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Shortcut Trick

Age difference between A and B is always (a + 4) − a = 4 years.

4 years ago, ratio was 6 : 7. Difference 1 unit = 4 years ⇒ A = 24, B = 28.

Present ages: A = 28, B = 32, C = 32 + 10 = 42.

Average of A, B, C, D is 40 ⇒ Total = 160. D = 160 − (28 + 32 + 42) = 58.

Ratio B:C, (32 − x)/(42 − x) = 3/4 ⇒ 128 − 4x = 126 − 3x ⇒ x = 2.

D 4x years ago (8 years ago) = 58 − 8 = 50 years.

∴ The correct answer is 50 years.

Alternate Method

Given: Present age A = a, B = a + 4, C = B + 10.

⇒ 4 years ago: (a − 4) / (a + 4 − 4) = 6 / 7

⇒ 7(a − 4) = 6a ⇒ 7a − 28 = 6a ⇒ a = 28.

⇒ Present Ages: A = 28, B = 32, C = 42.

⇒ Age of B and C x years ago: (32 − x) / (42 − x) = 3 / 4

⇒ 128 − 4x = 126 − 3x ⇒ x = 2.

⇒ Total present age = 40 × 4 = 160 years.

⇒ Age of D = 160 − (28 + 32 + 42) = 160 − 102 = 58 years.

⇒ Age of D 4x years (4 × 2 = 8 years) ago = 58 − 8 = 50 years.

∴ The correct answer is 50 years.

Additional Information

Ratio and Age Difference

The numerical difference between the ages of two persons remains constant at any point in time.

Average Age Concept

Total age = Average Age × Number of persons. If everyone's age increases by 'n' years, the average also increases by 'n'.

General solution:

Given:

 

Calculations:

June 2023 = 8,000 ⇒ July 2023 = 8,000 × 1.25 = 10,000

June 2022 = 6,000 ⇒ July 2022 = 6,000 × 0.80 = 4,800

Total (July 2022 + July 2023) = 4,800 + 10,000 = 14,800

Hence, the correct answer is 14,800.

General solution:

Given:

 

Calculations:

Feb 2022 revenue = 6,000 × 500 = ₹3,000,000

Apr 2022 revenue = 7,000 × 500 = ₹3,500,000

Feb 2023 revenue = 4,000 × 600 = ₹2,400,000

Apr 2023 revenue = 6,000 × 600 = ₹3,600,000

Total revenue = 3,000,000 + 3,500,000 + 2,400,000 + 3,600,000 = ₹12,500,000

Hence, the correct answer is ₹12,500,000.

General solution:

Given:

Calculations:

Total 2022 (Jan–Jun) = 4,000 + 6,000 + 5,000 + 7,000 + 8,000 + 6,000 = 36,000

Average 2022 = 36,000 / 6 = 6,000

Total 2023 (Jan–Jun) = 5,000 + 4,000 + 7,000 + 6,000 + 9,000 + 8,000 = 39,000

Average 2023 = 39,000 / 6 = 6,500

Difference = 6,500 − 6,000 = 500

Hence, the correct answer is 500.

General solution:

Given:

 

Calculations:

Jan: 2023 (5,000) > 2022 (4,000) ⇒ 2023 greater

Feb: 2022 (6,000) > 2023 (4,000) ⇒ 2022 greater

Mar: 2023 (7,000) > 2022 (5,000) ⇒ 2023 greater

Apr: 2022 (7,000) > 2023 (6,000) ⇒ 2022 greater

May: 2023 (9,000) > 2022 (8,000) ⇒ 2023 greater

Jun: 2023 (8,000) > 2022 (6,000) ⇒ 2023 greater

Counts → 2022 > 2023 : 2 months (Feb, Apr). 2023 > 2022 : 4 months (Jan, Mar, May, Jun).

Hence, the correct answer is 2, 4.

General solution:

Given:

 

Calculations:

Sum (Mar–May 2023) = 7,000 + 6,000 + 9,000 = 22,000

Sum (Mar–May 2022) = 5,000 + 7,000 + 8,000 = 20,000

Increase = 22,000 − 20,000 = 2,000

Percentage increase = (2,000 / 20,000) × 100 = 10.00%

Hence, the correct answer is 10.00%.

Shortcut Trick

Equating total work: (P + Q) × 10d = (1.5P + 0.75Q) × 8d

⇒ 10P + 10Q = 12P + 6Q ⇒ 2P = 4Q ⇒ P : Q = 2 : 1

Since P takes 20 hr, Q takes 20 × (2/1) = 40 hr. Then R takes 40 − 10 = 30 hr.

P + R time = (20 × 30) ÷ (20 + 30) = 600 ÷ 50 = 12 hr.

∴ The correct answer is 12 hr.

Alternate Method

Given: P takes 20 hr. Work done in Case 1 = (P + Q) × 10d. Case 2 = (1.5P + 0.75Q) × 8d.

⇒ 10dP + 10dQ = 12dP + 6dQ

⇒ 4dQ = 2dP ⇒ P = 2Q

⇒ Since Time ∝ 1/Efficiency, Time taken by Q = 2 × Time taken by P = 2 × 20 = 40 hr.

⇒ Time taken by R = 40 − 10 = 30 hr.

⇒ Total Work = LCM(20, 30) = 60 units.

⇒ Efficiency of P = 60 ÷ 20 = 3 units/hr.

⇒ Efficiency of R = 60 ÷ 30 = 2 units/hr.

⇒ Combined Efficiency (P + R) = 3 + 2 = 5 units/hr.

⇒ Time taken by P and R together = 60 ÷ 5 = 12 hr.

∴ The correct answer is 12 hr.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a task. If the ratio of efficiencies of two people is x : y, the ratio of time taken by them is y : x.

Combined Work Formula

If person A takes 'a' hours and person B takes 'b' hours, they together complete the work in (a × b) ÷ (a + b) hours.

Work Invariance Principle

Total Work = Efficiency × Time. When the same work is done with different efficiencies, (E₁ × T₁) = (E₂ × T₂).

I: 14x2 + 21x - 10x - 15 = 0

⇒ 7x(2x + 3) - 5(2x + 3) = 0

⇒ (2x + 3) × (7x - 5) = 0

⇒ x = -3/2, 5/7

II: 20y2 - 31y + 12 = 0

⇒ 20y2 - 15y - 16y + 12 = 0

⇒ 5y(4y - 3) - 4(4y - 3) = 0

⇒ (5y - 4) × (4y - 3) = 0

⇒ y = 4/5, 3/4

∴ x < y

Shortcut Trick

I. x2 + 74x – 480= 0

There are three terms = a, b, and c

Where 'a' = value of x2, 'b' = Value of x, 'c' = 480

'c' is negative here.   

II. y2 + 64y – 1280 = 0

There are also three terms = a, b, and c

Where 'a' = value of y2, 'b' = Value of y, 'c' = 1280

'c' is also negative here.

If the value of 'c' is negative in both statements, then the answer always is "x = y or the relation between x and y can’t be established."

The basic and conventional method is given below:

Given:

I. x2 + 74x – 480= 0       

II. y2 + 64y – 1280 = 0

Calculation:

I. x2 + 74x – 480= 0       

⇒ x² + 80x – 6x – 480 = 0

⇒ x(x + 80) – 6(x + 80) = 0

⇒ (x + 80)(x – 6) = 0

⇒ x = (-80), 6    

II. y2 + 64y – 1280 = 0

⇒ y² + 80y – 16y – 1280 = 0

⇒ y(y + 80) – 16(y + 80) = 0

⇒ (y + 80)(y – 16) = 0

⇒ y = (-80), 16

When we compared the values of X and Y in the table above, we found that there are THREE relations between X and Y i.e. >, = and <. So, a relation cannot be defined.

∴ After comparison all the values of x and y, the relation is “x = y or the relation between x and y can't be established.”

Given:

I. 4x² – 17x + 15 = 0

II. 3y² + 13y - 38 = 0

Calculation:

From I,

4x² – 17x + 15 = 0

⇒ 4x² – 12x - 5x + 15 = 0

⇒ 4x(x - 3) - 5(x - 3) = 0

⇒ (x - 3)(4x - 5) = 0

Taking,

⇒ (x - 3) = 0 or  (4x - 5) = 0

⇒ x = 3 or (5/4)

From II,

3y² + 13y - 38 = 0

⇒ 3y² + 19y - 6y - 38 = 0

⇒ y(3y + 19) - 2(3y + 19) = 0

⇒ (y - 2)(3y + 19) = 0

Taking,

⇒ (y - 2) = 0 or (3y + 19) = 0

⇒ y = 2 or (-19/3)

Comparison between x and y (via Tabulation):

 

∴ x = y or relation between x and y can not be established.