Question

sum of the digits of a two digit number is 9 when we interchange the digits it is found that the resulting number is greater than the original number by 62.Find the numbers .
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Answer 1

Correct question :

Sum of the digits of a two digit number is 9 when we interchange the digits it is found that the resulting number is greater than the original number by 63 .Find the numbers .

Given :

• Sum of the digits of a two digit number = 9
• When interchanging the digits the resulting number is greater than original number by 63.

To find :

• Original number and interchanged number

Solution :

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

∴ Original number = x + 10y

Now according to first condition,

⇒ x + y = 9

⇒ x = 9 - y .............. (i)

Now if we interchange digits, then interchanged number :

∴ Interchanged number = y + 10x

Now according to 2nd condition,

⇒ y + 10x = x + 10y + 63

⇒ y + 10x - x - 10y = 63

⇒ 9x - 9y = 63

⇒ 9(x - y) = 63

⇒ 9(9 - y - y) = 63

⇒ 9(9 - 2y) = 63

⇒ 81 - 18y = 63

⇒ -18y = 63 - 81

⇒ y = -18/-18

⇒ y = 1

Now putting value in (i) :

⇒ x = 9 - 1

⇒ x = 8

∴ Original number = x + 10y = 8 + 10(1) = 8 + 10 = 18

∴ Interchanged number = y + 10x = 1 + 10(8) = 1 + 80 = 81

Therefore,

Original number is 18 and interchanged number is 81.

Answer 2

Answer :-

$\: \: ➫ \: \: \boxed{\rm{Understanding \: the \: concept \: }}$

Here the concept of formation of Number system is used along with Linear Equations in two Variables. According to this, the number system says, when we expand a number, we write digits according to multiple of 10 by taking its place.

Also, by using Linear Equations in Two Variables here, we can find the value of both the terms and both the numbers by making the value of one depend on other. Standard form of Linear Equations in Two Variables is given as :-

ax + by + c = 0

px + qy + d = 0

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★ Question :-

Sum of the digits of a two digit number is 9. When we interchange the digits, it is found that the resulting number is greater than the original number by 63. Find the numbers .

★ Solution :-

Given,

» Sum of the digits = 9

» New number after interchanging the digits = Original number + 63

• Let the unit digit be 'x' and the tens place digit be 'y'. Then,

$\: \: ➠ \: \: \boxed{\rm{Original \: number \: = \bold{10y \: + \: x}}}$

$\: \: ➠ \: \: \boxed{\rm{New \: number \: after \: changing \: digits \: = \: \bold{10x \: + \: y}}}$

Now using the given things, we form equations as :-

~ Case I :-

➮ x + y = 9

➮ x = 9 - y ... (i)

~ Case II :-

➮ 10x + y = 10y + x + 62

➮ 10x + y - 10y - x = 62

➮ 9x - 9y = 62 ... (ii)

From equations (i) and equations (ii), we get,

➮ 9(9 - y) - 9y = 63

➮ 81 - 9y - 9y = 63

➮ -18y = 62 - 81

➮ -18y = -18

➮ 18y = 18

$\: \longrightarrow \: \: \huge{\bold{y \: = \: \dfrac{18}{18}}}$

➮ y = 1

• So, tens place digit = y = 1

Now using equation (i) and value of y, we get,

➮ x = 9 - y

➮ x = 9 - 1

➮ x = 8

• So, unit place digit = x = 8

Then, using bot values of x and y, we get,

»Original number = 10y + x = 10(1) + 8 = 18

» New Number = 10x + y = 10(8) + 1 = 81

$\: \: \boxed{\sf{Hence \: the \: original \: number \: is \: \underline{18} \: and \: the \: new \: number \: after \: changing \: the \: digits \: is \: \underline{81}}}$

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$\boxed{\tt{Confused ?, \: Don't \: worry \: let's \: verify \: it}}$

For verification, we must simply apply the values we got into the equations we formed.

~Case I :-

:⟹ x + y = 9

:⟹ 8 + 1 = 9

:⟹ 9 = 9

Clearly, LHS = RHS

Here the condition satisfies, so our answer is correct.

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$\: \: \boxed{\tt{A \: piece \: of \: supplementary \: counsel}}$

• Linear Equations in One Variable is the form of equations where we use to find the value of one variable using constant terms.

• Linear Equations are the form of linear polynomial with degree is equal to 1.