# How old is Jim? (A problem)

Many years ago (6?) a friend of mine called me and asked me if I could solve this problem. Her son - seventh grade at the time - had this as a bonus question in his algebra class.

I won’t post the answer for a bit. If you’re really hurting for the answer, a clue or validation, let me know here and I’ll e-mail you.

Here goes…

Good God. My brain hurts.

First: What did they have for dinner?

Now: Jim is…
A. 17 because he and his brother art twins.

B. 4,928 and 3 months old.

C. Dead because he actually touched his brother which is actually him from another dimention.

D. The semantics made my brain melt.
-Rue.

Jim is 17, but his brother is 9.

Here’s how I solved it:

Let B=the brothers age NOW and B+x=Jim’s age now. X would obviously be the age difference.

The sentece converts to B+x=17(B-x).

I did some quick solutions for x and B, assuming that they would be whole numbers.

When x=8, B=9.

So, Jim is 17. His brother is 9. When Jim was 9, is brother was 1. Jim’s age is 17 times 1 year.

I bet if I had just guessed Jim was 17, I would’ve gotten the answer faster.

I love stuff like this.

Nope

so far so good…

Hmmm, I think you missed a clause in the sentance :

If Jim is 17 and his brother (Bob) is 9 then the sentance becomes

Jim was 9 eight years ago so Bob is 1 therefore :

But when Jim was 1 his brother wasn’t born. I think you need the equation to be

B + x = 17(B - 2x)

Now to find solutions for that.

• SpaceDog -

Who really should be working

SpaceDog Central

Um, I think that there is an extra phrase or something in there. Let’s do it backwards. Here’s the original question:

So if the brother is now 9, we have

Jim is 17 times as old as his brother was when Jim was as old as his brother was when Jim was 9.

If the age difference is 8, we now have

Jim is 17 times as old as his brother was when Jim was 1.

Again, if the age difference is 8, we how have

Jim is 17 times -7

Which is -119.

We also have the second sentence in the question. Why is it relevant that they had dinner with their father last week? This is one of those sentences that gives some information that allows you to choose between two equally likely answers, so at some point in the solution you should have to make a choice and this sentence would help you.

Jim is 51, his brother is 35.

Jim’s age divided by 17 is equal to his brother’s age minus 2 times the difference between them.
51 / 17 = 35 - 2 * 16

Jim is 17 times as old as his brother was when Jim was as old as his brother was when Jim was as old as his brother is now. His brother is 35.

Jim is 17 times as old as his brother was when Jim was as old as his brother was when Jim was 35. His brother was 19 then.

Jim is 17 times as old as his brother was when Jim was 19. His brother was 3 then.

Jim is 17 times as old as a 3-year-old.

Jim is 51.

That’s what I get for posting before double-checking my answer.

I think the trick is to work backwards from the end of the first sentence:

(1) When Jim (current age “J”) was as old as his brother is now (“B”), Jim’s age was B. (The tenses in the sentence indicate that Jim is older. The difference in ages is J-B.)

(2) When Jim was B years old, his brother was (J-B) years younger: i.e., his brother’s age was B-(J-B) = 2B-J.

(3) When Jim was (2B-J) years old, his brother was (2B-J) - (J-B) = 3B-2J.

(4) Jim’s age is now 17 times (3B-2J), so –

J = 17(3)B - 17(2)J = 51B -34J

0 = 51B - 35J

51B = 35J

Jim is 51 and his brother is 35 (or Jim is 102, his brother is 70, and their father is .really old.)

What did they have for dinner?

“Jim is 17 times as old as his brother was when Jim was as old as his brother was when Jim was as old as his brother is now.”

Let Jim’s age now be ‘j’ and brother’s age now be ‘b’. The difference in their ages is always j-b. Then, “as old as his brother is now” is b.

“Jim is 17 times as old as his brother was when Jim was as old as his brother was when Jim was b*.”

“as old as his brother was when Jim was b” is Jim’s age (at that time) minus the difference in their ages, or b-(j-b), or 2b-j.

“Jim is 17 times as old as his brother was when Jim was 2b-j.”

“as old as his brother was when Jim was 2b-j” is that age minus the difference in their ages, or 2b-j-(j-b), or 3b-2j.

“Jim is 17 times as old as 3b-2j.”

So j=17(3b-2j)., which means that the brother is 35j/51. The ratio b/j is 35/51.

Choose an age for Jim. Brother’s age is 35/51 of that age. For instance, if Jim is 51, then Brother is 35. If Jim is 8 then brother is around 5.5. Since they had dinner with their father, and if we make the following reasonable assumptions:

• Father was at least 10 years old when Jim was born.
• Father didn’t live longer than 110 years.

We’re sort of assuming that these guys are all human, but they might be tortoises or parrots or something. It could also be the case that their father is no longer alive, even though they “had dinner with him”. And, we can assume that the difference in Jim’s and his brother’s ages is either zero or greater than 9 months (though this is a leap; we can’t really assume that they have the same mother–it could even be the case that one of them is adopted; or even that his brother is “having dinner” in the womb).

Then, we get an age for Jim anywhere between 9 months and 100 years.

(kgmm)/(s*s)

D’oh. Tullius beat me to it, I screwed up the coding, and I forgot to say that Jim and brother could be twins and be nursing for the very first time, both are zero years old.

(kgmm)/(s*s)

Same here. I’m glad I don’t have to deal with Algebra anymore.

I figured it out I think, I just wanna see if I’m right.

How old is Jim? Pick an age. That’s how old he is.

The problem statement only contains enough information to learn the ratio of his age to his brother’s. You can make some assumptions (they’re humans who’ve both been born to the same birth mother, not twins, their father is also a human who is still living and not Methuseleh, his and his brother’s ages are best expressed as integral numbers of years, etc.) which might lead you to conclude that Jim is 51 years old.

But if you’re going to do that, you might as well just make the assumption that he’s 51 years old. Or 12. Or 17. Or 21. Or you get the picture.

I think anywhere between 9 months and 100 years could be fairly “reasonable”.

(kgmm)/(s*s)

From the problem, a matrix can be set up that looks kinda like this.

``````

Jim's age	Brother's age
Now       17Y	      X
Was	   Z	      Y
Before	   X	      Z

``````

It should be noted that the variables X, Y, and Z were chosen randomly, as were the titles for the rows. The placement of the variables in the matrix comes from the problem.

The problem states “…when Jim was as old as is brother is now.” This shows that Jim is older (and always was) than his brother. From this we can add a third column to the matrix showing that this is the case

``````

Jim's age	Brother's age		Inequality
Now	  17Y	      X			17Y > X
Was	   Z	      Y			Z  >  Y
Before	   X	      Z			X  >  Z

``````

These inequalities can be combined to form one that orders the variables:

``````

17Y  >  X  >  Z  >  Y

``````

Since Jim was born some number of years before the brother, he is always that number of years older. Thus the difference between Jim’s age at any time and his brother’s age at that time is constant. This means that the “spacing” between the terms in the long inequality is the same. This can be written as:

``````

17Y  >  X  >  Z  >  Y
\__C_/ \_C_/ \_C_/

``````

This relationship can be written algebraically like so:

``````

17Y = Y + 3C

``````

Solving for C:

``````

3C = 16Y
C = 16 Y
3      This is 16/3

``````

Assuming that Jim’s and the Brother’s ages are whole numbers (not unreasonable) the easiest solution for Y is for Y to equal 3. This causes C to equal 16 i.e. Jim was 16 when his brother was born. Similarly, if the brother was 3 at the youngest (in the above matrix), then Jim is currently 17 x 3 or 51 years old.

Using the facts that Jim was 16 when brother was born and. all of the values can be filled in the matrix. Since all of the values fit the matrix, we can say with gobs of satisfaction that Jim is currently 51 years old and his brother is currently 51 - 16 or 35 years old. This can be validated by noting that 35 + 16 (the age difference) = 51.

Incidentally, this is reasonable given that the father is currently alive, even though it wasn’t necessary to solve the problem.

Yeah, you can bring up points regarding obscure possibilities (age = 0, picnic at the gravesite, etc.) but we should be reasonable here.
tullius had the first correct answer with correct explanation. (Sorry, Protesilaus, I didn’t get your explanation.) However, this line didn’t make sense:

``````
Jim is 51 and his brother is 35 (or Jim is 102, his brother is 70, and their father is .really old.)

``````

only because the difference in their ages (32 years) would not fit with the rest of the matrix.

Getting a mathematical anser is fine, but it must be checked within the structure of the question.

Kudos and braggin’ rights to tullius . If you’re ever near Annapolis, MD I’ll buy the first beer.

I worked my way backwards on this. I set it up this way:

Jim’s brother is now X.

When Jim was X, his brother was Y.

When Jim was Y, he was 17 times his brother’s age.

So I put in age 1 for his brother when Jim was 17. Jim is therefore 16 years older than his brother. So when his brother was 17, Jim was 33, and when his brother got to his present age of 33, Jim would be 49.

If we start out with his brother being 2, Jim would be 34, 32 years older. By the end, he would be 98 and his brother 66, but the problem said his father was still alive, so I’ll go with the first one.

But I see that it’s different from the posted answer. Where did I go wrong?