Given

Average age of husband and wife = 24 years

Age of their son is 1/9th of the age of the husband

Wife is 6 years younger than the husband

Concept:

Average age = Total age/Number of individuals

Solution:

⇒ Total age of husband and wife = 24 years × 2 = 48 years

⇒ Age of husband = (48 years + 6 years)/2 = 27 years

⇒ Age of son = 27 years/9 = 3 years

⇒ Total age of all three = 48 years + 3 years = 51 years

⇒ Average age of all three = 51 years/3 = 17 years

Hence, the average age of all three members of the family is 17 years.

Given

Three numbers: 8, 14, 24

Concept: In the fourth proportion, if a:b = c:d, then d is the fourth proportion to a, b, c.

Solution:

⇒ a/b = c/d

⇒ 8/14 = 24/d

⇒ d = (24 × 14)/8

⇒ d = 42.

Therefore, the fourth proportion to 8, 14, and 24 is 42.

Given:

A student scores 40% of marks and fails by 40 marks in the examination.

Another student scores 60% and scores 20 more marks.

Calculation:

The maximum marks for examination = x

According to question

(m × 40%) + 40 = (m × 60%) - 20

Now, m × 60% - m × 40% = 40 + 20

⇒ 20m% = 60 

⇒ m/100 = 3 

⇒ m = 300 

∴ The maximum marks for the examination is 300.

Given:

Height (h) of the cylinder = 3.5 cm

Radius (r) of the cylinder = 3 × h = 3 × 3.5 cm = 10.5 cm

Formula:

Volume of the cylinder (V) = πr²h

Calculation:

⇒ V = π × (10.5 cm)² × 3.5 cm

V = 1212.75 cm³

Therefore, the volume of the cylinder is 1212.75 cm³.

Given:

The side of the cube is 8 cm.

Formula used:

Length of diagonal of a cube = √3a

Where, a = side of a cube

Calculation:

We know, largest diagonal of cube = √3 × side = 8√3

Now, the shortest distance from the centre of the spherical ball to vertices of a cube = Half the length of diagonal = 8√3/2 = 4√3

Given

Average age of husband and wife = 24 years

Age of their son is 1/9th of the age of the husband

Wife is 6 years younger than the husband

Concept:

Average age = Total age/Number of individuals

Solution:

⇒ Total age of husband and wife = 24 years × 2 = 48 years

⇒ Age of husband = (48 years + 6 years)/2 = 27 years

⇒ Age of son = 27 years/9 = 3 years

⇒ Total age of all three = 48 years + 3 years = 51 years

⇒ Average age of all three = 51 years/3 = 17 years

Hence, the average age of all three members of the family is 17 years.

Given

Three numbers: 8, 14, 24

Concept: In the fourth proportion, if a:b = c:d, then d is the fourth proportion to a, b, c.

Solution:

⇒ a/b = c/d

⇒ 8/14 = 24/d

⇒ d = (24 × 14)/8

⇒ d = 42.

Therefore, the fourth proportion to 8, 14, and 24 is 42.

Given:

Height (h) of the cylinder = 3.5 cm

Radius (r) of the cylinder = 3 × h = 3 × 3.5 cm = 10.5 cm

Formula:

Volume of the cylinder (V) = πr²h

Calculation:

⇒ V = π × (10.5 cm)² × 3.5 cm

V = 1212.75 cm³

Therefore, the volume of the cylinder is 1212.75 cm³.

Given:

The side of the cube is 8 cm.

Formula used:

Length of diagonal of a cube = √3a

Where, a = side of a cube

Calculation:

We know, largest diagonal of cube = √3 × side = 8√3

Now, the shortest distance from the centre of the spherical ball to vertices of a cube = Half the length of diagonal = 8√3/2 = 4√3

Given:

A student scores 40% of marks and fails by 40 marks in the examination.

Another student scores 60% and scores 20 more marks.

Calculation:

The maximum marks for examination = x

According to question

(m × 40%) + 40 = (m × 60%) - 20

Now, m × 60% - m × 40% = 40 + 20

⇒ 20m% = 60 

⇒ m/100 = 3 

⇒ m = 300 

∴ The maximum marks for the examination is 300.