Question

the sum of the 3 digits number is 22. The middle digit is less by 10 than the sum of extreme digits. The number obtained on interchanging the digits is less than the original number by 198. Find the number​

Answer 1

Also, given a+c=b−1

Now as a+c=b−1 and c=a+2

⇒a+a+2=b−1

⇒b=2a+3

now in eq(i)

substitute the values of b and c  

83a=7b+16c

⇒83a=7(2a+3)+16(a+2)

⇒53a=21+32

⇒a=1

⇒c=a+2=3

⇒b=2a+3=5

So the number is 153

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Answer 2

Given: Sum of the digits of a 3 digit number = 22

          Middle digit is less by 10 than sum of extreme digits

          Number obtained by interchanging digits = original number - 198

To find: The number

Let: The given number = 100x + 10y + z

       The number is xyz where x,y,z have hundredth, tenth and unit place respectively.

Solution: According to the given question,

     x + y + z = 22      OR     x + z = 22 - y                     ...(1)

also,      y = x + z - 10                                                ...(2)

from equation (1) and (2),

  y = 22 - y - 10

2y =  12

  y = 6

and x + z = 22 - 6 = 16                                      ...(3)

when number is reversed, it becomes zyx i.e., 100z + 10y + x

Now,

(100x + 10y + z) - (100z + 10y + x) = 198

100x + 10y + z - 100z - 10y - x = 198

99x - 99z = 198

99(x - z) = 198

x - z = 198/99 = 2                                                     ...(4)

Adding equation (3) and (4)

x + z + x - z = 16 + 2

2x = 18

x = 9

putting value of x in equation 3

9 + z = 16

z = 16 - 9 = 7

The required number is xyz i.e., 967.