Question

The sum of the digits of a two- digit number is 13. The number obtained by interchanging it's digits exceeds the given number by 27. Find the original number.​

Answer 1

$\huge\mathfrak\blue{Answer:}$

Given:

• We have been given a number such that sum of its digits is 13
• The number obtained by interchanging its digit exceed the number by 27

To Find:

• We have to find the original number

Concept Used:

This Question can be solved by using the concept of linear equation in two variables which can be further solved by using any of the following technique

• Substitution
• Elimination
• Cross Multiplication

Solution:

Let the unit digit be = y

And Tens digit = x

$\boxed{\sf{Original \: Number = 10x + y}}$

When the digits are interchanged

$\boxed{\sf{Reversed \: Number = 10y + x}}$

$\sf{ }$

$\odot \:$According to first Condition :

Sum of digits of number is 13

$\implies \sf{x + y = 13}$

$\implies \boxed{\sf{x = 13 - y}}$ ---------------------------(1)

$\sf{ }$

$\odot \:$According to second Condition

The number obtained by interchanging its digit exceed the number by 27

$\implies \boxed{\sf{Reversed - Original = 27}}$

$\implies \sf{(10y + x) - (10x + y) = 27}$

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

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

$\implies \sf{y - x = 3 }$

Putting value of x from equation (1)

$\implies \sf{y - (13-y) = 3 }$

$\implies \sf{y - 13 + y = 3 }$

$\implies \sf{2y = 3 + 13}$

$\implies \sf{y = \dfrac{16}{2}}$

$\implies \boxed{\sf{y = 8 }}$

$\sf{ }$

Putting value of y in Equation (1)

$\implies \sf{x = 13 - 8}$

$\implies \boxed{\sf{x = 5}}$

_______________________________

$\odot \:$Original Number :

$\implies \sf{10 \times 5 + 8}$

$\implies \sf{50 + 8}$

$\implies \sf{58}$

Original Number = 58

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$\huge\underline{\sf{\red{A}\orange{n}\green{s}\pink{w}\blue{e}\purple{r}}}$

$\large\boxed{\sf{Original \: Number = 58}}$

________________________________

$\huge\mathtt\green{Verification:}$

Sum of digits of number is 13

➡️Sum = 5 + 8

➡️Sum = 13

The number obtained by interchanging its digit exceed the number by 27

• Original Number = 58
• Reversed Number = 85

➡️Difference = 85 - 58

➡️Difference = 27

Hence Verified ...... ✍️