Friday, June 13, 2008

A Math Riddle that gets better with 'Age'!

[Don't forget the Mystery Math Anagram for this month. Only two correct replies have been received thus far. I will announce the winners in a few days.]

Have you been wondering where the math challenges have gone on this blog? Here's one that I came across while reading David Baldacci's recent best seller, Simple Genius, just your usual tale of the dark world of mathematicians, codes and spies. Gee, math has become such an integral part of novels, TV shows and movies over the past few years, our students are going to think the life of a mathematician is really cool and exciting (which, as we all know, it is!).

Anyway, here is a paraphrasing of the problem (as long as I'm not copying the problem verbatim, the publisher granted me permission to discuss this):

Alex is as many months old as his grandpa is in years and about as many days old as his dad is in weeks. If the sum of their 3 ages is 140, how old is each?

Hint: This is a wonderful problem demonstrating the power of ratios. If you can solve it less than 20 seconds, then you're either an honorary member of Mensa or you could be the subject of Baldacci's next book!


(1) Like all riddles, the wording is somewhat convoluted and the mathematical assumptions are not explicitly stated. But that's part of the intrigue here. I will say that one needs to assume the ages are integers, but that's about it.

(2) In the novel, the problem is posed to a young mathematical prodigy named Viggie. While another mathematician in the room takes some time to solve it algebraically, Viggie comes up with the solution mentally in a few seconds. Can you!

(3) You may want to give this to middle school students, although the wording might frustrate them. You could demonstrate the idea with a concrete example or make it into a simpler problem:
Let's say that Alex is 96 months old, then his grandpa would be 96 years old. Now ask them to determine how old Alex's dad would be. This may be challenging enough...

(4) I'm naturally wondering what the source of this problem is. If anyone out there recognizes it, let us know its source!


Florian said...

1) 12a=g // months -> years conversion
2) 51*7a = 51*d // days - > weeks -> year conversation
3) a+g+d = 140

Substitute 1 and 2 into 3:

a+12a+7a = 140 <=> 20a = 140 <=> a=7, g=84, d = 49

Solution: Alex is 7, Dad is 49 and Grandpa 84 years old.

The time critical factor seems to
be to get the conversion right.
I'm not a member of mensa, but
made it in about 20 secs (yay)
...but then I figured you meant
if middle schoolers can solve it
in 20 secs they were smart.

Totally_clueless said...

I agree with Florian (except for the 51(?) part). The trick is interpreting the words to get the correct ratios.

If the question had been worded: Grandpa is 12 times as old as Alex, and Pa is 7 times as old as Alex, and the sum of their ages is 140, then anyone would do it much faster.


Florian said...

I also don't agree with the 51. Yet I posted it to illustrate my thinking
process when I solved the problem.
Getting everything in the same unit
makes it easier to put things into
perspective (and find ways to solve
a problem). Here it wasn't necessary
of course.

Dave Marain said...

Nice comments, Florian and TC...

I personally love the convoluted wording of this riddle. Concealing a 12:7:1 ratio in the phrasing is clever and I am curious who thought of it.

Florian, it took me longer than 20 secs to decipher the 'code' so I'm not in your league! Of course the young lady in the novel who solved it almost immediately is fictitious but, after all, she is named after someone famous. You'll have to read the book to appreciate this. It's a good read. Baldacci seems to enjoy the ideas behind RSA and cryptography in general. Cool stuff...

Maria Miller said...

I solved it this way:

a) I deduced Alex's grandpa is 12 times as old as Alex

b) I deduced Alex's dad is 7 times as old as Alex.

c) Mark Alex's age as x. Then his dad is 7x and his grandpa is 12x.

x + 7x + 12x = 140.
x = 7.

Dave Marain said...

Nice, Maria--
Do you believe the deductions you made require more maturity than most middle schoolers have? I feel they can handle the ratio method but how many could 'decode' the ratio idea from the language? I think some could but one never knows until we try it out in the classroom! Perhaps I'm making too much of the wording and many would figure it out.

Also, I wonder if the language in this riddle would be difficult for kids raised with Singapore math, which contains harder ratio problems...

The Rielle Deal said...

I think this is a very interesting problem, although trying to get every thing into the same unit first is really kind of a hassle. Just get things in terms of Alex's age, g=12a, d=7a and a=1a and then add them up. Sum of the ages (20a) equals 140, so a = 7. Then just multiply out to see what g and d equal.

Dave Marain said...

Nice comment and easy solution for most students -- thanks!

Anonymous said...

Hi all,

Singapore Maths is really a challeague to students for their PSLE preparation. This question with regards to age is not uncommon.

Below is one of the question set by renowned primary school in Singapore. Singapore primary 6 is equivalent to grade 6.

However, I believe that many students can solve this kind of question using algebra without any hassle. Algebra is not new to anyone and it was part of students' forte.

Take a look at this question below and answer them without using algebra. Using Singapore model drawing to help you solve this question.

The question goes like this:

When Mrs Lee was 40 years old, her son was twice her daughter’s age. Mrs Lee will be twice her son’s age when her daughter is 28 years old. How old will Mrs Lee be when her daughter is 20 years old?