Expressing relationship between two-numbers as a percentage based on their quotient

Or, numbers are fun and here is a way to wow someone

I love finding relationships in number-sets. A favorite memory from high school deals with a game we would play in Algebra class. The teacher would give us a set of numbers and one target number. The goal was to use the set of numbers and arithmetic operations to get to the target. Here is an example:

Set: \({1, 2, 4, 6}\)

Target: \(49\)

A couple solutions: \([1 + (2 * 4) * 6], [1 + 2 * (4 * 6)]\)

You can find problems like this in a handy desk calendar (non-affiliate link).

Fast forward several years and I started thinking about age as a percentage between a parent and child. It seemed interesting that a **child is half the age** *of their parent when the* **child is the same age** *as the parent when the* **child is born**. At first, I thought this was a fluke of the number-world, but gave it more thought, given both child and parent age at the same numerical-rate. I was interested in finding a general-purpose equation–one that could be applied at any age and came up with: \(a = (b + a)c\), where \(a\) is the child's age, \(b\) is the age of the parent when the child is born, and \(c\) is the relationship between both as a percentage. Let us go through an example:

*My Dad is thirty years old when I am born. When I am ten, what percentage of my Dad's age am I?*

Solution:

\(10 = (30 + 10)c\)

\(10 = 40c\)

*Divide both sides by 40*\(c = 10 / 40\)

\(c = 0.25\)

*or 25%*

What I really want to do, though, is find age based on percentage. Now, the question becomes *what is my age when I am twenty-five percent of my Dad's age?* Taking the original equation, we can work to get \(a\) on one-side:

\(a = (b + a)c\)

*Distribute the right-side*\(a = bc + ac\)

*Subtract \(ac\) from both-sides*\(a - ac = bc\)

*Factor \(a\) from the left-side*\(a(1 - c) = bc\)

*Divide both sides by \((1 - c)\)*\(a = (bc) / (1 - c)\)

And plugging our numbers into the equation:

\(a = (30 * 0.25 ) / (1 - 0.25)\)

\(a = 7.5 / 0.75\)

\(a = 10\)

But, we can do even better–we can also answer the question *what is the age of my Dad when I am twenty-five percent of his age?* Again, looking at the original equation, we can create a new variable \(d\) and put it in place of \((b + a)\):

\(a = dc\)

*Divide both sides by \(c\)*\(d = a / c\)

We already solved for \(a\), so we can put it into the equation:

\(d = ((cb) / (1 - c))/c\)

*Let us convert the outer division problem into multiplication for better readability*\(d = ((cb) / (1 - c))(1 / c)\)

*Multiply both numerators and denominators*\(d = (cb) / (c - c^2)\)

*Factor \(c\) from the denominator*\(d = (cb) / (c(1 - c))\)

*We can now remove \(c\) from both numerator and denominator*\(d = b / (1 - c)\)

And plugging our numbers into this equation:

\(d = 30 / (1 - 0.25)\)

\(d = 30 / 0.75\)

\(d = 40\)

Of course, these equations are not limited to comparing age. They can be applied to any two-numbers. The key is to represent \(b\) relative to zero. So, if \(a\) is not zero, then \(b\) becomes \(b-a\). Keep in mind, however, \(c\) cannot be one-hundred percent. This leads to division by zero. Lastly, for some extra visual flaire, we can graph the results using \(x = 30 / (1 - y)\). Keeping with our theme, this plots my Dad's age on the x-axis with my percentage (in decimal-form) of his age on the y-axis. Notice the asymptotic-nature of the line:

*Graphs plotted using https://www.desmos.com/calculator.*