Math Formula for Decay
2026-02-28 10:54 Diff
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Last updated on October 6, 2025

In mathematics and physics, the decay formula is used to model the decrease of a quantity over time. This can pertain to various contexts, such as radioactive decay, depreciation of assets, or diminishing returns in investments. In this topic, we will learn about the decay formula and how it is applied in different scenarios.

List of Math Formulas for Decay

Decay is a process where a quantity decreases over time. Let's learn the formula to calculate decay in various contexts.

Math Formula for Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value.

It is calculated using the formula:\( [ N(t) = N_0 \cdot e^{-kt} ]  \)where \(( N(t))\) is the quantity at time \(( t ), ( N_0 ) \)is the initial quantity, k  is the decay constant, and  e  is the base of the natural logarithm.

Math Formula for Radioactive Decay

Radioactive decay is a specific type of exponential decay.

The formula for radioactive decay is: \([ N(t) = N_0 \cdot e^{-\lambda t} ]\) where\( ( \lambda )\) is the decay constant specific to the substance.

The half-life\( ( t_{1/2} )\) is related to \(( \lambda ) \) by the formula\( ( t_{1/2} = \frac{\ln(2)}{\lambda} ).\)

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Math Formula for Depreciation

Depreciation models the decrease in value of an asset over time. One common method is straight-line depreciation, calculated as:\( [ \text{Depreciation} = \frac{\text{Cost} - \text{Salvage Value}}{\text{Useful Life}} ]\)

In exponential decay terms, depreciation can also be modeled with: \([ V(t) = V_0 \cdot (1 - r)^t ] \)where \(( V(t) )\) is the value at time\( ( t ), ( V_0 )\) is the initial value, and \(( r ) \)is the depreciation rate.

Importance of Decay Formulas

Decay formulas are crucial in many scientific and financial fields. Here are some important uses of decay formulas:

  • They are used to predict how substances transform over time, such as radioactive materials.
  • In finance, they help in understanding asset depreciation and investment returns.
  • In environmental science, they model the decay of pollutants and other materials.

Tips and Tricks to Memorize Decay Formulas

Students often find decay formulas tricky. Here are some tips to master them:

  • Relate decay scenarios to everyday life, like depreciation of a car's value or the cooling of coffee
  • Use mnemonic devices to remember constants and relationships, such as the natural logarithm base e in exponential decay.
  • Practice with different examples to understand the underlying principles better.

Common Mistakes and How to Avoid Them While Using Decay Formulas

Students often make errors when using decay formulas. Here are some mistakes and ways to avoid them:

Problem 1

A radioactive substance has a half-life of 10 years. If the initial quantity is 100 grams, how much is left after 20 years?

Okay, lets begin

The remaining amount is 25 grams.

Explanation

Using the half-life formula, we know after one half-life (10 years), 50 grams remain. After another half-life (20 years total), 25 grams remain.

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Problem 2

A car worth $20,000 depreciates at an annual rate of 10%. What is its value after 3 years?

Okay, lets begin

The car's value is approximately $14,580.

Explanation

Using the depreciation formula: \([ V(t) = 20,000 \cdot (1 - 0.10)^3 = 20,000 \cdot 0.9^3 = 14,580 ]\)

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Problem 3

A radioactive isotope decays with a decay constant of 0.693 per year. What is its half-life?

Okay, lets begin

The half-life is 1 year.

Explanation

Using the formula \(( t_{1/2} = \frac{\ln(2)}{\lambda} ): [ t_{1/2} = \frac{\ln(2)}{0.693} = 1 \text{ year} ]\)

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Problem 4

A machine costs $50,000 and has a salvage value of $10,000 after 10 years. What is the straight-line depreciation per year?

Okay, lets begin

The straight-line depreciation is $4,000 per year.

Explanation

Using the straight-line depreciation formula:\( [ \text{Depreciation} = \frac{50,000 - 10,000}{10} = 4,000 ]\)

Well explained 👍

Problem 5

If a population decreases from 500 to 250 in 5 years, what is the decay constant assuming exponential decay?

Okay, lets begin

The decay constant is approximately 0.1386.

Explanation

Using the formula\( ( N(t) = N_0 \cdot e^{-kt} ): [ 250 = 500 \cdot e^{-5k} ] [ e^{-5k} = 0.5 ] [ -5k = \ln(0.5) ] [ k = -\frac{\ln(0.5)}{5} \approx 0.1386 ]\)

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FAQs on Decay Formulas

1.What is the exponential decay formula?

The exponential decay formula is\( ( N(t) = N_0 \cdot e^{-kt} )\), where \(( N(t) )\) is the quantity at time  t , \(( N_0 )\) is the initial quantity, and \(( k ) \)is the decay constant.

2.What is the half-life formula?

The half-life formula is \(( t_{1/2} = \frac{\ln(2)}{\lambda} )\), where \(( \lambda )\) is the decay constant.

3.How do you calculate depreciation using decay formulas?

Depreciation can be calculated using the formula \(( V(t) = V_0 \cdot (1 - r)^t ),\) where \(( V(t) )\) is the value at time\( ( t ), ( V_0 ) \) is the initial value, and \( ( r )\) is the depreciation rate.

4.What is the decay constant?

The decay constant is a parameter in the exponential decay formula that indicates the rate at which a substance decays. It is denoted as  k  or\( ( \lambda ).\)

5.How do you find the remaining quantity after a certain time in exponential decay?

To find the remaining quantity, use the formula\( ( N(t) = N_0 \cdot e^{-kt} ), \)where you input the initial quantity \(( N_0 ),\) the decay constant\( ( k )\), and the time\( ( t ).\)

Glossary for Decay Formulas

  • Exponential Decay: A process where a quantity decreases at a rate proportional to its current value, often modeled with\( ( N(t) = N_0 \cdot e^{-kt} ).\)
  • Half-Life: The time required for a quantity to reduce to half of its initial value, calculated using\( ( t_{1/2} = \frac{\ln(2)}{\lambda} ).\)
  • Decay Constant: A parameter representing the rate of decay in an exponential decay model, denoted by\( ( k ) \)or\( ( \lambda ).\)
  • Depreciation: The reduction in value of an asset over time, often calculated using straight-line or exponential decay methods.
  • Natural Logarithm: A logarithm to the base\( ( e ), \)where\( ( e ) \)is an irrational constant approximately equal to 2.718.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.