Question

The number obtained by adding 12 to a natural number is 160 times of the multiplicative inverse of that natural number.Find the number.​

Answer 1

Answer :-

8 is the required natural number

Solution :-

Let the natural number be x

12 is added to a natural number = x + 12

Multiplicative inverse of the given natural number = 1/x

160 times of the multiplicative inverse of that natural number = 160 * 1/x = 160/x

Given that

12 is added to a natural number = 160 times of the multiplicative inverse of that natural number

⇒ x + 12 = 160/x

Transpose x to LHS

⇒ x(x + 12) = 160

⇒ x² + 12x = 160

⇒ x² + 12x - 160 = 0

Split the middle term

⇒ x² + 20x - 8x - 160 = 0

⇒ x(x + 20) - 8(x + 20) = 0

⇒ (x + 20)(x - 8) = 0

⇒ x + 20 = 0 , x - 8 = 0

⇒ x = - 20 or x = 8

As per the condition

8 is the required natural number

Verification :-

x + 12 = 160/x

Let us check

Substitute x = 8 in the above equation.

⇒ 8 + 12 = 160/8

⇒ 20 = 20

Answer 2

Answer:

Let the Number be n and, reciprocal be 1 /n.

$\underline{\bigstar\:\textsf{According to the Question Now :}}\\\\\implies\tt n + 12 = 160 \times \dfrac{1}{n} \\\\\\\implies\tt n + 12 = \dfrac{160}{n}\\\\\\\implies\tt n(n + 12) = 160\\\\\\\implies\tt {n}^{2} + 12n = 160\\\\\\\implies\tt {n}^{2} + 12n - 160 = 0\\\\\\\implies\tt {n}^{2} + (20 - 8)n - 160 = 0\\\\\\\implies\tt {n}^{2} + 20n - 8n - 160 = 0\\\\\\\implies\tt n(n + 20) - 8(n + 20) = 0\\\\\\\implies\tt (n - 8)(n + 20) = 0\\\\\\\implies\tt \green{n = 8} \quad or \quad \red{n = - 20} \\\\\\ \therefore \underline{\textsf{Hence, Natural Number will be \textbf{8.}}}$