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March 31, 2017

Will there be a point in your life when your child is older than you? The answer is no. You were born first. As she grows older, you do too. If you are willing to ignore the tiny effects that the theory of relativity has on human ageing, then you would agree that both you and your child grow at the same rate. When a year passes for your child, a year passes for you as well.

But here is a related question:

Will your child's age at any given time be half of your age at that time? If so, then when?

Let's take one of my daughters as an example. She is now 8 and I'm 43. There are 35 years between us. Indeed, she was born when I was 35 years old. Will she ever be half my age?

When she was 1 year old, I was 36. She was 1/36 of my age (0.0277 of my age). When she was 5 years old, I was 40. That is 1/8 of my age (0.125 of my age). Now at 8 and me at 43 she is 0.186 of my age. When I'm 50, she will be 15. That is 0.3 of my age.

Is she closing in on me? As you clearly see, we'll always have 35 years between us. But as we both grow older, the gap of 35 years becomes less significant. What if I am still alive at 100? She will be 65 at the time. That is 0.65 of my age. And just for the fun of it, what if we were immortal; how do my daughter and me compare when I'm 1000? She is then 0.965 of my age. Very close to me in relative terms.

Back to the original question:

When is my daughter going to be half my age?

This is by no means a deep mathematical question. Nevertheless, it can be very rewarding to discuss with your children. Speaking about it doesn't only bring mathematical thought, it can also stimulate discussion about childhood vs. adulthood, years, times, plans, aspirations and pretty much anything else.

On the basic mathematical front, it can be fun to explore in various ways:

• With young children, it is a way to explore basic arithmetic together with trial and error calculations.

• With slightly older children, you can write an equation and use it to find a solution with basic Algebra.

• Finally, this question has a neat simple answer that doesn't require any calculation. Frankly, I wasn't aware of it until I sat down to write this blog post. For me it was a sweet little discovery. Maybe you want to give it a bit more thought to see if you can work it out yourself, before you read on...

Q and A with my 8 year old daughter...

Q: What is our age difference?
A: You daddy are 43, I am 8... (thinking), so 35.

Q: How old was I when you were born?

A: 35! (Looking at me with a "Duh Daddy!" look).

Q: How old will I be when you are 30?

A: 30+35 = 65, so you will be 65.

Q: Is that double your age at that time?

A: No. 2 x 30 = 60 < 65.

Q: How about when you are 31?

A: 66. But that is still not double because 2 x 31 = 62 < 66.

Q: How about when you are 32?

A: 67. It still not double (64 < 67).

Q: So when will you be half my age (or me double your age)?

A: (Thinking...) When I'm 33, you are 68. When I'm 34, you are 69..... Ahhh, I got it! When I'm 35, you are 70.

Quite magical actually, the thought of her being 35 and me 70. I hope that she'll still have patience for my mathematical exploration by that time. If not, maybe I'll have to try the grandchildren.

So is there a systematic way to find the solution? One straightforward way is to use basic equations. Let the variable C denote the age of the child and let P denote the age of the parent. Here both variables are measuring the ages in that special year where the child's age is half that of the parent. How can we express this with C and P? Mathematical expression of this type isn't hard actually. Just write,

2 x C = P.

But we also know that there is a fixed age difference between C and P. For example for my daughter and me as mentioned above,

C + 35 = P.

So now we are seeking the ages C and P that satisfy both of these equations. Does it make sense? Satisfying the first equation means that the parent's age is twice the child's age (or child is half the parent). Satisfying the second equation means that the parent had the child at age 35. There is always a difference of 35 years between them.

There are many ways of solving such (linear) equations. One way is to observe that the right hand side of both equations is P and thus equate the left hand sides of both equations. You see, the equal sign (=) in mathematics is incredible. It means that anything on the left hand side of it can replace anything on the right hand side and vice versa. Also, if X=Y and Y=Z, then guess what it says about X and Z?

We now get,

2 x C = C + 35.

This is now a single equation for the unknown variable C. We can now subtract C from both sides of the equation and this gives:

C = 35.

So you see, with algebra we can get the same answer we got with trial and error above. When the child (C) will be 35, the parent (P) will be twice the age.

But wait a minute! You may be looking at the algebra above and seeing that we didn't really do anything with the number 35. The algebra we did, works with any age difference.

Take my parents and me as an example. My mom and dad had me when they were both 22. Replace 35 with 22 in the above equations and what do you get?  Here I'm enjoying the prospect of being "C" instead of "P" for this example. Don't get me wrong, parenting is great. But being a child is even better, even if momentarily.

The algebra we did above for 35 works now for 22 in the exact same manner. We then get C = 22 (and P=44). By the way, at 43 and with parents at 65, I'm 0.66 of their age. I'm closing in!

Try this with your children. At what age did you have your child? At what age is she or he going to be half of your age?

Here is the rule we discovered:

Your child will be half your age when they reach the age you were on their birth.

Why is that? The algebra shows it, but are there other explanations? Well, think about it. It took me a bit of time to figure it out:

When your child reaches the age that was your age during their birth, then for every day that she lived till that age, you lived exactly two days!

The first half of those days were perhaps nice. But the second half is incredible! Can your child explain why?

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