Question

the sum of digits of a three digit number is 12 and the digits are in A.P. if the digits are reversed, then the number is diminished by 396. find the number.

Answer 1

let the 3 digit number be a-d,a,a+b
the it is given that it's sum is 12
(a-d)+(a)+(a+d)=12
a-d+a+a+d=12
3a=12
a=12/3
a=4.
now if the digits are reversed,
100(a+d)+ 10a+(a-d)=100(a-d)+10a+(a+d)-396
putting the value of a =4 the equation will be
400+100d+40+4-d=400-100d+40+4+d-396
99d=-99d-396
99d+99d=-396
198d=-396
d=-396/198
d=-2
with a=4 and d=-2
the number is (a-d )(a) (a+d)
[4-(-2)](4)[(4+(-2)]
642 ans

Answer 2

The original number for the given condition is 642  .

Step-by-step explanation:

Given as :

The sum of three digits numbers = 12

The digits are in Arithmetic progression

Let The digits in A.P  are  a - d ,  a , a + d

So, Original number = 100 (a - d) + 10 a + 1 (a + d)

Or, Original number = 100 (a - d) + 10 a +  (a + d)

According to question

The sum of digits = 12

i.e a - d +  a + a + d = 12

Or,  3 a - 0 = 12

Or,        3 a = 12

∴              a = $\dfrac{12}{3}$

i.e            a = 4                    .........1

Again

if the digits are reversed, then the number is diminished by 396

Now, Reserved number =  100 (a + d) + 10 a +  (a - d)

So,  100 (a + d) + 10 a +  (a - d) = [ 100 (a - d) + 10 a +  (a + d) ] - 396

Rearranging the equation

     100 (a + d) + 10 a +  (a - d) - [ 100 (a - d) + 10 a +  (a + d) ]  = - 396

     (a + d) (100 - 1) + (10 a - 10 a) + (a - d) (1 - 100 ) = - 396

Or,  99 (a + d) + 0 - 99  (a - d) = - 396

Or,                99 (a + d - a + d) = - 396

Or,                              99 (2 d) = - 396

Or,                                         d = $\dfrac{-396}{198}$

i.e                                          d  = - 2                         ............2

Now, From eq 1 and eq 2

∵ Original number =  100 (a - d) + 10 a + 1 × (a + d)

                           =  100 (4 - (- 2) ) + 10 × 4 + 1 × (4 + (-2) )

                           = 100 (4 + 2) +  10 × 4 + 1 × (4 - 2)

                           = 100 × 6 +  10 × 4 + 1 × 2

                           = 600 + 40 + 2

                           = 642

So, The original number = 642

Hence, The original number for the given condition is 642  . Answer