# Problem of the Day: 1/16/13

The ratio of Lauren’s current age to her age in two years is $\frac{5}{6}$. How old is she now?

Solution to yesterday’s problem:

Let’s use my favorite blanks method to solve this problem. Since the number is four digits long, we can write it like this:

____ ____ ____ ____

Multiples of 11 have a special characteristic: the sum of the odd-numbered digits equals the sum of the even-numbered digits. For instance, 286 is a multiple of 11 because 2+6=8. So in this problem, the first and third digits must add up to the same as the second and fourth digits. Also, the sum of all the digits must be 10. We can say that the (sum of the first and third digits) x 2 = 10. Dividing by 2, the sum of the first and third (and that of the second and fourth) digits must be 5. What are the possible digits that could add up to 5?

0, 5; 1, 4; 2, 3

Now let’s fill in the blanks. For the first blank, the number could be any of the numbers above except zero (it would then be three digits). That makes 5 possible digits:

__5__ ____ ____ ____

For the second blank, the number could be any of the numbers above, or 6 possible digits:

__5__ __6__ ____ ____

The digit in the third blank must make 5 when added to the first digit, so it can ONLY be the other number in the pair from which the first digit came. Same goes for the fourth blank.

__5__ __6__ __1__ __1__

Multiplying these numbers together, we find that there are 30 numbers that satisfy these conditions.