Question

A two_digit number is obtain by either multiplying the sum of the digits by 8 and the subtract 5 or multiplying the difference of the digits by 16 and then adding 3 .find the number.​

Answer 1

$\rule{400}4$

☆ ANSWER:-

▪︎ Given:-

• A two digit number.

• Which can be obtain either by Multiplying the sum of digits with 8 and then by subtracting by 5.

• Or by or multiplying the difference of the digits by 16 and then adding 3.

▪︎ To Find:-

• The number.

$\rule{400}2$

▪︎ So,

• Let the digits of the number be x and y.

• So the number would be (10×x)+(y×1) = 10x+y.

▪︎ Now,

• The question is why the number is 10x+y and not x+y.

• This is because here x and y are digits of the number and here x is at tenth place and y is at one's place.

• That is why the number is 10x + y and not x+y as x+y is the sum of digits but not the number.

$\rule{400}2$

▪︎ Therefore ATQ:-

▪︎ Case 1 :- When the number is obtained by multiplying the sum of digits by 8 and subtract 5 from the result .

$\\ \tt\mapsto((x + y) \times 8) - 5 = 10x + y \\ \\ \tt\mapsto(8x + 8y) - 5 = 10x + y \\ \\ \tt\mapsto8x + 8y - 5 = 10x + y \\ \\ \tt\mapsto0 = 10x - 8x + y - 8y + 5 \\ \\ \sf(by \: transporting \: to \: rhs) \\ \\ \tt\mapsto0 = 2x - 7y + 5 \\ \\ \tt\mapsto2x - 7y = - 5 ...eq(1)$

$\rule{400}1$

▪︎ Case 2:- Multiplying the difference of digits by 16 and adding 3 to it.

$\\ \tt\mapsto((x - y) \times 16) + 3 = 10x + y \\ \\ \tt\mapsto16x - 16y + 3 = 10x + y \\ \\ \tt\mapsto16x - 10x - 16y - y = - 3 \\ \\ \tt\mapsto6x - 17y = - 3...eq(2)$

$\rule{400}2$

▪︎ Now,

• By Multiplying eq(1) by 3 we get:-

$\\ \tt\mapsto3 \times (2x - 7y = - 5) \\ \\ \tt\mapsto6x - 21y = - 15...eq(3)$

$\rule{400}2$

▪︎ By subtracting eq(2) from eq(3) we get:-

$\\ \sf \mapsto(6x - 21y) - (6x - 17y) = (-1 5) - ( - 3) \\ \\ \sf \mapsto\cancel{6x} - 21y \cancel{- 6x} + 17y = - 12 \\ \\ \sf(by \: ope ning \: brackets) \\ \\ \sf \mapsto - 21y + 17y = - 12 \\ \\ \sf \mapsto\cancel{ - }4y = \cancel{-} 12 \\ \\ \sf \mapsto4y = 12 \\ \\ \sf \mapsto \: y = \cancel \frac{12}{4} \\ \\ \sf \mapsto \: y = 3.$

$\rule{400}2$

▪︎ So by putting the value of y in eq(1) we get,

$\\ \sf \mapsto2x - 7y = - 5 \\ \\ \sf \mapsto2x - 7(3) = - 5 \\ \\ \sf \mapsto2x - 21 = - 5 \\ \\ \sf \mapsto2x = - 5 + 21 \\ \\ \sf \mapsto2x = 16 \\ \\ \sf \mapsto \: x = \cancel \frac{16}{2} \\ \\ \sf \mapsto \: x = 8.$

$\rule{400}2$

▪︎ Therefore, the number is,

$\\ \tt\leadsto10x + y \\ \\ \tt = (10 \times 8) + 3 \\ \\ \tt = 80 + 3 \\ \\ = 83.$

▪︎ So the number is 83.

$\rule{400}4$

Answer 2

GIVEN:

• A two digit number is obtain by either multiplying the sum of the digits by 8 and the subtract 5.
• A two digit number is obtain by multiplying the difference of the digits by 16 and then adding 3.

TO FIND:

• What is the number ?

SOLUTION:

Let the digit at units place be 'y' and the digit at ten's place be 'x'.

Then,

✯ Number = 10x + y ✯

A two digit number is obtain by either multiplying the sum of the digits by 8 and the subtract 5.

According to question:-

We have

$\sf{\mapsto 8(x + y) - 5 = 10x + y}$

$\sf{\mapsto 8x + 8y - 5 = 10x + y}$

$\sf{\mapsto - 5 = 10x + y -8x -8y}$

$\sf{\mapsto -5 = 2x - 7y }$

$\sf\red{\mapsto 2x - 7y = -5 ....1)}$

A two digit number is obtain by multiplying the difference of the digits by 16 and then adding 3.

According to question:-

We have

$\sf{\mapsto 16(x - y) + 3 = 10x + y}$

$\sf{\mapsto 16x - 16y + 3 = 10x + y}$

$\sf{\mapsto 16x - 16y - 10x - y = -3}$

$\sf\red{\mapsto 6x - 17y = -3 ....2)}$

Multiplying by '3' in equation 1)

$\sf{\mapsto 3 \times 2x - 3 \times 7y = - 5 \times 3 }$

$\sf{\mapsto 6x - 21y = - 15}$

By Eliminating equation 1) & 2), we get

$\large\sf{ 6x - 17y = -3 }$

$\large\underline\mathsf{ 6x - 21y = -15}$

$- \: \: \: \:\: + \: \:\:\:\:\:\:\:\:\:\:\:\: +$

______________

$\large\sf{4y = 12}$

$\bf\red{ \mapsto \:\star \: y = 3 \: \star }$

Put the value of 'y' in equation 1)

$\sf{\mapsto 2x - 7 \times 3 = -5 }$

$\sf{\mapsto 2x - 21 = -5 }$

$\sf{\mapsto 2x = -5 + 21 }$

$\sf{\mapsto 2x = 16 }$

$\bf\red{ \mapsto \:\star \: x= 8 \: \star }$

Number = 10x + y

➨ $\sf{ Number = 10 \times 8 + 3}$

➨ $\sf{ Number = 80 + 3}$

➨ $\bf\red{\star \: Number = 83 \: \star}$

❛ Hence, the required number is 83. ❜

_________________