How much more time must elapse before a trust fund kicks in, under the terms set by the will of an eccentric deceased and algebra-promoting relative? This article tells a story of Todd’s efforts to figure that out.

Elementary algebra moves beyond simple arithmetic, beyond the rules for addition, subtraction, multiplication, and division, only in one respect: it adds one or more unknown quantities.

When there is only one unknown quantity at stake, it is by convention called “x” (as below). This is nothing to be afraid of. x simply “marks the spot”, as in the maps of storybook pirates. It is the destination: the treasure sought.

Algebra Example: Trust Fund

Consider the case of a father and son, whom we will call Abe and Todd. Abe is 45 years old. His son is 18. A deceased relative has left a trust fund that provides them each with a large sum of money as soon as Todd is one-half the age of his father. How much longer must they wait?

That problem is open to a fairly straightforward solution. Todd’s first natural reaction might be to ask whether he is already at that point. Is his age one-half of Abe’s?
But then he multiplies 2 x 18 and finds, sadly, that their product is 36, well under 45.

So: what is it exactly that he wants to know? He writes it out numerically:
 
45 + x = 2(18 + x)
 
Notice that “x” is on both sides of the equation. This is simply because time is passing for both father and son – they are both getting older at the same rate.
 
If he wrote this: 45 = 2(18 + x) he would be asking the wrong question: in how many more years will Todd be half his father’s present age?
 
Suppose he solves that:

45 = 36 + 2x
9 = 2x
4.5 = x

He will have wasted his effort, because the passage of 4.5 years won’t satisfy the condition of the will. While he reaches the age of 22.5, his father will also be aging.

His true problem, in numeric terms is then:

45 + x = 2(18 + x)

He might learn the solution in simple steps. First, he wants to get rid of that parenthesis. Since he understands the rules for multiplication, he sees that the right hand part of the formula is equivalent to this: 36 + 2x
 
Restoring the left-hand side, then:
 
45 + x = 36 + 2x
 
Now he subtracts 36 from both sides: 9 + x = 2x

Running for the finish line now, Todd understands that two of anything, minus one of it, leaves one of it. So he subtracts 1x from each side of the equal sign:

9 = x

That’s his answer!
 
Disappointed (nine whoooole years!), he plugs the 9 back into our starting formula to check.

45 + 9 = 2(18 + 9)
Is that right?
54 = 2(27)

Yes, it clearly is right. In nine years, Todd will be 27 years old, so both he and his 54 year old father will qualify for their inheritance.