Percent Rates, and Reading Them Out of the Expression

In this lesson you will: Turn a percent increase or decrease into the factor you multiply by; Read the initial value and the percent rate out of a(1+r)^t or a(1-r)^t; Use the form to classify a function as growth or decay and to interpret it in context.

A percent change becomes a multiplier

Suppose a $200 item goes up by 6%. The new price is the old price plus six percent of it: 200+0.06(200)=200+12=212.

Factor the right side and the whole calculation collapses: 200(1+0.06)=200(1.06). The multiplier keeps the 100% you already had and adds the 6%. That is the 1 in 1.06, and it is the part students throw away.

A decrease works the same way from the other side: keeping all of it and removing 6% leaves 1-0.06=0.94 of it.

Running it backwards

F.IF.8b asks you to go the other way: given the expression, state the rate. Line the base up against 1+r or 1-r and read off r.

In f(t)=800 (1.07)^t the base is 1.07, which is 1+0.07, so the quantity increases by 7% each period, starting from 800.

In g(t)=500 (0.88)^t the base is less than 1, so it is decay. Since 0.88=1-0.12, the quantity falls by 12% each period. The common wrong answer is 88% — that is what remains, not what is lost.

Compound interest is this idea with a dollar sign

Put $1,000 in an account paying 5% a year, interest added once a year. Each year the balance is multiplied by 1.05, including the interest already earned.

Year by year: 1000, 1050, 1102.50, 1157.63, 1215.51, 1276.28. Simple interest would have added a flat $50 a year, reaching $1,250. The extra $26.28 is interest earned on interest.

Common mistake — Using 0.05 as the base for a 5% increase

A salary of $40,000 rises 5% a year. The model written down is S(t)=40000 (0.05)^t.

The thinking behind it is reasonable. The problem says five percent, five percent is 0.05, and 0.05 is the number that goes in the box. Nothing in the sentence mentions a 1.

But look at what the model claims for one year: 40000 (0.05)=2,000. A five percent raise does not shrink a salary to two thousand dollars. That $2,000 is the raise; the model kept the raise and discarded the salary.

The correct base keeps what you already had and adds the increase: 1+0.05=1.05, giving 40000(1.05)=42,000 after one year. The decay version has a matching trap — an 8% decrease is a base of 0.92, never -0.08 and never 1.08.

How to catch it: Test one time period against arithmetic you can do without algebra. One year of a 5% raise on $40,000 is 40,000+2,000=42,000. Now put t=1 into your model. If it does not return $42,000, the base is wrong — and you found out in ten seconds instead of after five years of compounding.