Quantitative Aptitude PYQs

General Solution:

The total employees in a company = Males + Females.

From the given data, we can calculate the total employees for each company using the percentage of female employees.

Formula: Total Employees = Males / (1 - Percentage of Females)

Company A:

Males = 240

Percentage of Females = 40%

Total Employees = 240 / (1 - 0.40) = 240 / 0.60 = 400

Females = 400 - 240 = 160 

Company B:

Males = 275

Percentage of Females = 45%

Total Employees = 275 / (1 - 0.45) = 275 / 0.55 = 500

Females = 500 - 275 = 225 

Company C:

Males = 270

Percentage of Females = 55%

Total Employees =270 / (1 - 0.55) = 270 / 0.45 = 600

Females = 600 - 270 = 330 

Company D:

Males = 120

Percentage of Females = 60%

Total Employees =120 / (1 - 0.60) = 120 / 0.40 = 300

Females = 300 - 120 = 180 

Company E:

Males = 180

Percentage of Females = 50%

Total Employees =180 / (1 - 0.50) = 180 / 0.50 = 360

Females = 360 - 180 = 180 

Calculations:

Company A:

Original males = 240

Increase by 25% = 240 × 1.25 = 300

Company B:

Original males = 275

Decrease by 20% = 275 × 0.80 = 220

Company C:

Original males = 270

Increase by 10% = 270 × 1.10 = 297

Company A:

Percentage of females = 40%

Total employees = 300 / (1 - 0.40) = 300 / 0.60 = 500

Company B:

Percentage of females = 45%

Total employees = 220 / (1 - 0.45) = 220 / 0.55 = 400

Company C:

Percentage of females = 55%

Total employees = 297 / (1 - 0.55) = 297 / 0.45 = 660

Total employees in A : B : C = 500 : 400 : 660 = 25 : 20 : 33

Thus, the new ratio is 25:20:33.

Shortcut Trick

Ratio of Profit share = (Investment_A × Time_A) : (Investment_B × Time_B) : (Investment_C × Time_C).

From A and B's profit ratio: (4500 × y) : (5500 × (y + 3)) = 63 : 110.

9y : 11(y + 3) = 63 : 110 ⇒ 9y ÷ (11y + 33) = 63 ÷ 110 ⇒ y ÷ (y + 3) = 7 ÷ 10.

10y = 7y + 21 ⇒ 3y = 21 ⇒ y = 7.

Total Profit = 79250, Charity = 20%, Remaining = 80% of 79250 = 63400.

Profit share of A = [63 ÷ (63 + 110 + 144)] × 63400 = (63 ÷ 317) × 63400 = 12600. Required ratio = 12600 ÷ 7 = 1800.

∴ The correct answer is 1800.

Alternate Method

Given → Investments: A = 4500, B = 5500, C = 6000. Time: A = y, B = y + 3, C = 12. Profit Ratio = 63 : 110 : 144. Total Profit = 79250.
Formulas: Profit Share = Investment × Time | Remaining Profit = Total × (100 − Charity%) ÷ 100.

⇒ Calculate y using A and C: (4500 × y) ÷ (6000 × 12) = 63 ÷ 144.
⇒ 4500y ÷ 72000 = 63 ÷ 144 ⇒ y ÷ 16 = 63 ÷ 144 ⇒ y = (63 × 16) ÷ 144 = 7.
⇒ Net Profit after charity = 79250 × 0.8 = 63400.
⇒ Profit of A = [63 ÷ (63 + 110 + 144)] × 63400 = (63 ÷ 317) × 63400 = 63 × 200 = 12600.
⇒ Required value = Profit of A ÷ y = 12600 ÷ 7 = 1800.

∴ The correct answer is 1800.

Additional Information

Partnership Basics

In a partnership, profits are distributed in the ratio of the product of Investment and Time period. Profit Share = Capital × Time.

Effective Profit

When a portion of profit is donated to charity or kept as a reserve, the remaining profit is distributed among partners based on their investment-time ratios.

Shortcut Trick

Interest from Scheme P = 1500 × 11.25% × 4 = 1500 × 45% = ₹675.

Interest from Scheme Q = 1200 × 25% × X = 300X.

Equating both: 300X = 675 ⇒ X = 2.25 years.

Rate for Scheme R = 4X% = 4 × 2.25 = 9%.

Interest from Scheme R = 2000 × 9% × 2 = ₹360.

∴ The correct answer is 360 Rs.

Alternate Method

Given: Principal P₁ = ₹1500, R₁ = 11.25%, T₁ = 4 years; Principal P₂ = ₹1200, R₂ = 25%, T₂ = X years; Principal P₃ = ₹2000, R₃ = 4X%, T₃ = 2 years.

Formula Used: Simple Interest (SI) = (P × R × T) ÷ 100

⇒ SI_P = (1500 × 11.25 × 4) ÷ 100 = 15 × 45 = ₹675

⇒ SI_Q = (1200 × 25 × X) ÷ 100 = 12 × 25 × X = 300X

⇒ Since SI_P = SI_Q, 300X = 675 ⇒ X = 675 ÷ 300 = 2.25

⇒ For Scheme R: Rate = 4X% = 4 × 2.25 = 9%

⇒ Interest in Scheme R = (2000 × 9 × 2) ÷ 100 = 20 × 18 = ₹360

∴ The correct answer is 360 Rs.

Additional Information

Simple Interest Property

Simple Interest is calculated only on the Principal amount. It remains constant for every year if the rate and principal are unchanged.

Effective Rate of Interest

The total interest percentage over a period is the product of Rate (R) and Time (T). Total Interest = (R × T)% of Principal.

Comparison of Interests

When interests are equal, P₁R₁T₁ = P₂R₂T₂. This relation can be used to find any missing variable directly.

Shortcut Trick

Let Upstream speed be U and Downstream speed be D.

Given U + D = 30. Using trial values from the sum: If U = 9 and D = 21 (since 9+21=30):

Case 1: 72/9 + 180/21 = 8 + 8.57 ≠ 16. If U = 12 and D = 18: 72/12 + 180/18 = 6 + 10 = 16 hours (Satisfied).

Case 2: 108/12 + 126/18 = 9 + 7 = 16 hours (Satisfied). U = 12 kmph, D = 18 kmph.

Speed of stream = (D − U) ÷ 2 = (18 − 12) ÷ 2 = 3 kmph.

∴ The correct answer is 3 kmph.

Alternate Method

Given:

Distance Upstream₁ = 72 km, Distance Downstream₁ = 180 km, Time₁ = 16 hours

Distance Upstream₂ = 108 km, Distance Downstream₂ = 126 km, Time₂ = 16 hours

Formula Used: Time = Distance ÷ Speed; Stream Speed = (D − U) ÷ 2

⇒ 72/U + 180/D = 16 ---- (i)

⇒ 108/U + 126/D = 16 ---- (ii)

⇒ Since both equal 16: 72/U + 180/D = 108/U + 126/D

⇒ 180/D − 126/D = 108/U − 72/U

⇒ 54/D = 36/U ⇒ D/U = 54/36 = 3/2

⇒ Let D = 3k and U = 2k. Given U + D = 30 ⇒ 5k = 30 ⇒ k = 6

⇒ D = 18 kmph, U = 12 kmph

⇒ Speed of stream = (18 − 12) ÷ 2 = 6 ÷ 2 = 3 kmph

∴ The correct answer is 3 kmph.

Additional Information

Boat Speed in Still Water

It is the average of downstream and upstream speeds: B = (D + U) ÷ 2.

Speed of Stream

It is half the difference between downstream and upstream speeds: S = (D − U) ÷ 2.

Relative Speed in Water

Downstream Speed (D) = Speed of Boat (B) + Speed of Stream (S); Upstream Speed (U) = B − S.

Shortcut Trick

Let Q's present age be 11x and R's present age be 15x.

From (P − 5) : (11x + 5) = 2 : 3 ⇒ P = (22x + 10)/3 + 5.

Using P + R = 120 ⇒ (22x + 25)/3 + 15x = 120 ⇒ 67x = 335 ⇒ x = 5.

P = 45 years, Q = 55 years. Since (P + y) : (Q + 5) = 1 : 1 ⇒ 45 + y = 60.

∴ The correct answer is 15 years.

Alternate Method

Given: Ratio (P − 5) : (Q + 5) = 2 : 3, Q : R = 11 : 15, P + R = 120.

⇒ Let Q = 11x and R = 15x.

⇒ (P − 5) / (11x + 5) = 2 / 3

⇒ 3P − 15 = 22x + 10 ⇒ 3P − 22x = 25 --- (Eq. 1)

⇒ P + 15x = 120 ⇒ P = 120 − 15x --- (Eq. 2)

⇒ Substitute Eq. 2 in Eq. 1: 3(120 − 15x) − 22x = 25

⇒ 360 − 45x − 22x = 25 ⇒ 335 = 67x ⇒ x = 5.

⇒ Present Age of Q = 11 × 5 = 55 years.

⇒ Present Age of P = 120 − 15(5) = 45 years.

⇒ (P + y) / (Q + 5) = 1 / 1 ⇒ 45 + y = 55 + 5

⇒ 45 + y = 60 ⇒ y = 15.

∴ The correct answer is 15.

Additional Information

Ratio of Ages

If the ratio of ages is a:b, we assume ages as ax and bx. The difference between ages remains constant over time.

Linear Equations in Ages

Age n years ago = (Present Age − n). Age n years hence = (Present Age + n).

Proportion Rule

If two quantities are in ratio 1:1, it implies they are equal to each other.

Considering the given series

9, 10, 22, 70, 280, 1405

The logic of the given series can be explained as

9 × 1 + 1 = 10

10 × 2 + 2 = 22

22 × 3 + 3 = 69

69 × 4 + 4= 280

280 × 5 + 5 = 1405

∴ Wrong term in given number series is 70.

GIVEN:

144, 2197, 196, 3375, 256, 4903

CALCULATION:

⇒ 17³ = 4913

⇒ So, 4903 is the wrong term.

Considering the given series

49, 110, 174, 244, 323, 412, 520

The logic of the given series can be explained as

 

∴ Wrong term in given number series is 412.

Shortcut Trick

Initial Savings: A = ₹9000 (75% of x), B = ₹10000 (66.67% of y).

New Savings: Since expenditure percentage remains same, savings percentage also remains same.

New Savings of A = 9000 × 1.5 = ₹13500 | New Savings of B = 10000 × 1.4 = ₹14000.

Difference = 14000 − 13500 = ₹500.

∴ The correct answer is Rs. 500.

Alternate Method

Given:

Savings of A = 75% of x = ₹9000 ⇒ x = 9000 ÷ 0.75 = ₹12000

Savings of B = 66.67% of y = ₹10000 ⇒ y = 10000 ÷ (2/3) = ₹15000

New Calculations:

⇒ New Income A = 12000 × 150% = ₹18000

⇒ New Expenditure A = 25% of 18000 = ₹4500

⇒ New Savings A = 18000 − 4500 = ₹13500

⇒ New Income B = 15000 × 140% = ₹21000

⇒ New Expenditure B = 33.333% of 21000 = 1/3 × 21000 = ₹7000

⇒ New Savings B = 21000 − 7000 = ₹14000

⇒ Difference = 14000 − 13500 = ₹500

∴ The correct answer is Rs. 500.

Additional Information

Income, Expenditure and Savings Relationship

The basic formula is Income = Expenditure + Savings. If the expenditure percentage remains constant, savings will grow at the same rate as income.

Percentage to Fraction Conversion

Useful conversions: 25% = 1/4, 75% = 3/4, 33.333% = 1/3, and 66.666% = 2/3.

Shortcut Trick

Total Work (LCM of 8, 12, 6) = 24 units.

Efficiencies: A = 3, B = 2, C = 4. Combined B+C = 6.

Work Equation: (B+C)×x + A×(x+2) = 24 ⇒ 6x + 3(x+2) = 24 ⇒ 9x = 18 ⇒ x = 2.

Work by A in (x+4) = 6 days: 3 × 6 = 18 units.

Percentage = (18/24) × 100 = 75%.

∴ The correct answer is 75%.

Alternate Method

Given: Time taken by A = 8 days, B = 12 days, C = 6 days.

Formula Used: Total Work = Efficiency × Time; Efficiency = Total Work ÷ Time.

Calculations:

⇒ Work done by B and C in x days = (2 + 4) × x = 6x units.

⇒ Work done by A in (x + 2) days = 3 × (x + 2) = 3x + 6 units.

⇒ Total Work: 6x + 3x + 6 = 24

⇒ 9x = 18 ⇒ x = 2.

⇒ If A alone works for (x + 4) days = (2 + 4) = 6 days.

⇒ Work done by A in 6 days = 3 × 6 = 18 units.

⇒ Percentage of work completed = (18 ÷ 24) × 100

⇒ Percentage = 0.75 × 100 = 75%.

∴ The correct answer is 75%.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete the work (E ∝ 1/T).

Remaining Work Calculation

Remaining Work = Total Work − Work already completed by previous group of workers.

Combined Efficiency

When people work together, their individual efficiencies are added to find the total rate of work (E_Total = E₁ + E₂ + ...).

General Solution:

Determine the value of 'x':

The sum of the percentages of profit from all items must equal 100%.

Therefore: (2x - 2) + (x + 1) + (x + 3) + (x - 1) + (x + 9) = 100

6x + 10 = 100

6x = 90

x = 15

Now, let's calculate the percentage of profit earned from each item:

Bag: (2 × 15 - 2)% = 28%

Bottle: (15 + 1)% = 16%

Box: (15 + 3)% = 18%

Pen: (15 - 1)% = 14%

Register: (15 + 9)% = 24%

Now, let's calculate the actual profit (in Rupees) earned from each item:

Bag: 28% of Rs. 25,000 = 0.28 × 25,000 = Rs. 7,000

Bottle: 16% of Rs. 25,000 = 0.16 × 25,000 = Rs. 4,000

Box: 18% of Rs. 25,000 = 0.18 × 25,000 = Rs. 4,500

Pen: 14% of Rs. 25,000 = 0.14 × 25,000 = Rs. 3,500

Register: 24% of Rs. 25,000 = 0.24 × 25,000 = Rs. 6,000

Calculations:

Profit from bags = Rs. 7,000

Percentage reinvested = 45%

Amount reinvested = (45/100) × 7,000 = Rs. 3,150

Profit from registers = Rs. 6,000

Percentage reinvested = 60%

Amount reinvested = (60/100) × 6,000 = Rs. 3,600

Total reinvested = 3,150 + 3,600 = Rs. 6,750

Thus, the correct answer is Rs. 6,750

General Solution:

Determine the value of 'x':

The sum of the percentages of profit from all items must equal 100%.

Therefore: (2x - 2) + (x + 1) + (x + 3) + (x - 1) + (x + 9) = 100

6x + 10 = 100

6x = 90

x = 15

Now, let's calculate the percentage of profit earned from each item:

Bag: (2 × 15 - 2)% = 28%

Bottle: (15 + 1)% = 16%

Box: (15 + 3)% = 18%

Pen: (15 - 1)% = 14%

Register: (15 + 9)% = 24%

Now, let's calculate the actual profit (in Rupees) earned from each item:

Bag: 28% of Rs. 25,000 = 0.28 × 25,000 = Rs. 7,000

Bottle: 16% of Rs. 25,000 = 0.16 × 25,000 = Rs. 4,000

Box: 18% of Rs. 25,000 = 0.18 × 25,000 = Rs. 4,500

Pen: 14% of Rs. 25,000 = 0.14 × 25,000 = Rs. 3,500

Register: 24% of Rs. 25,000 = 0.24 × 25,000 = Rs. 6,000

Calculations:

We are given:

The cost price CP of all registers = ₹19,000

Profit earned on registers = ₹6,000

The total selling price SP of all registers = CP + Profit = ₹19,000 + ₹6,000 = ₹25,000

The marked price MP of all registers is 40% above the cost price

Since the marked price is 40% more than the cost price, we calculate:

MP = CP + 40% of CP

MP = 19000 + (40 / 100 × 19000)

MP = 19000 + 7600 = 26600

The discount is given on the marked price to reach the selling price.

So, the discount amount is:

Discount = MP − SP = 26600 − 25000 = 1600

Now, the discount percentage is:

Discount % = Discount / Marked Price × 100 = 1600 / 26600 × 100 ≈ 6.02%

Thus, the approximate percentage discount applied to the marked price is 6%.

General Solution:

Determine the value of 'x':

The sum of the percentages of profit from all items must equal 100%.

Therefore: (2x - 2) + (x + 1) + (x + 3) + (x - 1) + (x + 9) = 100

6x + 10 = 100

6x = 90

x = 15

Now, let's calculate the percentage of profit earned from each item:

Bag: (2 × 15 - 2)% = 28%

Bottle: (15 + 1)% = 16%

Box: (15 + 3)% = 18%

Pen: (15 - 1)% = 14%

Register: (15 + 9)% = 24%

Now, let's calculate the actual profit (in Rupees) earned from each item:

Bag: 28% of Rs. 25,000 = 0.28 × 25,000 = Rs. 7,000

Bottle: 16% of Rs. 25,000 = 0.16 × 25,000 = Rs. 4,000

Box: 18% of Rs. 25,000 = 0.18 × 25,000 = Rs. 4,500

Pen: 14% of Rs. 25,000 = 0.14 × 25,000 = Rs. 3,500

Register: 24% of Rs. 25,000 = 0.24 × 25,000 = Rs. 6,000

Calculations:

Given that the profit percentage for the Notebook is:

(0.5x + 5)%

Substituting x = 15 from the general solution:

0.5 × 15 + 5 = 7.5 + 5 = 12.5%

Thus, the Notebook represents 12.5% of the total profit.

Now, the total profit percentage must still sum to 100%, but we have added the Notebook while keeping the total profit constant.

The original sum of the profit percentages was 100% from 5 items. Now, the Notebook takes 12.5%, meaning the remaining 87.5% must be redistributed among the previous five items in the same proportion.

The adjusted percentage for each previous item is calculated as:

New Percentage = Old Percentage / 100 × 87.5

Bag: 28 / 100 × 87.5 = 24.5%

Bottle: 16 / 100 × 87.5 = 14%

Box: 18 / 100 × 87.5 = 15.75%

Pen: 14 / 100 × 87.5 = 12.25%

Register: 24 / 100 × 87.5 = 21%

Thus, the new percentage of profit from the Pen is 12.25%.

Thus,

Notebook represents 12.5% of the total profit.

New profit percentage of the Pen: 12.25%.

General Solution:

Determine the value of 'x':

The sum of the percentages of profit from all items must equal 100%.

Therefore: (2x - 2) + (x + 1) + (x + 3) + (x - 1) + (x + 9) = 100

6x + 10 = 100

6x = 90

x = 15

Now, let's calculate the percentage of profit earned from each item:

Bag: (2 × 15 - 2)% = 28%

Bottle: (15 + 1)% = 16%

Box: (15 + 3)% = 18%

Pen: (15 - 1)% = 14%

Register: (15 + 9)% = 24%

Now, let's calculate the actual profit (in Rupees) earned from each item:

Bag: 28% of Rs. 25,000 = 0.28 × 25,000 = Rs. 7,000

Bottle: 16% of Rs. 25,000 = 0.16 × 25,000 = Rs. 4,000

Box: 18% of Rs. 25,000 = 0.18 × 25,000 = Rs. 4,500

Pen: 14% of Rs. 25,000 = 0.14 × 25,000 = Rs. 3,500

Register: 24% of Rs. 25,000 = 0.24 × 25,000 = Rs. 6,000

Calculations:

Profit from pens = Rs. 3,500

15% donation from pens = 15 / 100 × 3,500 = 525

Profit from bottles = Rs. 4,000

20% donation from bottles = 20 / 100 × 4,000 = 800

Remaining profit after donation = 3,500 − 525 = 2,975

30% of remaining profit from pens = 30 / 100 × 2,975 = 892.5

Final remaining profit = 2,975 − 892.5 = 2,082.5

Thus, the correct answer is ₹2,082.50

General Solution:

Determine the value of 'x':

The sum of the percentages of profit from all items must equal 100%.

Therefore: (2x - 2) + (x + 1) + (x + 3) + (x - 1) + (x + 9) = 100

6x + 10 = 100

6x = 90

x = 15

Now, let's calculate the percentage of profit earned from each item:

Bag: (2 × 15 - 2)% = 28%

Bottle: (15 + 1)% = 16%

Box: (15 + 3)% = 18%

Pen: (15 - 1)% = 14%

Register: (15 + 9)% = 24%

Now, let's calculate the actual profit (in Rupees) earned from each item:

Bag: 28% of Rs. 25,000 = 0.28 × 25,000 = Rs. 7,000

Bottle: 16% of Rs. 25,000 = 0.16 × 25,000 = Rs. 4,000

Box: 18% of Rs. 25,000 = 0.18 × 25,000 = Rs. 4,500

Pen: 14% of Rs. 25,000 = 0.14 × 25,000 = Rs. 3,500

Register: 24% of Rs. 25,000 = 0.24 × 25,000 = Rs. 6,000

Calculations:

From the general solution:

Profit on Box = Rs. 4,500

Profit on Pen = Rs. 3,500

Selling Price of Box:

Cost Price (CP) of Box = Rs. 15,000

Profit = Rs. 4,500

Selling Price (SP) of Box = CP + Profit = 15,000 + 4,500 = Rs. 19,500

Selling Price of Pen:

Cost Price CP of Pen = Rs. 20,000

Profit = Rs. 3,500

Selling Price (SP) of Pen = CP + Profit = 20,000 + 3,500 = Rs. 23,500

Box: Selling price increases by 20%

New SP of Box = 19,500 × 1.20 = Rs. 23,400

Pen: Selling price decreases by 10%

New SP of Pen = 23,500 × 0.90 = Rs. 21,150

New Profit on Box:

New SP of Box = Rs. 23,400

CP of Box = Rs. 15,000

New Profit = 23,400 15,000 = Rs. 8,400

New Profit on Pen:

New SP of Pen = Rs. 21,150

CP of Pen = Rs. 20,000

New Profit = 21,150 - 20,000 = Rs. 1,150

From the general solution, the original profits on other items are:

Bag = Rs. 7,000

Bottle = Rs. 4,000

Register = Rs. 6,000

New profits:

Box = Rs. 8,400

Pen = Rs. 1,150

Total new profit = 7,000 + 4,000 + 8,400 + 1,150 + 6,000 = Rs. 26,550

Thus, the new total profit earned by the shopkeeper is Rs. 26,550.

General Solution:

Determine the value of 'x':

The sum of the percentages of profit from all items must equal 100%.

Therefore: (2x - 2) + (x + 1) + (x + 3) + (x - 1) + (x + 9) = 100

6x + 10 = 100

6x = 90

x = 15

Now, let's calculate the percentage of profit earned from each item:

Bag: (2 × 15 - 2)% = 28%

Bottle: (15 + 1)% = 16%

Box: (15 + 3)% = 18%

Pen: (15 - 1)% = 14%

Register: (15 + 9)% = 24%

Now, let's calculate the actual profit (in Rupees) earned from each item:

Bag: 28% of Rs. 25,000 = 0.28 × 25,000 = Rs. 7,000

Bottle: 16% of Rs. 25,000 = 0.16 × 25,000 = Rs. 4,000

Box: 18% of Rs. 25,000 = 0.18 × 25,000 = Rs. 4,500

Pen: 14% of Rs. 25,000 = 0.14 × 25,000 = Rs. 3,500

Register: 24% of Rs. 25,000 = 0.24 × 25,000 = Rs. 6,000

Calculations:

We are given:

Total bottles sold = 500

Cost price per bottle = Rs. 50

Originally, all bottles were sold at a 20% profit.

Selling price per bottle = Cost price + 20% profit = 50 + 20 / 100 × 50 = 50 + 10 = 60 

20% of bottles are defective

20 / 100 × 500 = 100 defective bottles

These defective bottles were sold at a 15% loss.

Selling price per defective bottle = Cost price − 15% loss = 50 − 15 / 100 × 50 = 50 − 7.5 = 42.5

Total loss from defective bottles:

Total loss = 50 − 42.5 × 100 = 7.5 × 100 = 750 

We initially had a profit of Rs. 4,000 from bottles.

Net profit = Initial profit − Loss from defective bottles = 4,000 − 750 = 3,250

Thus, the net profit earned from bottles after accounting for defective items is Rs. 3,250.

Shortcut Trick

Let speed of Car A be S_A and Car B be S_B.

From the second condition: Total Distance = (Relative Speed) × Time

⇒ 160 = [S_A + (S_B + 10)] × 2 ⇒ S_A + S_B = 70 kmph.

Using the first condition (distance 120 km, 1 hr difference): Try values that sum to 70.

If S_A = 30 and S_B = 40: (120/30) − (120/40) = 4 − 3 = 1 hour. This satisfies all conditions.

∴ The correct answer is 30 kmph.

Alternate Method

Given: Distance = 120 km, Time difference = 1 hr, Distance apart = 160 km, Meeting time = 2 hr.

Case 1: Movement in opposite directions

⇒ Relative Speed = S_A + (S_B + 10)

⇒ 160 / 2 = S_A + S_B + 10

⇒ 80 = S_A + S_B + 10 ⇒ S_A + S_B = 70 --- (i)

Case 2: Time difference for 120 km

⇒ (120 / S_A) − (120 / S_B) = 1

⇒ 120 × (S_B − S_A) = S_A × S_B

Substitute S_B = 70 − S_A from (i):

⇒ 120 × (70 − S_A − S_A) = S_A(70 − S_A)

⇒ 120(70 − 2S_A) = 70S_A − S_A²

⇒ 8400 − 240S_A = 70S_A − S_A²

⇒ S_A² − 310S_A + 8400 = 0

⇒ (S_A − 30)(S_A − 280) = 0

⇒ Since speed < 100 kmph, S_A = 30 kmph.

∴ The correct answer is 30 kmph.

Additional Information

Relative Speed

When two objects move in opposite directions, relative speed is the sum of their individual speeds (u + v). When moving in the same direction, it is the difference (u − v).

Time, Speed, and Distance Relation

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

Meeting Time Formula

Meeting Time = (Initial distance between objects) ÷ (Relative Speed).

Shortcut Trick

Let n be the number of students and A be the original average.

Case 1: 30 − A = 2(n + 1) ⇒ 30 − A = 2n + 2 ⇒ 2n + A = 28

Case 2: 54 − 2A = 3(n + 2) ⇒ 54 − 2A = 3n + 6 ⇒ 3n + 2A = 48

Multiply Case 1 by 2: 4n + 2A = 56. Subtract Case 2: (4n − 3n) = 56 − 48 ⇒ n = 8.

Substitute n = 8 in Case 1: 2(8) + A = 28 ⇒ 16 + A = 28 ⇒ A = 12 kg.

∴ The correct answer is 12 kg.

Alternate Method

Given: Original average = A, Total students = n.

Formula: Total Weight = Average × Number of students.

⇒ Case 1: (nA + 30) = (n + 1)(A + 2)

⇒ nA + 30 = nA + 2n + A + 2 ⇒ 2n + A = 28 ...

(i) ⇒ Case 2: (nA + 2 × 27) = (n + 2)(A + 3)

⇒ nA + 54 = nA + 3n + 2A + 6 ⇒ 3n + 2A = 48 ...

(ii) ⇒ From (i), A = 28 − 2n. Substitute in (ii):

⇒ 3n + 2(28 − 2n) = 48 ⇒ 3n + 56 − 4n = 48

⇒ −n = −8 ⇒ n = 8 students

⇒ A = 28 − 2(8) = 28 − 16 = 12 kg

∴ The correct answer is 12 kg.

Additional Information

Net Deviation Method

In average problems, the sum of deviations from the original average is equal to the total increase or decrease in weight across all members (Net Deviation = Change in Average × New Total Count).

Weighted Average

When two groups are combined, the new average A_w = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick
Using Total Surface Area (TSA) formula: 2πr(r + h) = 960 with π = 3.
⇒ 2 × 3 × (x − 4)(x − 4 + x) = 960 ⇒ (x − 4)(2x − 4) = 160.
⇒ (x − 4)(x − 2) = 80. By inspection, if x = 12, then 8 × 10 = 80.
Breadth (b) = x = 12 cm. CSA = 2 × 3 × 8 × 12 = 576 cm2.
Perimeter (P) = 576 − 492 = 84 cm. 2(L + 12) = 84 ⇒ L = 30 cm.
Area = 30 × 12 = 360 cm2.
∴ The correct answer is 360 cm2.
Alternate Method
Given: TSA = 960 cm2, π = 3, Height (h) = x, Radius (r) = x − 4, Breadth (b) = x.
Formula Used: TSA = 2πr(h + r); CSA = 2πrh; Perimeter = 2(L + b); Area = L × b.

⇒ 2 × 3 × (x − 4)(x + x − 4) = 960
⇒ 6 × (x − 4)(2x − 4) = 960
⇒ (x − 4) × 2(x − 2) = 160
⇒ x2 − 6x + 8 = 80 ⇒ x2 − 6x − 72 = 0
⇒ (x − 12)(x + 6) = 0 ⇒ x = 12 cm (as x > 0)
⇒ Radius (r) = 12 − 4 = 8 cm; Height (h) = 12 cm; Breadth (b) = 12 cm
⇒ CSA of Cylinder = 2 × 3 × 8 × 12 = 576 cm2
⇒ Perimeter of Rectangle = CSA − 492 = 576 − 492 = 84 cm
⇒ 2(L + 12) = 84 ⇒ L + 12 = 42 ⇒ L = 30 cm
⇒ Area of Rectangle = 30 × 12 = 360 cm2
∴ The correct answer is 360 cm2.
Additional Information
Cylinder Formulas
Curved Surface Area = 2πrh. Total Surface Area = 2πr(h + r). Volume = πr2h.
Rectangle Properties
Perimeter = 2(Length + Breadth). Area = Length × Breadth. Diagonal = √(L2 + B2).
Quadratic Roots Shortcut
For ax2 + bx + c = 0, find two numbers that multiply to ac and add to b, then change signs and divide by a.

Shortcut Trick

Equate the sum of individual honey parts to the total honey in the final mixture.

Total Honey = (7/10)×(x+20) + (3/5)×(x−30) + (5/8)×(2x)

Total Water = (3/10)×(x+20) + (2/5)×(x−30) + (3/8)×(2x)

Ratio 20:11 ⇒ 11×Honey = 20×Water. Solving gives x = 80.

Calculate 2.5x = 2.5 × 80 = 200 ml.

∴ The correct answer is 200 ml.

Alternate Method

Given: Ratio of Honey : Water in Y = 7:3, Z = 3:2, T = 5:3. Final Ratio = 20:11.

⇒ Honey from Y = 0.7(x + 20) = 0.7x + 14

⇒ Honey from Z = 0.6(x − 30) = 0.6x − 18

⇒ Honey from T = (5/8) × 2x = 1.25x

⇒ Total Honey = 0.7x + 14 + 0.6x − 18 + 1.25x = 2.55x − 4

⇒ Total Mixture = (x + 20) + (x − 30) + 2x = 4x − 10

⇒ (Total Honey) / (Total Mixture) = 20 / (20 + 11) = 20/31

⇒ (2.55x − 4) / (4x − 10) = 20 / 31

⇒ 31(2.55x − 4) = 20(4x − 10)

⇒ 79.05x − 124 = 80x − 200

⇒ 0.95x = 76 ⇒ x = 76 / 0.95 = 80

⇒ 2.5x = 2.5 × 80 = 200

∴ The correct answer is 200 ml.

Additional Information

Mixture Rule

When multiple mixtures are combined, the total quantity of a specific component is the sum of that component in each individual mixture.

Ratio to Fraction

If the ratio of A:B is m:n, then the quantity of A in a total volume V is [m/(m+n)] × V.

Weighted Average of Ratios

Final concentration C = (q₁c₁ + q₂c₂ + ... + q_nc_n) / (q₁ + q₂ + ... + q_n).