Expert: Josh - 9/10/2006

Question
This is a question for my 4th grader's gifted and talented class.  The teacher stated that we are allowed to use any and all resources available to solve it. Please help us, we are stuck.

Now I am 4 times older than my sister was when she was 1/2 as young as I was.  In 15 years our combined age will be 100. How old are we now?

You must give my age and the age of my sister along with the strategy used to develop your solution.

Answers are due Sept. 11...what a joke, we received this Friday.  

Answer
Hi Judy,

I recommend tackling this question using algebra. Yes, I know your child is in 4th grade; but any reasonable strategy for solving this problem basically amounts to doing the same thing.

The idea is to assign a variable for each unknown quantity, then, try and relate the unknowns using the facts we have been given. Usually, we pick an english alphabet, such as "x", "y", or "z" as the variable, where "x" ("y" or "z") is an abstract representation of the unknown quantity whose value is yet to be determined.

In the context of your problem, let "x" denote my age, "y" denote my sister's age.

Initially, at some earlier time "-T", "sister was half as young as I was". So, y(-T)=0.5*x(-T).

Now (at time t=0), I am four times older than my sister was back then (at time t=-T). This means that x(0)=4*y(-T)=2*x(-T)......[#1]  This is saying that I am now (at time t=0) twice the age I was back then.

At present, my sister must age y(0)=y(-T)+T, where T is the number yeas elapsed since then......[#2]

Fact: "In 15 years, combined age equals 100".
We simply add 15 years to both my current age and my sister's current age. [*]

In fifteen years' time, t=15,
my age will be x(15)=x(0)+15 ...[#3]
while my sister's age will be y(15)=y(0)+15......[#4]

Substituting [#1] into [#3]
x(15)=2*x(-T)+15 ...call this [#5]
Substituting [#2] into [#4]
y(15)=0.5*x(-T)+T+15 ...call this [#6]

Finally, using the fact in [*],
x(15)+y(15)=100 ...call this [#7].

Solving equations [#5],[#6],[#7] simultaneously, equation [#7] becomes
x(15)+y(15)=100
[2*x(-T)+15]+[0.5*x(-T)+T+15]=100
simplifying this gives
2.5*x(-T)+T=70 ...[#8]

Focus on equation [#8], we have two unknowns: viz., x(-T) my age back then, and "T", the number of years elapsed since then. If we can eliminate the number of unknowns, we can work everything out. Have we exploited all known facts yet? Revisiting my present age, it turns out that [#1] can be rewritten as x(0)=x(-T)+T. In words, my current age is my age back then, plus T years (the number of years elapsed since then). Reconciling this expression with [#1], (since x(-T)+T=x(0) and 2*x(-T)=x(0) are both equal to my current age, x(0)) we must have x(-T)+T=2*x(-t)..[#9]. This is exactly what we need to eliminate the number of unknowns down to one variable. We now know, from [#9], that T=x(-T)......[#10] Plugging this into equation [#8], we finally obtain (through substitution of T)

2.5*x(-T)+x(-T)=70
x(-T)=70/3.5=20.

This reveals that my age back then was 20 years.
Using other relations we have derived, [#2] gives my sister's age back then, y(-T)=0.5*x(-T)=0.5*20=10 years.

From [#10], we know that T=20 years have elapsed since then. Thus, my current age is x(-T)+T=40, while my sister's current age is y(-T)+20=30.

Just to verify, in 15 years time, my age will be x(0)+15=55, while my syster's age will be y(0)+15=45. And they add up to 100.

COMMENTS: Ok, so this takes a fair bit of work. I'm not sure it's reasonable to demand this from a fourth grader, no matter how gifted he/she is. I was neither gifted nor talented at that age, but with time, you learn a great deal. All I can say, is that algebra provides a "systematic" way of solving the problem. Perhaps your child is not "expected" to find the solution. May be, the point of the exercise is to get him/her to rationalize ideas and think creatively about the problem. Abstract reasoning can be learnt, just as algebraic techniques can be learnt. I know as parents, you are trying to do what you can to provide the best opportunities for your children. Just try not to push your child too hard, and make sure that he/she enjoys it. I never received any private tuition during my high school years, in hindsight, I think I did alright.