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The sum of the digits of a two-digit number is 7. When the digit are reversed,the numbers is increase by 27. How do you find the numbers?

2 Answers

May 16, 2018

The digits are $2$ and $5$

Explanation:

If we let the digits be $color(blue)a$ and $color(blue)b$
then the possible numbers composed of those two digits are
$color(blue)(10a+b)$ and $color(blue)(10b+a)$

We are told
[1] $color(white)("XXX")color(blue)a+color(blue)b=7$
and (assuming $10a+b$ is the larger composite number)
[2] $color(white)("XXX")(color(blue)(10a+b))-(color(blue)(10b+a))=27$

[2] simplifies into
[3] $color(white)("XXX")9a-9b=27$
or
[4] $color(white)("XXX")a-b=3$

Adding [1] and [4], we get
[5] $color(white)("XXX")2a=10$
which implies
[6] $color(white)("XXX")color(blue)a=5$

Substituting $5$ for $color(blue)a$ back in [1]
we see that
[7] $color(white)("XXX")color(blue)b=2$

May 16, 2018

The two digit number is $25$

Explanation:

Let tens digit and units digit of the number be

$x and y ; x+y=7; (1)$

The two digit number is $10 x+y$ , when reversed,

the two digit number becomes $10 y+x$ , by given condition,

$10 y+x =10 x+y+27 or 9 y -9 x =27$ or

$9 (y - x) =27 or (y-x)=3 or y = x+3$ Putting

$y=x+3$ in equation (1) we get, $x+x +3=7;$ or

$2 x= 4 :. x =2 ; y= 2+3=5$

Hence the two digit number is $25$ [Ans]

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