Question

a number consists of two digits whose sum is 9 if 27 is subtracted from the number its digits are interchanged find the original number ​

Answer 1

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Original Number = 63

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GiVeN :

• A number consists of two digits whose sum is 9
• If 27 is subtracted from the number its digits are interchanged.

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

$\sf{\underline{\underline{As\:PeR\:tHe\:FiRsT\:cOnDiTiOn:}}}$

• Sum of the digits is 9

Constituting it mathematically,

$\leadsto$ $\sf{x\:+\:y\:=\:9}$ ----> 1

$\sf{\underline{\underline{As\:PeR\:tHe\:SeCoNd\:cOnDiTiOn:}}}$

• Subtracting 27 from the number, the digits get interchanged.

Constituting it mathematically,

$\leadsto$ $\sf{10x\:+\:y\:-\:27\:=\:10y\:+\:x}$

$\leadsto$ $\sf{10x\:+\:y\:=\:10y\:+\:x\:+\:27}$

$\leadsto$ $\sf{10x\:-\:x\:=\:10y\:-\:y\:+\:27}$

$\leadsto$ $\sf{9x\:=\:9y\:+\:27}$

$\leadsto$ $\sf{9x\:-\:9y\:=\:27}$

$\leadsto$ $\sf{9(x\:-\:y)\:=\:27}$

$\leadsto$ $\sf{x\:-\:y\:=\:{\dfrac{27}{9}}}$

$\leadsto$ $\sf{x\:-\:y\:=\:3}$ ----> 2

Solve equation 1 and 2 simultaneously by elimination method.

Add equation 1 to 2,

$\leadsto$ $\sf{x\:+\:y\:+\:x\:-\:y\:=\:9\:+\:3}$

$\leadsto$ $\sf{2x\:=\:12}$

$\leadsto$ $\sf{x\:=\:{\dfrac{12}{2}}}$

$\leadsto$ $\sf{x\:=\:6}$

Substitute x = 6 in equation 1,

$\leadsto$$\sf{x\:+\:y\:=\:9}$

$\leadsto$ $\sf{6\:+\:y\:9}$

$\leadsto$ $\sf{y\:=\:9\:-\:6}$

$\leadsto$ $\sf{y\:=\:3}$

$\sf{\large{\boxed{\tt{\red{\therefore{Tens\:digit\:=\:x\:=\:6}}}}}}$

$\sf{\large{\boxed{\tt{\red{\therefore{Units\:digit\:=\:y\:=\:3}}}}}}$

$\sf{\large{\boxed{\tt{\red{\therefore{Original\:Number\:=\:10x\:+\:y\:=\:10\:\times\:6\:+\:3\:=\:60\:+\:3\:=\:63}}}}}}$

Answer 2

ANSWER:-

Given:

A number consists of two digits whose sum is 9, if 27 is subtracted from the number its digits are interchanged.

To find:

The original number.

Explanation:

Let the unit's digit be R &

Let the ten's digit be M.

We have,

The two digit whose sum is 9

R+M=9........................(1)

Therefore,

• The original number is 10R+M

• The reversed number is 10M+R

If 27 is subtracted from the original number. Its digit are reversed.

So,

→ 10R+M-27=10M+R

→ 10R-R+M-10M=27

→ 9R -9M= 27

→ 9(R-M= 3)

→ R-M=3..........................(2)

Using Substitution Method:

From equation (1),we get;

⇒ R+M=9

⇒ R= 9-M.....................(3)

Putting the value of M in equation (2),we get;

⇒ 9-M-M=3

⇒ 9-2M= 3

⇒ -2M= 3-9

⇒ -2M= -6

⇒ M= $\frac{\cancel{-6}}{\cancel{-2}} =3$

Putting the value of M in equation (1),we get;

⇒ R+3=9

⇒ R= 9-3

⇒ R= 6

Thus,

The original number is 10(6)+3

The original number is 60+3= 63.