# The difference between two natural numbers is 6. Adding 10 to the twice of the larger number, we get 2 less than 3 times of the smaller number. Find these numbers.

Let the two natural numbers be x and y such that x is the larger number and y is the smaller number.

According to the question,

x – y = 6

and 2x + 10 = 3y – 2

x – y – 6 = 0 …(i)

And 2x – 3y + 12 = 0 ..(ii)

Multiply (i) by 2,

2x – 2y – 12 = 0 .. (iii)

2x – 3y + 12 = 0 ..(ii)

Subtract (iii) from (ii),

2x – 3y + 12 – 2x + 2y + 12 = 0

– y + 24 = 0

y = 24

Putting this value in (i),

x – 24 – 6 = 0

x = 30

So the numbers are 30 and 24.

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