Question 1161973
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Let t and u be the tens and units digits of her age.  Then<br>
10t+u = her age
10u+t = her age with the digits reversed
(10u+t+17)/2 = her age, with the digits reversed, and 17 added, and the whole thing divided by 2<br>
The problem says that expression is equal to her age:<br>
{{{(10u+t+17)/2 = 10t+u}}}
{{{10u+t+17 = 20t+2u}}}<br>
This is a single equation with two variables -- an example of a Diophantine equation.  There are an infinite number of solutions; however, there is only one solution that satisfies the conditions of the problem -- that t and u are single digit positive integers.<br>
To solve a Diophantine equation, solve the equation for one variable in terms of the other:<br>
{{{10u+t+17 = 20t+2u}}}
{{{8u = 19t-17}}}
{{{u = (19t-17)/8}}}<br>
To find single digit positive integer solution(s) for that equation, perform the division on the right to write the expression as a whole number and a remainder:<br>
{{{u = (19t-17)/8 = ((16t-16)+(3t-1))/8 = (2t-2)+(3t-1)/8}}}<br>
In that equation, u and (2t-2) are integers, so (3t-1)/8) has to be an integer.<br>
The only single digit positive integer t for which that expression is an integer is t=3.  So<br>
{{{t = 3}}}
{{{u = (2t-2)+(3t-1)/8 = 4+1 = 5}}}<br>
ANSWER: Emma's age is 35<br>
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NOTE: You can solve part or all of this problem by trial and error, knowing that there are very few options for the tens and units digit of her age.  Other tutors have provided responses to your post in which they do that.<br>
There is of course nothing wrong with solving the problem that way; if getting an answer as quickly as possible is important, then that is probably the best way to solve the problem.<br>
However, knowing how to solve linear Diophantine equations is a useful skill to have; so I thought it useful to show this formal method for solving the problem.<br>