Question

sum of digits of two digit number is 9.When the digits are interchanged the new number is greater than the original number by 9.Find the original number.​

Answer 1

Given :

• Sum of digits of two digit number is 9.
• When the digits are interchanged the new number is greater than the original number by 9.

To Find :

• The original number.

Solution :

Let the digit at the tens place be x.

Let the digit at the units place be y.

Original Number = (10x + y)

Case 1 :

The sum of tens digit and units digit is 9.

Equation :

$\implies$ $\sf{x+y=9}$

$\sf{x=9-y\:\:\:\:\bold{1}}$

Case 2 :

When the digits of the two digit number are interchanged, the new number is greater than original number by 9.

Reversed Number = (10y +x)

Equation :

$\implies$ $\sf{10y+x=10x+y+9}$

$\implies$ $\sf{10y-y=10x-x+9}$

$\implies$ $\sf{9y=9x+9}$

$\implies$ $\sf{9x+9=9y}$

$\implies$ $\sf{9x-9y=-9}$

$\implies$ $\sf{9(9-y) -9y=-9}$

$\implies$ $\sf{81-9y-9y=-9}$

$\implies$ $\sf{-18y=-9-81}$

$\implies$ $\sf{-18y=-90}$

$\implies$ $\sf{y=\dfrac{-90}{-18}}$

$\implies$$\sf{y=\dfrac{90}{18}}$

$\implies$ $\sf{y=5}$

Substitute, y = 5 in equation (1),

$\implies$ $\sf{x=9-y}$

$\implies$ $\sf{x=9-5}$

$\implies$ $\sf{x=4}$

$\large{\boxed{\bold{Ten's\:digit\:=\:x\:=\:4}}}$

$\large{\boxed{\bold{Unit's\:digit\:=\:y\:=\:5}}}$

$\large{\boxed{\bold{\purple{Original\:Number\:=\:10x+y=10(4)+5=40+5}}}}$

Answer 2

Answer:

$\underline{\boxed{\red{\bf Original \: Number = 45}}}$

Step-by-step explanation:

Let the digits at ten's place be x. And, the digit at unit's place be y.

★ $\boxed{\bf Original \: Number = 10x + y}$

Also, it is given that the sum of the digits is 9.

⇒ x + y = 9..... (i)

According to the question,

If the digits are interchanged, the new number is greater than the original number by 9.

$\implies \sf 10y + x = 10x + y + 9\\\\\implies \sf 10y - y + x - 10x = 9\\\\\implies \sf 9y - 9x = 9\\\\\implies \sf -9(x - y) =9\\\\\implies \sf x - y =- \frac{9}{9}\\\\\implies \sf x - y = - 1 \: \: \:..... (ii)$

Adding equation (i) and (ii),

$\implies \sf x + y + x - y = 9 - 1\\\\\implies \sf 2x = 8\\\\\implies \sf x =\frac{8}{2}\\\\\boxed{\bf \therefore x = 4}$

Substituting the value of x in (i),

$\implies \sf x + y = 9 \\\\\implies \sf 4 + y = 9\\\\\implies \sf y = 9-4\\\\\boxed{\therefore \bf y = 5}$

Now, Original number = 10x + y

                                     = 10 * 4 + 5

                                     = 40 + 5

                                     = 45

Hence, the required number is 45.