Question

a number consists of two digits when the number is divided by the sum of digit the quotient if 7if 27 is subtracted from the numbersthe did=gits interchange their places find the number

Answer 1

CORRECT QUESTION:

❶ A number consists of two digits when the number is divided by the sum of digit the quotient is 7. If 27 is subtracted from the numbers, the digits interchange their places find the number.

GIVEN:

• A number consists of two digits when the number is divided by the sum of digit the quotient is 7.
• If 27 is subtracted from the numbers, the digits interchange their places.

TO FIND:

• Find the number ?

SOLUTION:

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

Then,

✰ Number = 10x + y ✰

CASE:- ❶

➣ A number consists of two digits when the number is divided by the sum of digit the quotient is 7

According to given conditions:-

$\rm{\dashrightarrow \dfrac{10x + y}{x + y} = 7 }$

Use cross product

$\rm{\dashrightarrow 10x + y = 7(x + y)}$

$\rm{\dashrightarrow 10x + y = 7x + 7y }$

$\rm{\dashrightarrow 10x - 7x = 7y - y }$

$\rm{\dashrightarrow 3x = 6y}$

Divide by '3' on both sides

$\large\bf{\dashrightarrow x = 2y....1)}$

CASE:- ❷

➣ If 27 is subtracted from the numbers, the digits interchange their places.

❐ Number obtained by reversing the digits = 10y + x

❐ Original number - 27 = Number obtained by reversing the digits

According to given conditions:-

$\rm{\dashrightarrow 10x + y - 27 = 10y + x }$

$\rm{\dashrightarrow 10x + y - 10y - x = 27}$

$\rm{\dashrightarrow 9x - 9y = 27 }$

Divide by '9' on both sides

$\large\bf{\dashrightarrow x - y = 3....2)}$

Put the value of 'x' from equation 1) in equation 2)

$\rm{\dashrightarrow 2y - y = 3 }$

$\large\bf{\star \: y = 3 \: \star}$

Now, put the value of 'y' in equation 1)

$\rm{\dashrightarrow x = 2 \times 3 }$

$\large\bf{\star \: x = 6 \: \star}$

➠ Number = 10x + y

➠ Number = 10(6) + 3

➠ Number = 60 + 3

✫ Number = 63 ✫

❝ Hence, the number formed is 63 ❞

______________________

Answer 2

$\huge{\underline{\pink{\tt{Given,}}}}$

• A Number Consists of Two Digits when the Number is Divided by the sum of Digit the Quotient is 7.
• If 27 is Subtracted from the Number then the Digits Interchange their Place.

$\huge{\underline{\pink{\tt{To \ Find,}}}}$

• The Two Digit Numbers

$\huge{\underline{\pink{\tt{Solution :}}}}$

$\longrightarrow$ Suppose the Digit at the One's Place be x

And, Suppose the Digit at the Ten's Place be y

Therefore,

• The Two Digit Number - (10y + x)
• The Interchange Number - (10x + y)

$\mapsto \underline{\underline{\purple{\mathfrak{According\:to\:the\:First\:Condition:}}}}$

• A Number Consists of Two Digits when the Number is Divided by the sum of Digit the Quotient is 7.

$\longrightarrow \sf{\dfrac{10y + x}{x + y} = 7}$

$\longrightarrow \sf{10y + x = 7x + 7y}$

$\longrightarrow \sf{10y - 7y = 7x - x}$

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

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

$\longrightarrow \boxed{\sf{y = 2x}}$...1)Equation

$\mapsto \underline{\underline{\purple{\mathfrak{According\:to\:the\:Second\:Condition:}}}}$

• If 27 is Subtracted from the Number then the Digits Interchange their Place.

$\longrightarrow \sf{10y + x - 27 = 10x + y}$

$\longrightarrow \sf{10y - y - 9x = 27}$

$\longrightarrow \sf{9y - 9x = 27}$

║Now Put the Value of y From the First Equation ║

$\longrightarrow \sf{9(2x) - 9x = 27}$

$\longrightarrow \sf{18x - 9x = 27}$

$\longrightarrow \sf{9x = 27 }$

$\longrightarrow \sf{x = \dfrac{27}{9}}$

$\longrightarrow \boxed{\sf{x = 3}}$

Now Put the Value of x in First Equation :

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

$\longrightarrow \sf{y = 2(3)}$

$\longrightarrow \boxed{\sf{y = 6}}$

Therefore,

$\mapsto \boxed{\pink{\mathfrak{The\:Original\:Number = 10y + x = 10(6)+ 3 = 63}}}$

$\mapsto \boxed{\pink{\mathfrak{The\:Interchange\:Number = 10x + y = 10(3) + 6 = 36}}}$

$\rule{200}2$