Question

A two digit number is 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number.
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Answer 1

Step-by-step explanation:

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Answer 2

Given :

• A two digit number is 4 times the sum of its digits.
• If 18 is added to the number, the digits are reversed.

To Find :

• The 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 number, (10x + y) is 4 times the sum of the digits (x+y).

Equation :

$\longrightarrow$ $\sf{10x+y=4(x+y)}$

$\longrightarrow$ $\sf{10x+y=4x+4y}$

$\longrightarrow$ $\sf{10x-4x=4y-y}$

$\longrightarrow$ $\sf{6x=3y}$

$\sf{x\:=\dfrac{3y}{6}\:\:\:(1)}$

Case 2 :

If we add 18 to the number (10x + y) the places of digits are reversed.

•°• Reveresed Number = (10y + x)

Equation :

$\longrightarrow$ $\sf{10x+y+18=10y+x}$

$\longrightarrow$ $\sf{10x+y-10y-x=-18}$

$\longrightarrow$ $\sf{9x-9y=-18}$

Divide throughout by 9,

$\longrightarrow$ $\sf{\cancel{\dfrac{9}{9}}x\:-\:\cancel\dfrac{9}{9}y\:=\:\cancel\dfrac{-18}{9}}$

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

From (1), x = 3y/6

$\longrightarrow$ $\sf{\dfrac{3y}{6}\:-\:y=-2}$

$\longrightarrow$ $\sf{\dfrac{3y-6y}{6}=-2}$

$\longrightarrow$ $\sf{\dfrac{-3y}{6}=-2}$

$\longrightarrow$ $\sf{-3y=-12}$

$\longrightarrow$ $\sf{y=\dfrac{-12}{-3}}$

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

Substitute, y = 4 in equation (1),

$\longrightarrow$ $\sf{x=\dfrac{3y}{6}}$

$\longrightarrow$ $\sf{x=\dfrac{3\:\times\:4}{6}}$

$\longrightarrow$ $\sf{x=\cancel\dfrac{12}{6}}$

$\longrightarrow$ $\sf{x=2}$

Number :

$\large{\boxed{\sf{\red{Ten's\:digit\:=\:x\:=\:2}}}}$

$\large{\boxed{\sf{\purple{Unit's\:digit\:=\:y\:=\:4}}}}$

$\large{\boxed{\sf{\red{Original\:Number\:=\:10x+y\:=\:10(2)+4=20+4=24}}}}$