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Mathematics

The difference between two numbers is 12 and the difference between their squares is 456. Find the numbers.

Linear Equations

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Answer

Let two numbers be x and y, where x > y.

Given,

Difference between two numbers = 12.

⇒ x - y = 12     ………(1)

Given,

Difference between their squares = 456

⇒ x2 - y2 = 456

By identity,

⇒ x2 - y2 = (x + y)(x - y)

⇒ (x + y)(x - y) = 456

Substituting value of x - y from equation (1) in above equation, we get :

⇒ (x + y)(12) = 456

⇒ (x + y) = 45612\dfrac{456}{12}

⇒ x + y = 38     ……..(2)

Adding equations (1) and (2),

⇒ x + y + x - y = 38 + 12

⇒ 2x = 50

⇒ x = 502\dfrac{50}{2}

⇒ x = 25.

Substituting value of x in equation (1), we get :

⇒ x - y = 12

⇒ 25 - y = 12

⇒ y = 25 - 12

⇒ y = 13.

Hence, the numbers are 25 and 13.

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