Question

A number consists of three digit, the right hand being zero.if the left-hand and the middle digits be interchanged the number is diminished by 180.if the left-hand digit be halved and the middle and right hand digits be interchanged the number is diminished by 454.find the number.​

Answer 1

Answer:

860

Step-by-step explanation:

Given :

A number consists of three digit,

the right hand being zero.

If the left-hand and the middle digits be interchanged the number is diminished by 180.

If the left-hand digit be halved and the middle and right hand digits be interchanged the number is diminished by 454.

find the number.​

Solution :

Statement 1 :

A number consists of three digit,

the right hand being zero.

⇒ One's / unit digit is 0,.

Let the digits of the numbers be,

x y 0

Then, it an be written in expanded form as,

100x + 10y + 0    (E.g., Expanded form of 121 = 100 × 1 + 10 × 2 + 1 × 2)

Statement 2 :

If the left-hand and the middle digits be interchanged the number is diminished by 180.

⇒ If the number changes to,

y x 0, then it gets reduced by 180

i.e., In expanded form,

⇒ 100y + 10x + 0 + 180 = 100x + 10y + 0

⇒ 100y + 10x + 180 = 100x + 10y

⇒ 10x - 100x + 100y - 10y + 180 = 0

⇒ -90x + 90y = -180

Dividing both the sides by 90,

We get,

⇒ -x + y = -2

⇒ x - y = 2 ...(i)

Statement 3 :

If the left-hand digit be halved and the middle and right hand digits be interchanged the number is diminished by 454.

⇒ when it happens,

The digits will be ,

$\frac{x}{2}$ 0 y ,

The number gets reduced by 454,.

i.e., In expanded form,

⇒ $100(\frac{x}{2} ) + 10 \times 0 + y + 454 = 100x + 10y + 0$

⇒ $50x +y + 454 = 100x + 10y$

⇒ 50x - 100x + y - 10y + 454 = 0

⇒ -50x - 9y +454 = 0

⇒ 50x + 9y = 454 ..(ii)

By adding (i) × 9 and (ii),

We get,

⇒ 9(x - y) + (50x + 9y) = 9(2) + 454

⇒ 9x - 9y + 50x + 9y = 18 + 454

⇒ 59x = 472

⇒ x = 8,.

By substituting the value of x, in (i)

We get,

⇒ x - y = 2

⇒ 8 - y = 2

⇒ -y = 2 - 8

⇒ -y = -6 ⇒ y = 6,.

The number is x y 0 = 8 6 0