Question

The digits of a 2-digit number differ by 5. If the digits are interchanged and the resulting number is added to the original number, we get 99. Find the original number.

Answer 1

STATEMENT
The digits of a 2-digit number differ
by 5.

_______CASE 1. _____
Let the smallest number be = X
Then the second number (greatest)
Y = X + 5
Number formed = XY or YX
We can write it as 10X + Y or 10Y + X as per tens or unit place.

DIGITS INTERCHANGE
Then the number becomes = YX or XY
We can write it as 10Y + X or 10X + Y as per tens or unit place.

SUM
XY + YX = 99
means
(10X + Y) + (10Y + X) = 99
11 X + 11Y = 99
11X + 11(X + 5) = 99
11X + 11X + 55 = 99
22X = 99 - 55
22X = 44
X = 44 / 22
X = 2
Then Y = 2 + 5 = 7
Number = 27

or

YX + XY = 99
means
(10Y + X) + (10X + Y) = 99
11 X + 11Y = 99
11X + 11(X + 5) = 99
11X + 11X + 55 = 99
22X = 99 - 55
22X = 44
X = 44 / 22
X = 2
Then Y = 2 + 5 = 7
Number = 27

________CASE 2. ____

Let the largest number be = X
Then the second number (smallest )
Y = X - 5
Number formed = XY or YX
We can write it as 10X + Y or 10Y + X as per tens or unit place.
DIGITS INTERCHANGE
Then the number becomes = YX or XY
We can write it as 10Y + X or 10X + Y as per tens or unit place.
SUM
XY + YX = 99
means
(10X + Y) + (10Y + X) = 99
11 X + 11Y = 99
11X + 11(X - 5) = 99
11X + 11X - 55 = 99
22X = 99 + 55
22X = 154
X = 154 / 22
X = 7
Then Y = 7 - 5 = 2
Number = 72

or

YX + XY = 99
means
(10Y + X) + (10X + Y) = 99
11 X + 11Y = 99
11X + 11(X - 5) = 99
11X + 11X - 55 = 99
22X = 99 + 55
22X = 154
X = 154 / 22
X = 7
Then Y = 7 - 5 = 2
Number = 72

****NOTE****
1. The digit of number depends on its status what we assumed greatest or smallest.
2. That digit is in unit or tens place.

CONCLUSION

The two digits in number are 2 & 7
But the original number is 27 or 72 as per our assumption.

Answer 2

Given -

The digits of a 2-digit number differ by 5. If the digits are interchanged and the resulting number is added to the original number, we get 99.

To find -

Original number

Solution -

Let the tens digit be x and the ones digit be y

Original number = 10x + y

✰ According to the first condition ✰

The digits of a 2-digit number differ by 5.

→ x - y = 5 ----(i)

✰ According to the second condition ✰

If the digits are interchanged and the resulting number is added to the original number, we get 99.

Reversed number = 10y + x

→ 10x + y + 10y + x = 99

→ 11x + 11y = 99

→ 11(x + y) = 99

→ x + y = 9 ----(ii)

Add both the equations

→ (x - y) + (x + y) = 5 + 9

→ x - y + x + y = 14

→ 2x = 14

→ x = 7

Put the value of x in eqⁿ (ii)

→ x + y = 9

→ 7 + y = 9

→ y = 9 - 7

→ y = 2

•°• Original number = 10x + y = 72

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