Question

A certain number between 10 and 100 is 4 times the sum of its digits and if 18 is added to it the digit will be reversed find the number ​

Answer 1

✍ Gɪᴠᴇɴ :-

A certain number between 10 and 100 is 4 times the sum of its digits and if 18 is added to it the digit will be reversed .

✍ Tᴏ Fɪɴᴅ :-

• The number

✍ Sᴏʟᴜᴛɪᴏɴ :-

➠ Let the number at tens place be 'x'

And,

➠ Number at ones place be 'y'

Then ,

➠ Original number = ( 10x + y)

Now,

• ✞ According to 1st condition :-

⇒ 10x + y = 4 (x + y)

⇒ 10x + y = 4x + 4y

⇒ 10x - 4x = 4y - y

⇒ 6x = 3y

⇒ 6x - 3y = 0

⇒ 3( 2x - y) = 0

⇒ 2x - y = 0

⇒ y = 2x. ----(1)

• ✞ According to 2nd Condition :-

⇒ 10x + y + 18 = 10y + x

⇒ 10x - x + y - 10y = -18

⇒ 9x - 9y = -18

⇒ 9 ( x - y) = -18

⇒ x - y = -18/9 = -2

⇒ x - y = -2. --(2)

Now,

On putting the value of y from 1st in 2nd, We get

⇒ x - (2x) = -2

⇒ x - 2x = -2

⇒ -x = -2

⇒ x = 2

So,

Put the value of x in (1) , We get,

⇒ y = 2x

⇒ y = 2×2 = 4

⇒ y = 4

Hence,

Original number = 10x + y = 10 × 2 + 4 = 20+4 = 24

Reversed number = 10y + x = 10×4+2 = 40 +2 = 42

So,

• Original Number = 24

• Reversed Number = 42

Answer 2

Answer:

The two - digit number is 24.

Step-by-step-explanation:

NOTE: Kindly refer to the attachment first.

We have given that, the number is between 10 and 100.

It means the number must be a two - digit number.

Let the digit at the units place be x.

And the digit at the tens place be y.

$\therefore$The original number = $\sf\:10y+x$

And the number obtained by reversing the digits = $\sf\:10x+y$

Now, from the first condition,

$\sf\:Two\:digit\:number\:=\:4\:\times\:Sum\:of\:the\:digits$

$\therefore\:\sf\:10y+x=4\times\:(x+y)\\\\\therefore\sf\:10y+x=4x+4y\\\\\therefore\sf\:10y+x-4x-4y=0\\\\\therefore\sf\:10y-4y-4x+x=0\\\\\therefore\sf\:6y-3x=0\\\\\therefore\sf\:2y-x=0\:\:\:\:...[\sf\: Dividing\:both\:sides\:by\:3.\:]\:\:-(1)$

Now, from second condition,

$\sf\:Two\:digit\:number\:+\:18\:=\:Number\:obtained\:by\:reversing\:digits$

$\therefore\sf\:10y+x+18=10x+y\\\\\therefore\sf\:10y+x-10x-y=-18\\\\\therefore\sf\:9y-9x=-18\\\\\therefore\sf\:y-x=-2\:\:\:...[\sf\:Dividing\:both\:sides\:by\:9]\:\:-(2)$

Now, subtracting equation ( 2 ) from equation ( 1 ), we get,

$\sf\:2y-\cancel{x}=0\:\:-\:-\:-(1)\\\\\sf\:-\:y-\cancel{x}=-2\:\:-\:-\:-(2)\\\\\therefore\sf\boxed{\sf\:y\:=\:2}$

Now, by substituting $\sf\:y=2$ in equation ( 1 ), we get,

$\sf\:2y-x=0\\\\\therefore\sf\:2\times(2)-x=0\\\\\therefore\sf\:4-x=0\\\\\therefore\sf\:-x=0-4\\\\\therefore\sf\:\cancel{-}x=\cancel{-}4\\\\\therefore\sf\boxed{\sf\:x\:=\:4}$

$\therefore$ The digit at units place $\sf\:(x)\:=\:4$

$\therefore$ The digit at tens place $\sf\:(y)\:=\:2$

$\therefore\sf\:The\:two\:-\:digit\:number\:=\:10y+x\\\\\implies\:(10\:\times\:2)+4\\\\\implies\:20+4\\\\\implies\:24$

$\boxed{\sf\:The\:two\:digit\:number\:=\:24}$

Additional Information:

1. Linear Equations in two variables:

The equation with the highest index

( degree ) 1 is called as linear equation. If the equation has two different variables, it is called as 'linear equation in two variables'.

The general formula of linear equation in two variables is

$\sf\:ax\:+\:by\:+\:c\:=\:0$

Where, a, b, c are real numbers and

a ≠ 0, b ≠ 0.

2. Solution of a Linear Equation:

The value of the given variable in the given linear equation is called the solution of the linear equation.

3. Stpes to solve word problems based on linear equations in two variables:

1. Understand the information given in the problem.

2. Identify the required quantities given and to find.

3. Convert the words into symbols by using variables ( x, y, a, b ).

4. Solve the equations formed step-by-step.

5. Write the value of variables ( solution of equation ) in words ( as asked in the question ).