Percentage Decay Rate Calculator

Percentage Decay Rate Calculator

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What is a Percentage Decay Rate?

A percentage decay rate represents the consistent proportional decrease of a value over a specific period of time. Unlike linear reduction, where a fixed amount is removed every step, exponential decay involves removing a fixed percentage of the remaining value. This concept is fundamental in physics (radioactive half-life), finance (asset depreciation), and biology (medicine metabolization).

The Exponential Decay Formula

To find the percentage decay rate per time period, we use the following mathematical formula:

r = 1 – (N_t / N₀)^(1/t)

Where:

• r is the decay rate (multiply by 100 for percentage).
• N₀ is the initial starting quantity.
• N_t is the final quantity after time has passed.
• t is the total number of time periods elapsed.

Real-World Example

Imagine a piece of machinery is purchased for 10,000 units. After 5 years, its value has dropped to 4,000 units. To find the annual percentage decay rate (depreciation):

1. Initial Value (N₀): 10,000
2. Final Value (N_t): 4,000
3. Time (t): 5 years
4. Calculation: r = 1 – (4,000 / 10,000)^(1/5)
5. Result: r ≈ 0.1674 or 16.74% per year.

Common Applications

• Pharmacology: Calculating how fast a drug leaves the bloodstream.
• Environmental Science: Measuring the rate of deforestation or population decline in endangered species.
• Data Science: Applying "decay" to historical data to give more weight to recent events in machine learning models.
• Retail: Estimating the rate at which inventory loses value or appeal over time.

function calculateDecayRate() { var initial = parseFloat(document.getElementById('initialValue').value); var final = parseFloat(document.getElementById('finalValue').value); var time = parseFloat(document.getElementById('timeElapsed').value); var resultDiv = document.getElementById('decayResult'); if (isNaN(initial) || isNaN(final) || isNaN(time) || initial <= 0 || time <= 0) { resultDiv.style.display = 'block'; resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'Error: Please enter valid positive numbers. The initial value and time must be greater than zero.'; return; } if (final > initial) { resultDiv.style.display = 'block'; resultDiv.style.backgroundColor = '#fff3cd'; resultDiv.style.color = '#856404'; resultDiv.innerHTML = 'Note: The final value is higher than the initial value, indicating growth rather than decay.'; } // Formula: r = 1 – (Nt/N0)^(1/t) var rate = 1 – Math.pow((final / initial), (1 / time)); var percentageRate = rate * 100; resultDiv.style.display = 'block'; resultDiv.style.backgroundColor = '#d4edda'; resultDiv.style.color = '#155724'; resultDiv.innerHTML = '

Calculated Decay Rate

' + '' + percentageRate.toFixed(4) + '%' + 'The value decreases by ' + percentageRate.toFixed(4) + '% per time period.'; }