The following question has three statements. Study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. Find the average of the two numbers. I) 6 times the smaller number is 4 more than 4 times the larger one. II) 1/2 of the smaller number is 3 more than 1/5 of the larger one. III) 2 times their difference is 2 less than the smaller number.?

Option 2 : II and either I or III

Let the smaller and larger number be ‘x’ and ‘y’ respectively

Considering statement I,

⇒ 6x = 4 + 4y

⇒ 3x - 2y = 2      ----(1)

Considering statement II,

⇒ (1/2)x = 3 + (1/5)y

⇒ 5x - 2y = 30      ----(2)

Considering statement III,

⇒ 2y - 2x = x - 2

⇒ 3x - 2y = 2      ----(3)

Considering statements I and II,

Subtracting (1) from (2),

⇒ 5x - 2y - 3x + 2y = 30 - 2

⇒ 2x = 28

⇒ x = 28/2 = 14

Substituting in (1),

⇒ y = (42 - 2)/2 = 20

⇒ Average of numbers = (14 + 20)/2 = 17

Considering statements II and III,

Subtracting (3) from (2),

⇒ 5x - 2y - 3x + 2y = 30 - 2

⇒ 2x = 28

⇒ x = 28/2 = 14

Substituting in (3),

⇒ y = (42 - 2)/2 = 20

⇒ Average of numbers = (14 + 20)/2 = 17

∴ The question can be answered using statement II and either statement I or III

In statement III, the Difference mentioned needs to be positive and we can clear that confusion as we already established the smaller and larger number as ‘x’ and ‘y’ respectively.

So, Difference = (y -x) not ( x-y).