Seeing a Chunk as One Thing

In this lesson you will: Treat a bracketed expression as a single quantity and count copies of it; Identify the initial amount and the growth factor in P(1+r)^t; Say what a growth factor greater or less than 1 means in a situation.

Count the copies before you expand anything

Look at 5(x+3)-2(x+3). Whatever (x+3) happens to equal, you have five of it and then take away two of it, so you are left with three of it: 3(x+3).

You never needed to know what x was. The chunk (x+3) behaved exactly like a single letter, and that is the whole idea of this lesson. Some students find it helps to write u in place of the chunk, do the easy algebra, and put the chunk back at the end.

The growth factor is the whole parenthesized factor

P(1+r)^t is the expression Ohio's standard names directly, so it is worth taking apart slowly. P is what you start with. r is the rate per period, written as a decimal. And (1+r) — the parentheses, not the r — is the growth factor.

A rate of 6% gives a growth factor of 1.06, because keeping all of what you had and adding six percent more is multiplying by 1.06. The 1 is the part you keep.

What it looks like over time

Multiplying by the same factor over and over does not add the same amount each period. Each year's increase is computed on a larger balance than the year before, so the gaps between consecutive values widen.

That is the practical difference between 500+30t and 500(1.06)^t. Both start at 500 and both gain 30 in the first year. Only one of them gains more than 30 in the second.

Common mistake — Using the rate where the growth factor belongs

An account grows 4% a year from $1,200. A student writes 1200(4)^t, or sometimes 1200(1+4)^t.

It is a reasonable-looking translation. The problem said four, the formula has a slot, and the four goes in the slot.

But r is a rate written as a decimal, so 4% is 0.04, and the factor is 1+0.04=1.04. Writing 4 says the money quadruples every year; writing 5 says it quintuples. After ten years 1200(1.04)^10 is about $1,776, while 1200(4)^10 is over a billion dollars.

How to catch it: Substitute t=1 and read the result out loud. One year of 4% growth on $1,200 has to be $1,248. If your expression gives $4,800 or $6,000, the factor is wrong, and you will catch it in one line rather than at the end of a page.