Question

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

Answer 1

Answer:

$\textsf{The Number is 83}$

Step-by-step explanation:

$\textsf{\underline{\underline{Solution :}}}$

$\textsf{Let the digits of the number be a and b.}$

$\textsf{The two-digit number is = 10a + b}$

✯ $\textbf{\underline{It is given that,}}$

$\tt{\longrightarrow}\:10a + b = 8(a + b) - 5 \\ \\\tt{\longrightarrow}\:10a +b = 8a + 8b- 5 \\ \\\tt{\longrightarrow} \: 7b - 2a = 5 .....{\rm{\gray{(Equation \: 1)}}}$

$\rule{300}{1.5}$

✯ $\textbf{\underline{Also given that,}}$

$\tt{\longrightarrow} \: 10a + b = 16(a - b) + 3 \\ \\ \tt{\longrightarrow} \: 10a + b = 16a - 16b + 3 \\ \\ \tt{\longrightarrow} \: 17b - 6a = 3......\rm{\gray{(Equation \: 2)}}$

$\rule{300}{1.5}$

✯$\textbf{\underline{Multiply Equation 1 by 3 -}}$

$\tt{\longrightarrow} \: 21b - 6a = 15 --- \: \rm{\gray{(Equation \: 3)}}$

$\rule{300}{1.5}$

✯ $\textbf{\underline{Subtract Equ. 2 and Equ. 3 -}}$

$\tt{\longrightarrow} \:4b = 12 \\ \\ \tt{\longrightarrow} \:b = \dfrac{12}{4} \\ \\ \tt{\longrightarrow} \:b = 3$

$\rule{300}{1.5}$

✯ $\textbf{\underline{Substitute value of b in equ. 1 -}}$

$\tt{\longrightarrow} \:7b - 2a = 5 \\ \\ \tt{\longrightarrow} \:(7 \times 3) - 2a = 5 \\ \\ \tt{\longrightarrow} \:21 - 2a = 5 \\ \\ \tt{\longrightarrow} \: 2a = 21 - 5 \\ \\ \tt{\longrightarrow} \:2a = 16 \\ \\ \tt{\longrightarrow} \:a = \dfrac{16}{2} \\ \\ \tt{\longrightarrow} \:a = 8$

$\rule{300}{1.5}$

✯ $\textbf{\underline{The two digit number -}}$

$\tt{\longrightarrow} \:10a + b \\ \\ \tt{\longrightarrow} \:10(8) + 3 \\ \\ \tt{\longrightarrow} \:80 + 3 \\ \\ \tt{\longrightarrow} \:83$

$\therefore$ $\textsf{The Number is 83}$

Answer 2

||✪✪ QUESTION ✪✪||

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

|| ✰✰ ANSWER ✰✰ ||

Let us assume that the Required Two digit Number is (10x+y) ..

❦❦ Case ⓵ :--

it has been said that , when we multiplying the sum of the digits by 8 then subtract 5 we get our two - digit Number.

So,

➺ 8(x+y) - 5 = 10x+y

➺ 8x + 8y - 10x - y = 5

➺ 7y - 2x = 5 ------------------------ Equation (1) .

_____________________

✭✭ Case ❷ :-

Now, also said that, when we multiplying the difference of the digits by 16 and then add 3 , we gets same Two digit Number .

So,

➳ 16(x-y) + 3 = 10x + y

➳ 16x - 16y + 3 = 10x + y

➳ 16x - 10x - 16y - y = (-3)

➳ 6x - 17y = (-3)

➳ 17y - 6x = 3 ( Taking (-1) common)

➳ 17y - 6x = 3 -------------------------- Equation (2)

________________________

Now, Multiply Equation (1) by 3 and than , subtracting it from Equation (2) , we get,

☞ (17y - 6x) - 3(7y - 2x) = 3 - 3*5

☞ 17y - 6x - 21y +6x = 3 - 15

☞ 17y - 21y - 6x + 6x = (-12)

☞ (-4)y = (-12)

Dividing both sides by (-4) ,

☞ y = 3 .

Putting This value in Equation (1) Now, we get,

☛ 7y - 2x = 5

☛ 7*3 - 2x = 5

☛ 21 - 5 = 2x

☛ 16 = 2x

Dividing both sides by 2 ,

☛ x = 8

______________

So, our Required Two digit Number is :-

➾ 10x + y

➾ 10*8 + 3

➾ 80 + 3

➾ 83 (Ans).