

⇒ N = 1020

∴ The correct answer is 1020.

Additional Information

Alligation Method

It is a rule used to find the ratio in which two or more ingredients at their respective prices must be mixed to produce a mixture of a specified price. Ratio = (Higher Value − Mean) : (Mean − Lower Value).

Average Salary Formula

Total Salary = Average Salary × Number of Employees. This base formula is used to set up linear equations for solving population distribution problems.

Shortcut Trick

Ratio of quantities = 45 : 40 = 9 : 8.

Weighted Average CP = (9 × 140 + 8 × 155) ÷ (9 + 8) = (1260 + 1240) ÷ 17 = 2500 ÷ 17 ≈ 147.06.

SP for 20% loss = Weighted Average CP × 80 ÷ 100.

SP = (2500 ÷ 17) × 0.8 = 2000 ÷ 17 ≈ 117.64.

Approximating to the nearest integer, we get Rs. 118.

∴ The correct answer is Rs. 118.

Alternate Method

Given → Total quantity 1 = 45 kg, Rate 1 = Rs. 140/kg; Total quantity 2 = 40 kg, Rate 2 = Rs. 155/kg; Loss% = 20%.
Formulas: Total CP = (Q₁ × R₁) + (Q₂ × R₂) | Average CP = Total CP ÷ Total Quantity | SP = CP × (100 − Loss%) ÷ 100.

⇒ Cost of first variety = 45 × 140 = Rs. 6300.
⇒ Cost of second variety = 40 × 155 = Rs. 6200.
⇒ Total CP = 6300 + 6200 = Rs. 12500.
⇒ Total Quantity = 45 + 40 = 85 kg.
⇒ Selling Price to incur 20% loss = 12500 × (80 ÷ 100) = Rs. 10000.
⇒ SP per kg = 10000 ÷ 85 = Rs. 117.64...
⇒ Required approximate price = Rs. 118.

∴ The correct answer is Rs. 118.

Additional Information

Weighted Average Method

When two groups of items with different averages are combined, the new average is [n₁x₁ + n₂x₂] ÷ [n₁ + n₂].

Profit and Loss on SP

Loss is calculated on Cost Price unless specified otherwise. Loss = CP − SP. Loss% = (Loss ÷ CP) × 100.

Shortcut Trick

Total Work (LCM of 18, 24) = 72 units.

Efficiency: A = 4, B = 3.

Work done in first 6 days (A+B): (4 + 3) × 6 = 42 units.

Remaining work: 72 − 42 = 30 units.

Efficiency of (B+C) = 30 ÷ 6 = 5. ∴ C's efficiency = 5 − 3 = 2.

Time for C = 72 ÷ 2 = 36 days.

∴ The correct answer is 36 days.

Alternate Method

Given: Time taken by A = 18 days, B = 24 days. A worked for 6 days. B worked for (6 + 6) = 12 days. C worked for 6 days.

Formula Used: Total Work = ∑(Efficiency × Time) or ∑(Work done by each) = 1.

⇒ Work done by A in 6 days = 4 × 6 = 24 units

⇒ Work done by B in 12 days (6 initial + 6 with C) = 3 × 12 = 36 units

⇒ Work done by A and B = 24 + 36 = 60 units

⇒ Remaining work completed by C = 72 − 60 = 12 units

⇒ C finishes 12 units in 6 days

⇒ Efficiency of C = 12 ÷ 6 = 2 units/day

⇒ Time for C alone = Total Work ÷ Efficiency of C = 72 ÷ 2 = 36 days

∴ The correct answer is 36 days.

Additional Information

Work and Efficiency

Efficiency is inversely proportional to time. Efficiency = Total Work ÷ Time.

Pipes and Cisterns

Similar to Time & Work, but includes negative efficiency for outlet pipes (leakage).

Work & Wages

Wages are distributed in the ratio of the work done by each person.

Shortcut Trick

Time ratio (Upstream : Downstream) = 2 : 1

Speed ratio (Upstream : Downstream) = 1 : 2 (Inverse of time)

Ratio of Still Water Speed : Current Speed = (D + U) : (D − U)

Ratio = (2 + 1) : (2 − 1) = 3 : 1

∴ The correct answer is 3 : 1.

Alternate Method

Given: Time taken upstream (A to B) = 2 × Time taken downstream (B to A).

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

⇒ Let speed of boat in still water = u and speed of current = v.

⇒ Distance (d) is constant, so (u − v) × 2t = (u + v) × t.

⇒ 2u − 2v = u + v

⇒ 2u − u = v + 2v

⇒ u = 3v

⇒ u : v = 3 : 1

∴ The correct answer is 3 : 1.

Additional Information

Upstream and Downstream Speed

Downstream Speed (D) = u + v and Upstream Speed (U) = u − v, where u is boat speed and v is current speed.

Still Water Speed Calculation

The speed of the boat in still water (u) is the average of downstream and upstream speeds: u = (D + U) ÷ 2.

Current Speed Calculation

The speed of the current (v) is half the difference between downstream and upstream speeds: v = (D − U) ÷ 2.

From the Pie chart data,

Total number of employees in company D = 18%*3000 = 540

Number of Male in Company D = (3/5)*540 = 324

Number of Female in Company D = (2/5)*540 = 215

Similarly, 

 

Solution:

Total number of Employees in company C = 600

Total number of Employees in company F is 110% of Employees in company C

Therefore, total number of Employees in company F = 110%*600 = 660

Hence, number of Female Employees in company F = (2/5)*660 = 264

From the Pie chart data,

Total number of employees in company D = 18%*3000 = 540

Number of Male in Company D = (3/5)*540 = 324

Number of Female in Company D = (2/5)*540 = 216

Similarly, 

The total % distribution of Company D is 18% out of total number of Employees in all the company.

Total angle formed at the center of the pie chart is 360o 

Hence, Central Angle of Company D = (18/100)*360^o = 64.8

From the Piechart data,

 

Total number of Employees in company B = 16%*3000 = 480

number of males in company B = (3/5)*480 = 288

Total number of Employees in Company E = 22%*300 = 660

number of Male Employees in copmany E = (3/5)*660 = 396

Average number of Male Employees in company B and E = (288 + 396)/2 = 342

From the Pie chart data,

Total number of employees in company D = 18%*3000 = 540

Number of Male in Company D = (3/5)*540 = 324

Number of Female in Company D = (2/5)*540 = 215

Similarly, 

Therefore, Required % = (240-216)/216 = 1/9 = 11.11%

From the Pie chart data,

Total number of employees in company D = 18%*3000 = 540

Number of Male in Company D = (3/5)*540 = 324

Number of Female in Company D = (2/5)*540 = 216

Similarly, 

 

 difference between number of Male and Female in Company D = 324 - 216 = 108.

Shortcut Trick

Total Expenditure % = 13% + 27% + 35% + 5% = 80%

Remaining Savings % = 100% − 80% = 20%

20% of Income = ₹13200

100% (Total Income) = 13200 × 5 = ₹66,000

∴ The correct answer is ₹66,000.

Alternate Method

Given:

⇒ Traveling = 13%

⇒ Household expenses = 27%

⇒ Medical expenses = 35%

⇒ Insurance = 5%

⇒ Remaining Cash = ₹13200

Calculations:

⇒ Total spending = (13 + 27 + 35 + 5)% = 80%

⇒ Savings % = 100% − 80% = 20%

⇒ Let Total Income be 'X'

⇒ 20% of X = 13200

⇒ (20/100) × X = 13200

⇒ X = 13200 × 5

⇒ X = 66,000

∴ The correct answer is ₹66,000.

Additional Information

Percentage Base Change

When expenditures are given as a % of the total income, simply add the percentages. If given as % of remaining income, use successive multiplication: Remaining = Income × (1 − r₁) × (1 − r₂).

Income, Expenditure, and Savings Relationship

Income = Expenditure + Savings. If Income increases by x% and Expenditure by y%, the change in Savings depends on their initial ratio.

Shortcut Trick

Profit ratio Kamal : Raju = (3000 − 1400) : 1400 = 1600 : 1400 = 8 : 7.

Time ratio = 24 months : (42 − 24) months = 24 : 18 = 4 : 3.

Investment ratio = (Profit ÷ Time) ⇒ (8 ÷ 4) : (7 ÷ 3) = 2 : 7/3 = 6 : 7.

Since 6 units = Rs. 7500, then 1 unit = Rs. 1250.

Raju investment = 7 × 1250 = Rs. 8750.

∴ The correct answer is Rs. 8750.

Alternate Method

Given → Kamal Investment (C_k) = Rs. 7500, Time (T_k) = 24 months. Total time = 3.5 years = 42 months. Raju Time (T_r) = 42 − 24 = 18 months. Total profit = Rs. 3000, Raju Profit (P_r) = Rs. 1400.
Formulas: Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Steps:
⇒ Profit of Kamal (P_k) = Total Profit − Raju Profit = 3000 − 1400 = Rs. 1600.
⇒ Ratio of profits (P_k / P_r) = (C_k × T_k) / (C_r × T_r).
⇒ 1600 / 1400 = (7500 × 24) / (X × 18).
⇒ 8 / 7 = (7500 × 4) / (X × 3).
⇒ 8 × 3 × X = 7 × 7500 × 4.
⇒ 24X = 210000.
⇒ X = 210000 / 24 = Rs. 8750.

∴ The correct answer is Rs. 8750.

Additional Information

Partnership Basic Concept

In a partnership, the profit is distributed among partners in the ratio of the product of their invested capital and the duration of investment.

Investment Calculation

If profits and time periods are known, the investment ratio can be found by Investment = Profit ÷ Time.

Shortcut Trick

For 2 years, the percentage by which Compound Interest (CI) is greater than Simple Interest (SI) is given by the formula: (r ÷ 2)%.

Given Rate (r) = 10%.

Percentage difference = (10 ÷ 2)% = 5%.

∴ The correct answer is 5%.

Alternate Method

Given:

Simple Interest (SI) = ₹400, Rate (r) = 10%, Time (t) = 2 years.

Formula Used:

SI = (P × r × t) ÷ 100

CI = P[(1 + r/100)^t − 1]

Calculations:

⇒ 400 = (P × 10 × 2) ÷ 100

⇒ 400 = P × 0.2 ⇒ P = ₹2000

⇒ CI for 2 years = 2000 × [(1 + 10/100)² − 1]

⇒ CI = 2000 × [1.21 − 1] = 2000 × 0.21 = ₹420

⇒ Difference = CI − SI = 420 − 400 = ₹20

⇒ Percentage greater = (Difference ÷ SI) × 100

⇒ Percentage greater = (20 ÷ 400) × 100 = 5%

∴ The correct answer is 5%.

Additional Information

Effective Rate of Interest

The effective rate for Compound Interest for 2 years is r + r + (r × r)/100. For 10%, it is 10 + 10 + 1 = 21%.

Difference between CI and SI

For 2 years, the difference (D) is given by D = P(r/100)².

Simple Interest Property

Simple Interest remains constant every year if the principal and rate are unchanged. Total SI = r × t % of Principal.

Shortcut Trick

Total units = 150. Lost (40%) = 60. Remaining = 90.

Sold 50% of 90 (45 units) at 20% profit and 45 units at 5% loss.

Let CP of 1 t-shirt = x. Total SP = 45 × 1.20x + 45 × 0.95x = 7740.

54x + 42.75x = 96.75x = 7740.

x = 7740 ÷ 96.75 = 80.

∴ The correct answer is Rs. 80.

Alternate Method

Given: Total t-shirts = 150, Loss in transit = 40%, Profit on half remaining = 20%, Loss on other half = 5%, Total SP = Rs. 7740. Formulas: SP = CP × (100 + P%) ÷ 100 | SP = CP × (100 − L%) ÷ 100

⇒ Let Cost Price (CP) of each t-shirt = Rs. C.

⇒ Number of t-shirts lost = 40% of 150 = 60. Remaining = 150 − 60 = 90.

⇒ Half of remaining sold at 20% profit: 45 × C × 1.20 = 54C.

⇒ Half of remaining sold at 5% loss: 45 × C × 0.95 = 42.75C.

⇒ Total SP = 54C + 42.75C = 96.75C.

⇒ Given 96.75C = 7740 ⇒ C = 7740 ÷ 96.75 = 80.

∴ The correct answer is Rs. 80.

Additional Information

Total Cost vs Net Profit

Net Profit or Loss is calculated on the Total Cost Price of all 150 items, even those lost in transit. Total CP = 150 × 80 = 12000.

Effective Selling Price

Items that are lost/damaged have a Selling Price of 0, contributing to a 100% loss for those specific units.

Shortcut Trick

Let length of Train 1 be 3L and Train 2 be 4L.

From Train 2 crossing platform: Speed × 42 = 4L + 300 ⇒ (x − 6) × (5/18) × 42 = 4L + 300.

From Relative Speed: (x + x − 6) × (5/18) × 21 = 7L.

By substituting L or solving the ratio of distances, we find x = 70.

∴ The correct answer is 70.

Alternate Method

Given: Speed of Train 1 = x km/hr, Speed of Train 2 = (x − 6) km/hr. Length of Train 2 = (4/3) × Length of Train 1. Let length of Train 1 = 3L, then Train 2 = 4L.

Formula: Distance = Speed × Time

⇒ For Train 2 crossing platform: (x − 6) × (5/18) × 42 = 4L + 300

⇒ (x − 6) × (35/3) = 4L + 300 → (Eq 1)

⇒ For trains crossing each other (Relative speed): (x + x − 6) × (5/18) × 21 = 3L + 4L

⇒ (2x − 6) × (35/6) = 7L ⇒ (x − 3) × (35/3) = 7L ⇒ L = 5(x − 3)/3 → (Eq 2)

⇒ Substitute L in (Eq 1): (x − 6) × (35/3) = 4 × [5(x − 3)/3] + 300

⇒ 35x − 210 = 20x − 60 + 900

⇒ 15x = 1050

⇒ x = 70

∴ The correct answer is 70.

Additional Information

Relative Speed

When two objects move in opposite directions, relative speed = S₁ + S₂. In the same direction, relative speed = |S₁ − S₂|.

Unit Conversion

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

Crossing Distance

When a train crosses another train or platform, the total distance covered is the sum of their respective lengths.

Given,

Total number of boys including those from the five branches

= 500 + 500 + 450 + 200 + 300

= 1950

Required average

= 1950/5

= 390

Given,

Total number to girls including five branches

= 300 + 100 + 250 + 600 + 250

= 1500

Total number of boys including five branches

= 500 + 500 + 450 + 200 + 300

= 1950

Total number of students in College

= 1500 + 1950

= 3450

Required percentage

= (1500/3450) × 100 ≈ 43.48%

Total number of students in Computer branch

= 200 + 600

= 800

Total number of students in Mechanical and Electronics branch

= (500 + 100) + (300 + 250)

= 1150

Required ratio

= 800 : 1150

= 32 : 46

= 16 : 23

Given,

Total number of boys including five branches

= 500 + 500 + 450 + 200 + 300

= 1950

Number of boy’s students in Mechanical branch = 500

Required percentage

= (500/1950) × 100 ≈ 25.64%

Given,

Civil branch = 500 : 300 = 5 : 3 = 1.66

Mechanical branch = 500 : 100 = 5 : 1 = 5

Electrical branch = 450 : 250 = 9 : 5 = 1.8

Computer branch = 200 : 600 = 1 : 3 = 0.33

Electronics branch = 300 : 250 = 6 : 5 = 1.2

From this,

Computer branch has minimum ratio of boys and girls students in College.