Question

The sum of the digits of a certain two-digit number is 7. Reversing its digits increasesethe number by 9

Answer 1

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$\underline{\underline{\huge{\textbf{Solution:}}}}$

Given,

Statement- 1:
Sum of Digits of Two digit = 7

Statement-2:
When Digits are Reversed, Number increases by 9.

$\underline{\large{\textsf{Step-1:}}}$
Assume the digits of Two-digit number as variables.

Let Unit's digit be x
and Ten's digit be y

$\underline{\large{\textsf{Step-2:}}}$
Add both to get Sum of the digits of Two-digit Number.

Sum of Digits = x+y.

$\underline{\large{\textsf{Step-3:}}}$
Equate sum of Digits to 7 (as given by statement-1)

$\implies x+y = 7$ ---------(1)

$\underline{\large{\textsf{Step-4:}}}$
Using the Above equation, express one variable in terms of other.

$\implies y = (7-x)$

$\underline{\large{\textsf{Step-5:}}}$
Express the 2-digit No. in terms of single variable.

A Number is equal to sum of product of weight of the digit at each place and Face value at that place.

2-digit No. = 10(Ten's digit)+1(Unit's digit)

2-digit No. = 10(7-x)+x

$\underline{\large{\textsf{Step-6:}}}$
Express the 4-digit Number when the digits are reversed.

When digits of Two-digit Number are reversed,
Digit at Unit's place and Ten's place are Interchanged.

For Reversed Two-digit Number,
Unit's digit of Reversed = Ten's digit of Two-digit Number

Ten's digit of Reversed = Ten's digit of Two-digit Number

Reversed 2-digit No. = 10x + (7-x)

$\underline{\large{\textsf{Step-7:}}}$
Express the statement -2 in terms of equation.

Reversed 2-digit No. = (2-digit No.) + 9.

By Substituting ,

10x + (7-x) = [10(7-x) +x] + 9

$\underline{\large{\textsf{Step-8:}}}$
Solve the expressed equation to obtain the value of variable.

10x + (7-x) = [10(7-x) +x] + 9

$\implies$ 10x+ 7-x = 70-10x+x+9

$\implies$ 9x+ 7 = 79-9x

$\implies$ 18x = 79-7

$\implies$ 18x = 72

$\implies$ x = 4

$\therefore$
Unit's digit of two digit No. = 4.

$\underline{\large{\textsf{Step-9:}}}$
Find the Ten's digit of the Two-digit Number

From (1),
x + y = 7

$\implies$ y = 7-x

Substituting x = 4

$\implies$ y = 7-4

$\implies$ y = 3

$\therefore$
Ten's digit of two digit No. = 3

$\underline{\large{\textsf{Step-10:}}}$
Express the Two-digit Number with the known Unit's digit and Ten's digit.

Unit's digit of two digit No. = 4.
Ten's digit of two digit No. = 3

$\therefore$
Two - digit No. = 34.

$\underline{\large{\textbf{Verification:}}}$

Two - digit No. = 34

Sum of digits = 7 ✓

Reversed Two - digit No. = 43
43 = 34+9
43 = 43 ✓

Both the given statements satisfied accordingly.

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