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# The ratio of two numbers is 2:3. If 2 is subtracted from first and 8 from second, the ratio becomes reciprocal of the original ratio. Find the Numbers.

The given problem is an application of ratio and linear equations in one variable.

## Answer: The numbers are 8 and 12.

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## Explanation:

Let the numbers be 2x and 3x.

Given that if 2 is subtracted from first and 8 from second, the ratio becomes reciprocal of the original ratio.

⇒ (2x - 2) : (3x - 8) = 3: 2

⇒ (2x - 2) / (3x - 8) = 3: 2

⇒ 2(2x - 2) = 3(3x - 8) ------- [ cross multiplication ]

⇒ 4x - 4 = 9x - 24

⇒ 4x - 9x = 4 - 24

⇒ -5x = - 20

⇒ 5x = 20

⇒ x = 20/5

⇒ x = 4

Therefore, the numbers are

- 2x = 2(4) = 8
- 3x = 3(4) = 12

### Thus, the numbers are 8 and 12 respectively.

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