Question

8. The sum of the digits of a two-digit number is 9. When the digits are reversed, the number is increased by 9. Find the number.

Answer 1

Given :

• Sum of the digits of a two digit number is 9

• When digits are reversed,Number increased by 9

To Find :

• The number

Answer :

★ 45

Solution :

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

$\star \sf \: Original \: Number = 10y + x$

$\star \sf \: Interchanged \: Number = 10x + y$

As per the question :

$\longrightarrow \sf x + y = 9....(i)$

Again as per the question :

Reversed number = Original number + 9

10x + y = 10y + x + 9

10x + y - 10y - x = 9

9x - 9y = 9

$\longrightarrow \sf x - y = 1....(ii)$

Adding both equations we'll get,

x - y = 1

x + y = 9

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2x = 10

x = 10/2

$\sf \longrightarrow \boxed{ \red{ \sf \: x = 5}}$

Sub x in y + x = 9.

y + x = 9

y + 5 = 9

y = 9 - 5

$\sf \longrightarrow \boxed{ \red{ \sf \: y = 4}}$

________________________________

$\star \sf \: Original \: Number = 10y + x \\ \rightarrow \: 10(4) + 5 = \boxed{ \purple{45}}$

Answer 2

Given :

• The sum of the digits of a two-digit number is 9.
• When the digits are reversed, the number is increased by 9.

To Find :

• The two digit 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 ten's digit (x) and the unit's digit (y) adds up to 9.

Equation :

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

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

Case 2 :

If the digits of the two digit number is reversed, the new number formed is increased by 9.

Reversed Number = (10y+x)

Equation :

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

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

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

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

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

$\longrightarrow$ $\sf{\cancel\dfrac{9}{9}x\:-\:\cancel\dfrac{9}{9}y\:=\:\cancel\dfrac{-9}{9}}$

$\longrightarrow$ $\sf{x-y=-1}$

From equation (1), x = 9-y,

$\longrightarrow$ $\sf{9-y-y=-1}$

$\longrightarrow$ $\sf{9-2y=-1}$

$\longrightarrow$ $\sf{-2y=-1-9}$

$\longrightarrow$ $\sf{-2y=-10}$

$\longrightarrow$ $\sf{y=\cancel\dfrac{-10}{-2}}$

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

Substitute, y = 5 in equation (1),

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

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

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

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

$\large{\boxed{\sf{\purple{Originl\:Number\:=\:10x+y\:=\:10(4)\:+\:5\:=\:40\:+\:5\:=\:45}}}}$