Question

the sum of the digits of two-digit number is 7 and if 9 added to the number, the digits are reversed . find the number.

Answer 1

Let aa be the first digit and bb be the second digit. The two digit number ( which looks like abab ) value is 10a+b10a+b

Also it is given that a+b=7a+b=7 -- ( 1 )

When the digits are interchanged ( it looks like baba ) the number's value becomes 10b+a10b+a

It is given that adding 9 to the original number makes it 10b+a10b+a ( interchanged )

So, 10a+b+9=10b+a10a+b+9=10b+a

Moving "a" and "b" terms to left side and moving numericals to right side,

10a+b−10b−a=−910a+b−10b−a=−9

9a−9b=−99a−9b=−9

9(a−b)=−99(a−b)=−9

a−b=−1a−b=−1

a=b−1a=b−1

Substituting for aa in ( 1 ), we get,

b−1+b=7b−1+b=7

2b=7+1=82b=7+1=8

b=82=4b=82=4

Hence a=b−1=4−1=3a=b−1=4−1=3

Hence original number is 34 and new number is 43.

.

Brute force technique -

Possible numbers are 16, 25, 34, 70.

Since 16 reverse is 61 the difference is not 9. So ruled out.

Since reverse of 25 is 52, and difference ( 52 - 25 ) is not 9. So 25 is ruled out.

Since reverse of 70 is 7, and difference ( 70 - 7 ) is not 9. So 70 is ruled out.

Since reverse of 34 is 43, and difference ( 43 - 34 ) is 9. so 43 is the new number.

Answer 2

Let first digit be x and second digit be y .

Let original no be = 10 x X + Y

A/Q

1.   X + Y = 7

X= 7-Y

2.  10 x X + Y + 9 = 10 x Y + X

10X + Y + 9 = 10Y + X

10X - X + 9 = 10Y - Y

9X + 9 = 9Y

9X - 9Y = -9

X - Y = -1

7-Y-Y = -1

-2Y = -1- -7

-2Y = -8

Y = 4

X = 7-4

  = 3

Original no = 34