Question

two digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or multiplying the difference of the digits by 16 and then adding 3. Find the number.​

Answer 1

Answer :

Required number = 83

Solution :

Let the tens digit and unit digit of the number be x and y respectively .

Thus ,

Required Number = 10x + y

Now ,

According to the question , the number can be obtained by multiplying the sum of the digits by 8 and then subtracting 5 .

Thus ,

=> Number = 8(x + y) - 5

=> 10x + y = 8x + 8y - 5

=> 10x - 8x + y - 8y = -5

=> 2x - 7y = -5 ----------(1)

Also ,

The required number can be obtained by multiplying the difference of the digits by 16 and then adding 3 .

Thus ,

=> Number = 16(x - y) + 3

=> 10x + y = 16x - 16y + 3

=> 10x - 16x + y + 16y = 3

=> -6x + 17y = 3 ----------(2)

Now ,

Multiplying eq-(1) by 3 , we get ;

=> 3•(2x - 7y) = 3•(-5)

=> 6x - 21y = -15 ----------(3)

Now ,

Adding eq-(2) and (3) , we get ;

=> -6x + 17y + 6x - 21y = 3 + (-15)

=> -4y = 3 - 15

=> -4y = -12

=> y = -12/-4

=> y = 3

Now ,

Putting y = 3 in eq-(1) , we get ;

=> 2x - 7y = -5

=> 2x - 7•3 = -5

=> 2x - 21 = -5

=> 2x = 21 - 5

=> 2x = 16

=> x = 16/2

=> x = 8

Hence ,

Required number = 83

Answer 2

Step-by-step explanation:

$\bf{\underline{\underline\red{Given:-}}}$

• A two digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5.

• It can also be obtained by multiplying the difference of the digits by 16 and then adding 3.

$\bf{\underline{\underline\blue{To\: Find:-}}}$

• The number.

$\bf{\underline{\underline\green{Solution:-}}}$

Let the tens digit of the number be x

The ones digit of the number be y

The number = 10x + y

Sum of digits = x + y

Difference of digits = x - y

According to 1st condition:-

The number is obtained by multiplying the sum of the digits by 8 and then subtracting 5.

$\rm \implies8(sum \: of \: digits) - 5 = The \: number$

$\rm \implies8(x + y) - 5 =10x + y$

$\rm \implies8x + 8y - 5 =10x + y$

$\rm \implies8x + 8y - 10x - y = 5$

$\rm \implies - 2x + 7y = 5......(i)$

According to the 2nd condition:-

The number is obtained by multiplying the difference of the digits by 16 and then adding 3.

$\rm \implies16(difference \: of \:digits) + 3 = The \: number$

$\rm \implies16(x - y) + 3 = 10x + y$

$\rm \implies16x -16 y + 3 = 10x + y$

$\rm \implies16x - 10x -16 y - y = - 3$

$\rm \implies6x -17 y = - 3......(ii)$

Multiplying 3 to equation (i)

$\rm \implies3( - 2x + 7y = 5)$

$\rm \implies- 6x + 21y = 15......(iii)$

Adding equation (ii) and (iii)

$\rm \implies6x - 17y + (- 6x + 21y) = 15 + ( - 3)$

$\rm \implies6x - 17y - 6x + 21y = 15 - 3$

$\rm \implies4y = 12$

$\rm \implies y = 3$

Substituting y = 3 in equation (i)

$\rm \implies - 2x + 7y= 5$

$\rm \implies - 2x + 7(3)= 5$

$\rm \implies - 2x +21= 5$

$\rm \implies - 2x = 5 - 21$

$\rm \implies - 2x = - 16$

$\rm \implies x = 8$

The number

= 10x + y

= 10(8) + 3

= 80 + 3

= 83

$\large\underline{ \boxed{ \purple{\rm \therefore The \: number= 83}}}$