Question 1195485: Lola is 9 years older than Maggie, and in 1 year Maggie’s age will have the same two digits as Lola’s age, but in reverse order. How old is Lola now?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52810)   (Show Source):
You can put this solution on YOUR website! .
Lola is 9 years older than Maggie, and in 1 year Maggie’s age will have
the same two digits as Lola’s age, but in reverse order. How old is Lola now?
~~~~~~~~~~~~~~~~~
I will explain the solution to you in easy informal manner.
I will consider everything in next your, when Lola is still 9 years older than Maggie,
their ages are reversed two-digit numbers, and their difference is 9 years.
It is useful to know, that
(1) the difference of any two-digit number x and its reversed y
is ALWAYS multiple to 9: |x - y| = 9k,
and
(2) if "a" and "b" are the digits of such numbers, then |a-b| = k.
In our case, it means that the difference of the digits is 1: |a - b| = 1
(since |x - y| = 9 = 9*1).
+---------------------------------------------------------+
| So, our digits are two consecutive integer numbers. |
+---------------------------------------------------------+
And it is ALL the information, which we can extract from the given input.
Having only this information, we may conclude that their ages next your can be
(Lola,Maggie) = (21,12), (32,23), (43,34), . . . , (98,89).
So, 8 different solutions are possible, and the problem DOES NOT provide any additional info
to select a unique solution from these 8 possible solutions.
You can check it on your own, that all conditions,
imposed by the problem, are held.
Thus, it is  that in the given formulation the problem is  :
it gives 8 possible solutions, but it is IMPOSSIBLE
to select some a UNIOQUE soliton from these 8 solutions.
At this point, my explanation is complete.
======================
The solution can be done formally, and it would be appropriate, if you solve such problem
for the first time in your life.
But my interior voice says me that, under right educational curriculum, the student
obtains such problem, when he (or she) is just familiar to some degree with problems
that include reversed numbers.
Then such " lightened " explanation works better . . .
////////////////
For word problems on two-digit reversed integer numbers, see the lesson
- Word problems on reversing digits of numbers
in this site.
From this lesson, learn that properties of reversed numbers, which I used in my solution.
Happy learning ( ! )
\\\\\\\\\\\\\\\\
The fact that the given problem is INCOMPLETE and, THEREFORE, is DEFECTIVE,
does not surprise me : at this forum, I see defective incoming problems EVERY DAY,
and even SEVERAL TIMES per day.
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
The difference in their ages next year, when their two ages have the same digits but in reverse order, will still be 9 years.
When the digits of a 2-digit number are reversed, the difference between the original number and the new number is 9 times the difference between the two digits. This is easily proved algebraically; but you can see it with a few examples....
74-47 = 27 = 9(3) = 9(7-4)
82-28 = 54 = 9(6) = 9(8-2)
43-34 = 9 = 9(1) = 9(4-3)
Since in this problem the difference in their ages is 9, their ages 1 year from now will both have two digits whose difference is 1.
But there is no unique solution to that. Their ages next year could be 21 and 12, or 32 and 23, or 43 and 34, or..., or 98 and 89.
So the problem is defective....
ANSWER: no unique solution
We quickly see that there is no unique solution if we try to solve the problem with formal algebra.
1 year from now, Lola's age and Maggie's age will have the same two digits, so let Lola's age be "AB" and Maggie's age be "BA". Algebraically, then, their ages will be
Lola: 10A+B
Maggie: 10B+A
The difference in their ages then will be 9 years:
(10A+B)-(10B+A) = 9
10A+B-10B-A = 9
9A-9B = 9
A-B = 1
That's as far as we can go with the given information, so the problem has no unique solution.
|
|
|
