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Given:

Average of four consecutive odd numbers A, B, C, and D = 14

Formula used:

Average = (Sum of all numbers) / (Number of terms)

For four consecutive odd numbers: A, B, C, D, we can represent them as:

A = x, B = x + 2, C = x + 4, D = x + 6

Calculation:

Average = (A + B + C + D) / 4

⇒ 14 = (x + (x + 2) + (x + 4) + (x + 6)) / 4

⇒ 14 = (4x + 12) / 4

⇒ 14 × 4 = 4x + 12

⇒ 56 = 4x + 12

⇒ 4x = 56 - 12

⇒ 4x = 44

⇒ x = 11

Now, A = x = 11, D = x + 6 = 11 + 6 = 17

Product of A and D = A × D

⇒ Product = 11 × 17

⇒ Product = 187

∴ The correct answer is option (4).

Given:

Selling Price (SP) = ₹26,650

Loss Percentage = 18%

Formula used:

Cost Price (CP) = \(\frac{SP}{1-\frac{\text{Loss %}}{100}}\)

Calculation:

CP = \(\frac{26,650}{1-\frac{18}{100}}\)

⇒ CP = \(\frac{26,650}{1-0.18}\)

⇒ CP = \(\frac{26,650}{0.82}\)

⇒ CP = ₹32,500

∴ The correct answer is option (4).

Given:

A finishes work in 7 days of 9 h/day ⇒ 7 × 9 = 63 h

B finishes work in 6 days of 7 h/day ⇒ 6 × 7 = 42 h

Working time per day together = 4 1/5 h = 4.2 h

Formula used:

Rate = Work ÷ Time

Calculations:

LCM method: Take LCM(63, 42) as total work (units).

⇒ LCM(63, 42) = 126 units

⇒ A’s rate = 126 ÷ 63 = 2 units/h

⇒ B’s rate = 126 ÷ 42 = 3 units/h

⇒ Together = 2 + 3 = 5 units/h

⇒ Time for full work = 126 ÷ 5 = 25.2 h

⇒ Days at 4.2 h/day = 25.2 ÷ 4.2 = 6

∴ They will finish in 6 days.

Given:

Monthly income of Mukesh = ₹75,000

Increase in income = 29%

Formula used:

New income = Original income × (1 + Percentage increase/100)

Calculations:

New income = 75,000 × (1 + 29/100)

⇒ New income = 75,000 × (1 + 0.29)

⇒ New income = 75,000 × 1.29

⇒ New income = ₹96,750

∴ The correct answer is option (4).

Given:

Initial Bill = ₹30,000

First Discount = 10%

Second Discount = 20%

Additional Discount = 10% (if net payment after first two discounts > ₹20,000)

Formula used:

Net Price after Discount = Price × (1 - Discount/100)

Calculation:

Apply the first discount of 10% on ₹30,000

⇒ Net Price = ₹30,000 × (1 - 10/100)

⇒ Net Price = ₹30,000 × 0.9

⇒ Net Price = ₹27,000

Apply the second discount of 20% on ₹27,000

⇒ Net Price = ₹27,000 × (1 - 20/100)

⇒ Net Price = ₹27,000 × 0.8

⇒ Net Price = ₹21,600

Check if the net price after two discounts is greater than ₹20,000

Since ₹21,600 > ₹20,000, apply an additional discount of 10%

⇒ Final Price = ₹21,600 × (1 - 10/100)

⇒ Final Price = ₹21,600 × 0.9

⇒ Final Price = ₹19,440

∴ The correct answer is option (2).

Given:

First number = 27

Second number = 108

Third number = 165

Formula used:

If a, b, c are given, the fourth proportional (x) is calculated using the formula:

(a : b = c : x) ⇒ x = (b × c) / a

Calculation:

x = (108 × 165) / 27

⇒ x = 17820 / 27

⇒ x = 660

∴ The fourth proportional to 27, 108, and 165 is 660.

Given:

The sum of two numbers is 60.

HCF = 5

LCM = 175

Formula used:

If HCF × LCM = Product of two numbers, then the numbers can be expressed as:

Let numbers = 5a and 5b, where a and b are co-prime.

Sum of reciprocals of two numbers = \(\dfrac{1}{\text{Number 1}} + \dfrac{1}{\text{Number 2}}\).

Calculation:

HCF × LCM = Product of two numbers

⇒ 5 × 175 = 875

Let numbers = 5a and 5b

Product = 5a × 5b = 875

⇒ a × b = 35

Sum of numbers = 5a + 5b = 60

⇒ a + b = 12

From a × b = 35 and a + b = 12:

⇒ a and b are roots of the equation x² - 12x + 35 = 0

⇒ (x - 7)(x - 5) = 0

⇒ x = 7, 5

Numbers are 5a = 35 and 5b = 25.

Sum of reciprocals:

⇒ \(\dfrac{1}{35} + \dfrac{1}{25}\)

⇒ \(\dfrac{5 + 7}{175}\)

⇒ \(\dfrac{12}{175}\)

∴ The correct answer is option (1).

Given:

Dimensions of the cuboid:

Length (l) = 256 cm

Breadth (b) = 135 cm

Height (h) = 50 cm

The cuboid is melted and recast into a cube.

Formula used:

Volume of cuboid = l × b × h

Volume of cube = a³, where a = side of the cube

Surface area of cube = 6a²

Calculation:

Volume of cuboid = 256 × 135 × 50

⇒ Volume of cuboid = 1,728,000 cm³

Volume of cube = Volume of cuboid

⇒ a³ = 1,728,000

⇒ a = ∛1,728,000 = 120 cm

Surface area of cube = 6a²

⇒ Surface area = 6 × 120 × 120

⇒ Surface area = 86,400 cm²

∴ The correct answer is option (3).

Given:

Difference (CI − SI) = ₹964

Rate (r) = 20% per annum

Time (t) = 2 years

Formula used:

Difference for 2 years = P × (r/100)²

Calculations:

⇒ 964 = P × (20/100)²

⇒ 964 = P × (400/10000)

⇒ 964 = P × (1/25)

⇒ P = 964 × 25

⇒ P = 24,100

∴ The sum is ₹24,100.

Given:

Reeta finishes 28 m ahead of Sita.

Geeta finishes 40 m ahead of Reeta.

Geeta finishes 64 m ahead of Sita.

Formula used:

For uniform speeds, distances covered are proportional to times; ratios of distances remain constant.

Calculations:

Let total distance = y m.

When Geeta finishes: Geeta = y, Reeta = y − 40, Sita = y − 64.

Ratio Reeta : Sita = (y − 40) : (y − 64)

When Reeta finishes: Reeta = y, Sita = y − 28.

Ratio Reeta : Sita = y : (y − 28)

⇒ (y − 40)/(y − 64) = y/(y − 28)

⇒ (y − 40)(y − 28) = y(y − 64)

⇒ y² − 28y − 40y + (40 × 28) = y² − 64y

⇒ y² − 68y + 1120 = y² − 64y

⇒ −68y + 1120 = −64y

⇒ 1120 = 4y

⇒ y = 1120 ÷ 4 = 280

∴ The race distance y = 280 m.