Question

a number consists of three digits the right hand being zero if the left hand and the middle digit be interchange the number is diminished by 180 if the left hand digit be halved and the middle and right hand digits interchange the number is diminished by 454 find the number

Answer 1

Answer:

x = 8

y = 6

Step-by-step explanation:

Let's take the original number as-

100x +10y + 0 = 100x +10y

Case I

=> 100y + 10x = 100x +10y - 180

=> 90y - 90x = -180

=> 90x - 90y = 180

=> x - y =2  or [x=y+2] ------- (1)

Case II

=> (100x)/2 + 0 + y = 100x + 10y - 454

=> 50x + y = 100x + 10y - 454

=> 454= 50x + 9y

=> 454 = 50(y + 2) + 9y    [from (1)]

=> 454 - 100 = 50y + 9y

=> 354 = 59y

=> y = 6

=> x = y + 2 => x = 8

Answer 2

Answer:

The required three digit number is 860.

Step-by-step-explanation:

NOTE: Kindly refer to the attachment first.

We have given that the number is three digit number.

It means, it has hundreds place, tens place and ones place

But, the right hand digit is 0.

$\sf\:\therefore$ It's not considerable.

Let the digit at hundreds place be x.

And the digit at tens place be y.

$\sf\:\therefore\:The\:original\:number\:\\=\:\sf\:100x\:+\:10y\:+\:0\\\\=\sf\:100x\:+\:10y$

From the first condition,

$\sf\:100x\:+\:10y\:-\:(\:100y\:+\:10x\:)\:=\:180\\\\\implies\sf\:100x\:+\:10y\:-\:100y\:-\:10x\:=\:180\\\\\implies\sf\:100x\:-\:10x\:+\:10y\:-\:100y\:=\:180\\\\\implies\sf\:90x\:-\:90y\:=\:180\\\\\implies\pink{\sf\:x\:-\:y\:=\:2}\:\:\:[\sf\:Dividing\:both\:sides\:by\:90\:]\:-\:-\:(\:1\:)$

Now, from the second condition,

$\sf\:100x\:+\:10y\:-\:(\:\frac{\cancel100x}{\cancel2}\:+\:0\: +\:y\:)\:=\:454\\\\\implies\sf\:100x\:+\:10y\:-\:(\:50x\:+\:y\:)\:=\:454\\\\\implies\sf\:100x\:+\:10y\:-\:50x\:-\:y\:=\:454\\\\\implies\orange{\sf\:50x\:+\:9y\:=\:454}\:\:\sf-\:-\:(\:2\:)$

Now, by multiplying equation ( 1 ) by 9 and adding it to equation ( 2 ), we get,

$\sf\:9x\:-\:9y\:=\:18\\\\\sf\:50x\:+\:9y\:=\:454\\\\\therefore\sf\:50x\:+\:\cancel{9y}\:=\:454\\\\\sf\:9x\:-\:\cancel{9y}\:=\:18\\\\\implies\sf\:59x\:=\:472\\\\\implies\sf\:x\:=\:\frac{\cancel{472}}{\cancel{59}}\\\\\implies\boxed{\red{\sf\:x\:=\:8}}$

Now, by substituting $\sf\:x\:=\:8$ in equation ( 1 ), we get,

$\sf\:x\:-\:y\:=\:2\:\:-\:-\:(\:1\:)\\\\\implies\sf\:8\:-\:y\:=\:2\\\\\implies\sf\:-\:y\:=\:2\:-\:8\\\\\implies\sf\:\cancel{-}\:y\:=\:\cancel{-}\:6\\\\\implies\boxed{\red{\sf\:y\:=\:6}}$

Now,

$\sf\:The\:three\:digit\:number\:=\:100x\:+\:10y\:+\:0\\\\\implies\sf\:100\:\times\:8\:+\:10\:\times\:6\:+\:0\\\\\implies\sf\:800\:+\:60\:+\:0\\\\\implies\boxed{\red{\sf\:860}}$