Interpreting Remaining Fractions and Percent Decayed

Introduction

In a decay problem, a statement such as “40% has decayed” describes a different quantity from “40% remains.” Confusing these statements can lead to a large error in the amount calculated.

The key is to keep track of the amount still present relative to the initial amount. That ratio is the remaining fraction. From it, we can express the same information as a percentage remaining or as a percentage decayed.

The Remaining Fraction in an Exponential Model

For a continuous exponential decay model, let \(A_0\) be the initial amount and let \(k>0\) be the decay constant. Measuring time \(t\) from the initial measurement, the amount remaining is

Here \(A_0>0\), \(t\ge 0\), and \(k\) has units reciprocal to the time units used for \(t\). For example, if \(t\) is measured in hours, \(k\) is measured in inverse hours. The exponent \(kt\) is dimensionless.

A fraction is a ratio, so it has no units. It lies between 0 and 1 for this model: at \(t=0\), \(R(0)=1\), and for \(t>0\), \(0<R(t)<1\). The amount decayed by time \(t\) is \(A_0-A(t)\); dividing by \(A_0\) gives the fraction decayed.

How the Two Percentages Are Related

The initial amount is divided into two parts: the amount still present and the amount that has decayed. Their fractions of the initial amount must therefore add to 1. If \(D(t)\) denotes the fraction decayed, then

Multiplying these fractions by 100 expresses them as percentages. Thus, the percentage remaining and percentage decayed add to 100%. This relationship is a consequence of how the quantities are defined; it does not depend on the exponential formula.

The exponential model supplies the remaining fraction \(e^{-kt}\). The percentage statements are just different ways of reporting that fraction. In particular, “\(p\)% remains” corresponds to \(R(t)=p/100\), whereas “\(q\)% has decayed” corresponds to \(R(t)=1-q/100\).

Worked Example: Finding Both Percentages and Amounts

Converting a Percentage Decayed into an Amount

If a problem gives the percentage decayed, subtract that percentage from 100% to find the percentage remaining. Convert the resulting percentage to a fraction by dividing by 100, then multiply by the initial amount. The amount decayed can be found either by multiplying its own fraction by the initial amount or by subtracting the remaining amount from the initial amount.

Using a Percentage Remaining Directly

When the percentage remaining is given, convert it directly to a fraction by dividing by 100. Multiplying the initial amount by this fraction gives the amount remaining. Then use the complementary fraction, or subtract from the initial amount, to find the amount decayed.

Why the Conversion Works

The conversion follows from dividing each amount by the same initial amount. Since \(A(t)\) remains and \(A_0-A(t)\) has decayed,

The first ratio is the remaining fraction; the second is the decayed fraction. Their sum is exactly 1, so their percentages sum to 100%. For an exponential model, substituting \(A(t)=A_0e^{-kt}\) into the first ratio gives

where the cancellation is valid because \(A_0>0\). Consequently, the decayed fraction is \(1-e^{-kt}\), and the amount decayed is \(A_0(1-e^{-kt})\). These expressions describe the same amounts as subtracting the remaining amount from the initial amount.

Common Pitfall: Treating Percent Decayed as Percent Remaining

Suppose \(28\%\) has decayed from an initial amount of \(450\) grams. A common incorrect calculation is

This calculation uses the correct fraction for the decayed amount, but labels it as the amount remaining. It reverses the meanings of the two percentages. The correction is to recognize that \(0.28\) is the fraction decayed, while the fraction remaining is \(1-0.28=0.72\):

The check \(324+126=450\) confirms the corrected interpretation. A useful habit is to write “remaining fraction” or “decayed fraction” beside a decimal before multiplying it by the initial amount.

Choosing the Right Representation

A model may provide a rate constant and time, or a problem may directly state a percentage. In either case, identify which quantity the statement describes before calculating:

• If the model gives \(k\) and \(t\), calculate \(R(t)=e^{-kt}\).
• If a percentage remaining is given, divide it by 100 to obtain the remaining fraction.
• If a percentage decayed is given, subtract it from 100% to obtain the percentage remaining.
• Multiply the remaining fraction by \(A_0\) to obtain the amount remaining.
• Subtract the remaining amount from \(A_0\), or use the decayed fraction, to obtain the amount decayed.

Always preserve the distinction between a fraction and a percentage: \(0.28\) as a fraction corresponds to \(28\%\). Also keep the initial amount explicit in calculations. A percentage of the initial amount is not, by itself, an amount; multiplying by \(A_0\) supplies the units and the quantity.