PSAT Advanced Math — growth that multiplies rather than adds

Exponential change is the one topic in this domain where the mistake is almost always the same, and it is made in the first line rather than in the arithmetic. A quantity that grows by twelve percent a year is not multiplied by 0.12; it is multiplied by 1.12, because what it keeps is the whole of itself plus twelve percent more. A quantity that falls by fifteen percent is multiplied by 0.85, not by 0.15. Get that factor right and the rest of these questions take seconds.

This material is twelve questions built around it. Three ask which expression models a described situation — a plant population rising twelve percent a year, an asset depreciating fifteen percent, a savings account at four percent — and in each one the four options include the rate used as the factor, a linear model where the increase is fixed rather than growing, and the exponent confused with a multiplier.

Three work the other way, reading a model that is already written: the starting value at time zero, where the power of zero quietly becomes one; the growth factor in a model of a violin's value; and a straight evaluation at a stated time, where the wrong answers are the value a week earlier, the value a week later, and the factor multiplied by the exponent instead of raised to it.

Two give a short list of values and ask what kind of change it shows. One list doubles — 100, 200, 400, 800 — and one adds twenty-five at every step. The test is whether consecutive values have a constant difference or a constant ratio, and it is worth practicing on numbers small enough to see.

Two ask what a factor says in words: that 0.75 means three quarters of the quantity remains each period rather than that three quarters of it is lost, and that 1.20 means an increase of twenty percent rather than of a hundred and twenty.

The last two are doubling and halving, including one that takes two steps rather than one: a substance halving every three hours has halved twice after six, not once, so eight hundred grams become two hundred. Dividing the elapsed time by the interval before doing anything else is the habit that makes these safe.

Every situation here was invented for the exercise, all sums of money are written in words, and nothing is reproduced from any published test.

• Turn a percent increase or decrease into the factor a quantity is multiplied by each period
• Write an exponential model from a described situation, and reject the linear model that looks like it
• Read the starting value and the factor out of a model already written
• Evaluate a model at a stated time, raising the factor to the exponent rather than multiplying by it
• Tell exponential change from linear change by testing a list for a constant ratio or a constant difference
• Say in words what a factor such as 0.75 or 1.20 means for the quantity each period
• Divide the elapsed time by the doubling or halving interval before working out the amount

Every situation here was written for this exercise, and no two questions share one. Among them: five hundred plants rising twelve percent a year; an asset of eight thousand dollars depreciating fifteen percent a year; two thousand dollars at four percent compounded annually; the model $C = 400(0.90)^t$ read at $t = 0$; a violin valued by $V = 5000(1.08)^t$; the counts 100, 200, 400 and 800 against the counts 50, 75, 100 and 125; the models $P = 500(0.75)^t$ and $S = 1000(1.20)^t$ to be put into words; a hundred bacteria doubling hourly for four hours; eight hundred grams halving every three hours, read after six; and the colony $M = 60(3)^t$ after three weeks. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the PSAT/NMSQT, and it is not a test center.