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Given:

The expression is \(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\)

Formula Used:

(a³ + b³) = (a + b)(a² - ab + b²)

Calculation:

According to the question, we have

The given expression is

\(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\)        ----(1)

In the formula used

(a3 + b3) = (a + b)(a2 - ab + b2)

⇒ (a3 + b3)/(a2 - ab + b2) = (a + b)        ----(2)

Now, comparison between equation (1) and equation (2)

a = 0.73

b = 0.31

⇒ (a + b) = (0.73 + 0.31)

⇒ (a + b) = 1.04

∴ The value of given expression is 1.04.

Given:

Loss = 35%

If Anil sold the pen at Rs. 240 more he would Loss 20%

Formula Used:

S.P = C.P × {(100 - L%)/100}

S.P = C.P × {(100 + P%)/100}

S.P = Selling Price, C.P = Cost Price

L = Loss, P = Profit

Calculation:

Let the C.P of the pen be 100x

According to the question, we have

S.P = 100x × (100 - 35/100)

⇒ S.P = 65x

Now, he sold Rs. 240 more, then 20% loss

100x × (100 - 20/100) = 65x + 240

⇒ 80x = 65x + 240

⇒ 80x - 65x = 240

⇒ 15x = 240

⇒ x = 16

C.P = 100 × 16

⇒ 1600

Now, he gains 40%

S.P = 1600 × (100  + 40)/100

⇒ 1600 × (140/100)

⇒ 2240

∴ The selling price of the pen is Rs. 2240.

Formula Used:

The sum of first n natural numbers = {n × (n + 1)}/2

where, n = number of natural numbers

Average = The sum of the observations/The number of the observations

Calculation:

According to the question, we have

The sum of the first 19 natural numbers = {19 × (19 + 1)}/2

⇒ 19 × 10

⇒ 190

Average of the first 19 natural numbers = 190/19

⇒ 10

The first 6 positive odd numbers are 1, 3, 5, 7, 9 and 11

The sum of first 6 positive odd numbers = 1 + 3 + 5 + 7 + 9 + 11

⇒ 36

The average of the first 6 positive odd numbers = 36/6

⇒ 6

Now,

The sum of the average = 10 + 6

⇒ 16

∴ The sum of the average of first 19 natural numbers and the average of first 6 positive odd numbers is 16.

Given:

\(A=\dfrac{1}{30}+\dfrac{1}{20}+\dfrac{1}{42}:\)

\(Y=\dfrac{1}{56}+\dfrac{1}{72}+\dfrac{1}{90}\)

Calculation:

A = (1/30) + (1/20) + (1/42)

⇒ A = (14 + 21 + 10)/420

⇒ A = 45/420 = 3/28

B = (1/56) + (1/72) + (1/90)

⇒ B = (45 + 35 + 28)/2520

⇒ B = 108/2520 = 3/70

Now, 

A + B = (3/28) + (3/70)

⇒ A + B = 21/140

∴ The value of A + B is 21/140.

Given:

C.P/M.P = 5/6

Discount = 30%

Formula used:

Discount = [(D%)/100] × M.P

S.P = M.P - Discount

Loss = C.P - S.P

Calculation:

Let C.P and M.P be 500 and 600

Discount = 30%

Discount = [(D%)/100] × M.P

Discount = (30/100) × 600

⇒ Discount = 180

So, S.P = M.P - Discount

S.P = 600 - 180

⇒ S.P = 420

Loss = C.P - S.P

Loss = 500 - 420 = 80

Loss% = (80/500) × 100 = 16%

∴ The profit percentage is 16%.

Given:

MP : SP = 5 : 2

Where, MP = Marked Price, SP = Selling Price

Formula used:

Discount% = {(MP - SP)/MP} × 100

Calculation:

Let MP and SP be 5x and 2x

According to the formula used,

Discount% = {(MP - SP)/MP} × 100

⇒ Discount% = {(5x - 2x)/5x} × 100

⇒ Discount% = (3x/5x) × 100 = 60

∴ The discount percentage is 60%

Given:

Selling Price = Rs. 960

Profit earned = 50%

Formula used:

S.P = {(100 + Profit%)/100} × C.P

Where, S.P = Selling Price, C.P = Cost Price

Calculation:

According to the formula used,

S.P = {(100 + Profit%)/100} × C.P

⇒ 960 = {(100 + 50)/100} × C.P

⇒ 960 × (2/3) = C.P

⇒ C.P = 640

According to the question,

The new profit percentage is 30%

S.P = {(100 + Profit%)/100} × C.P

⇒ S.P = {(100 + Profit%)/100} × C.P

⇒ S.P = {(100 + 30)/100} × 640 = 832

∴ The new selling price of the article is Rs. 832

Given:

The ratio of age of A and B is 3 : 4

The Sum of their ages is 56

Calculation:

Let the ages of A and B is 3x and 4x

According to the given question,

⇒ 3x + 4x = 56

⇒ 7x = 56

⇒ x = 56/7

⇒ x = 8

The age of B = 4x = 4 × 8

⇒ 32

∴ The age of B is 32 years.

Given:

Selling price = Rs. 960

Profit = Rs. 160

Formula used:

Profit % = (P/C.P.) × 100

Profit = S.P. - C.P.

Where, Cost price = C.P, Selling price = S.P, Profit = P

Calculation:

Profit = S.P. - C.P.

⇒ 160 = 960 - C.P.

⇒ C.P. = Rs. 800      ----(1)

Profit % = (P/C.P.) × 100

⇒ P % = (160/800) × 100

⇒ P % = 20

∴ The total profit is 20%.

Given:

A = \(5\frac{1}{2} + 7\frac{3}{4} - 2\frac{1}{4}\)

B = \(11\frac{1}{4} - 3\frac{1}{2} - 5\frac{3}{4}\)

Calculation:

A = \(5\frac{1}{2} + 7\frac{3}{4} - 2\frac{1}{4}\)

⇒ A = (11/2) + (31/4) - (9/4)

⇒ A = {(22 + 31 - 9)/4}

⇒ A = 44/4

⇒ A = 11      ----(1)

B = \(11\frac{1}{4} - 3\frac{1}{2} - 5\frac{3}{4}\)

⇒ B = (45/4) - (7/2) - (23/4)

⇒ B = {(45 - 14 - 23)/4}

⇒ B = (8/4)

⇒ B = 2      ----(2)

Now, 

⇒ A² + B² 

From equation (1) and equation (2)

⇒ (11)² + (2)²

⇒ 125

∴ The value of A² + B² is 125.