Can you figure it out?

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crunkjosh

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Aug 29, 2005
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Lets see if you can crack this one whiz kids.Please explain how you figured it out.
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?
 
The two digits in Jack's age can be represented as such:
AB

The two digits in Bill's age can be represented as:
BA

Note that those are not multiplications, but a representation of the two digits of their ages.

Now, Jack's age is equal to 10A+B. This is because of the place values of each number. For example, 72 = 7*10 + 2. By the same logic Bill's age is 10B+A.

In five years Jack will be twice as old as Bill will be then:
Jack + 5 = 2(Bill + 5)

Substitute for Jack and Bill:
10A+B+5 = 2(10B+A+5)
10A+B+5 = 20B+2A+10
8A = 19B+5

Now it's just a matter of checking which values of A and B work here, because they both have to be integers between 0-9, right? To make it easier, divide each side by 8 to solve for A:

A = (19B+5)/8

Here is the key: Since A has to be an integer, the numerator of that fraction must be a multiple of 8 so that the fraction reduces properly. What values of B make this work? I just tried everything from 0-9. The only ones that produced a multiple of 8 on top are:
1: 24/8 = 3
9: 176/8 = 22

Of course, we could have stopped earlier because obviously A can't be 22, it has to be less than 10 because it is one digit of a number.

Now we have discovered that B is 1 and A is 3, so jack's age is 31 and Bill's age is 13. Try it: in five years, Jack will be 36 and Bill will be 18.

If anyone can do this faster or more clearly, please post a response because I know this was kind of messy.

Ted
 
The ages are 31 and 13.

Remember that when we take a number and reverse the digits, the difference of the two numbers = 9(Positive difference of the digits)

Let J's age = 10X+Y
B's age = 10Y +X

5 years hence J's age = 10X +Y +5
B's age = 10Y + X+ 5

J will be twice as old as Bill

10X +Y +5 = 2(10Y + X+ 5)
 
Sure, you can roll dice and check each result. Of course, you have to get 10-sided dice.
 
Well, j = Jack and b = Bill

In 5 years:
j + 5 = 2(b + 5)
j - 2b = 5 [1]

j - b = 9k where k is any positive integer (diff is multiple of 9)
b = j - 9k [2]

[2] in [1]:
j - 2(j - 9k) = 5
18k - j = 5

k=1: 18 - 13 = 5 ; nope (Bill = 31)
k=2: 36 - 31 = 5 ; yes! (Bill = 13)
k=3: 54 - 49 = 5 ; nope (Bill = 94)
k=4: 72 - 67 = 5 ; nope (Bill = 76)
k=5: 90 - 85 = 5 ; nope (Bill = 58; 85+5 <> 2(58+5))
k=6: 108 - 103 : whoa!

Jack = 31

Just another way...
 
Jack's age is 10x + y
Bill's age is 10y + x
In 5 years, 10x + 5 + 5 = 2(10y + x + 5) yielding 8x - 19y = 5
Dividing through by 8 yields x - 2y - 3y/8 = 5/8
(3y + 5)/8 must be an integer as does (9y + 15)/8
Dividing by 8 again yields y + y/8 = 1 + 7/8
(y + 7)/8 must be an integer k making y = 8k - 7
Substituting back into 8x - 19 y = 5 yields x = 19k - 16
While there are many solutions to this expression only k = 1 results in a single digit for x and y, namely x = 3 and y = 1.
Therefore, Jack is 31 and Bill is 13.
(31 + 5) = 2(13 + 5)

Just another viewpoint.
 
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