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Interactive Audio Lesson
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Population Growth Applications
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Today, we're going to explore how we can use the compound interest formula to predict population growth. Can anyone tell me why this might be important?
It helps us understand how many people will be in our city in the future!
Exactly! So, if a city's population was 20,000 in 1997 and grows at 5% each year, how would we calculate its population in 2000?
We would use the compound interest formula: Population in future = P * (1 + r)^n.
Great! So we substitute P = 20,000, r = 0.05, and n = 3 years. What does that give us?
It should be 20,000 * (1.05)^3, which is 20,000 times roughly 1.157625, right?
That's right! And what do you find?
The population would be approximately 23,153.
Excellent work! This demonstrates how we can model population growth effectively.
Bacterial Growth Applications
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Shifting gears, letβs talk about bacteria. If we start with 500,000 bacteria and they grow at a rate of 2.5% per hour, how can we predict how many there will be in 2 hours?
We would apply the same formula as before, but for a different context!
Correct! If we let P = 500,000, r = 0.025, and n = 2, how would it look?
It would be 500,000 * (1 + 0.025)^2.
And what does that calculate to?
Thatβs approximately 500,000 * 1.050625, which is around 525,313!
Great job! Bacteria can multiply quickly, and this formula helps us anticipate those numbers.
Depreciation of Assets
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Letβs discuss depreciation. If you buy a TV for 21,000 and it depreciates by 5% annually, how can we find out its value next year?
We calculate 5% of 21,000, right?
Correct! What is that?
Thatβs 1,050. So the value will be 21,000 - 1,050, which equals 19,950.
Excellent! We could also write it using our formula: Value = P * (1 - r). How does that look?
So it would be 21,000 * (1 - 0.05), giving 21,000 * 0.95?
Exactly! Youβre all catching on! Key point: understanding depreciation is as important as understanding growth.
Introduction & Overview
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Quick Overview
Standard
The section outlines situations where the compound interest formula can be applied, such as population growth, bacterial growth, and the changing value of items due to appreciation or depreciation. Key examples illustrate how to calculate using this formula effectively.
Detailed
Applications of Compound Interest Formula
The compound interest formula is not just a theoretical tool; it finds significant application in various real-world scenarios. This section elaborates on several practical situations where the compound interest formula proves to be useful. Among these applications are the calculations related to population growth, bacterial growth rates, and the valuation changes of items due to economic factors.
Key Applications:
- Population Growth: Population often increases at a constant percentage rate annually, meaning calculations can leverage compound interest to project future populations. For example, if a city has a starting population with a known annual growth rate, the formula can be used to determine future populations over the years.
- Bacterial Growth: In biological contexts, bacterial populations multiply at exponential rates, making the compound interest methods suitable for estimating counts after several time periods.
- Value Changes of Items: The formula is effective in assessing the appreciation or depreciation of items such as machinery or electronics. For instance, an item bought at a certain price will hold less value over time due to depreciation.
Importance in Education:
Understanding the applications of the compound interest formula equips students with a valuable mathematical tool applicable not only in finance but also in biology, economics, and everyday decision-making.
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Applications of Compound Interest
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There are some situations where we could use the formula for calculation of amount in CI. Here are a few.
(i) Increase (or decrease) in population.
(ii) The growth of a bacteria if the rate of growth is known.
(iii) The value of an item, if its price increases or decreases in the intermediate years.
Detailed Explanation
In this chunk, we introduce various practical scenarios where the compound interest formula is useful. The first application is in calculating population changes over time. When a population grows or shrinks at a consistent rate, we can use compound interest calculations to project future numbers. The second application speaks to scientific growth, such as bacteria, where knowing the growth rate allows for predicting future populations of these cells. Finally, the third application addresses economic situations where the value of an item fluctuates over time, requiring compound interest calculations to determine future prices accurately.
Examples & Analogies
Consider a community that has a population of 20,000 people. If the population grows at a steady rate of 5% each year, by simply applying the compound interest formula, we can foresee that community's future size instead of estimating randomly, making our predictions much more reliable. Similarly, in a science lab, if researchers note that a certain bacteria type doubles in number each hour, they could chart out future growth using these calculations.
Example of Population Increase
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Example 9: The population of a city was 20,000 in the year 1997. It increased at the rate of 5% p.a. Find the population at the end of the year 2000.
Solution: There is 5% increase in population every year, so every new year has new population. Thus, we can say it is increasing in compounded form.
Population in the beginning of 1998 = 20000 (we treat this as the principal for the 1st year)
Increase at 5% = Γ20000=1000
Population in 1999 =20000 + 1000 = 21000
Increase at 5% = Γ21000=1050
Population in 2000 =21000 + 1050 =22050
Increase at 5% = Γ 22050 =1102.5
At the end of 2000 the population =22050 + 1102.5 = 23152.5.
Detailed Explanation
This example outlines how to calculate the population increase using the compound interest method. Starting with the initial population, we apply a percentage increase each year. We treat the new totals as the new principal for subsequent calculations. Through a step-by-step approach, we move from 1997 through to 2000, adjusting the population according to the arranged increase, ultimately calculating the population at the end of the period.
Examples & Analogies
Imagine if a small town of 20,000 residents grows by 5% each year due to new families moving in and people having children. Knowing the initial amount and the percentage allowed town planners to predict how many services they would need to accommodate this growing population and plan accordingly.
Example of Value Depreciation
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Example 10: A TV was bought at a price of βΉ21,000. After one year the value of the TV was depreciated by 5%. Find the value of the TV after one year.
Solution:
Principal = βΉ21,000
Reduction = 5% of βΉ21,000 per year = βΉ21,000 Γ 5/100 = βΉ1,050
Value at the end of 1 year = βΉ21,000 β βΉ1,050 = βΉ19,950.
Alternatively, we may directly get this as follows:
Value at the end of 1 year = βΉ21,000 Γ (100 - 5)/100 = βΉ21,000 Γ (95/100) = βΉ19,950.
Detailed Explanation
This example illustrates how compound interest can also be applied to situations where values depreciateβmeaning their worth drops over time. This is important in finance for understanding asset values. The calculation shows both the step-by-step reduction by applying the depreciation percentage and a shortcut method illustrating how to derive the new value directly by multiplying the original price with a depreciation factor.
Examples & Analogies
Think of buying a brand new television. After one year, it's no longer brand new; its value falls due to usage and new model releases. If you knew its depreciation rate, you could easily determine how much itβs worth for resale, ensuring you donβt lose money unintentionally if you decide to sell it.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Compound Interest: It is calculated on the principal plus accumulated interest, showing true financial growth.
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Population Growth: Often modeled using the compound interest formula to predict future populations.
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Asset Depreciation: Can also use compound-like calculations to determine future value reductions of assets.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example 1: Calculating future population using the formula: 20,000 * (1 + 0.05)^3 = 23,153.
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Example 2: Bacteria propagation after two hours would result in a total of approximately 525,313 from an initial population of 500,000 growing at 2.5%.
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Example 3: A TVβs depreciation from 21,000 to 19,950 after one year with a 5% depreciation rate.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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Values grow fast, and numbers last, with compound interest, you'll have a blast!
π Fascinating Stories
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Once there was a town that wanted to track its population. Each year they multiplied their citizens, and the numbers flourished just as the trees in spring, thanks to the magic of compound interest.
π― Super Acronyms
GIR - Growth In Rates captures the essence of compounding growth!
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Compound Interest
Definition:
Interest calculated on the initial principal which also includes all the accumulated interest from previous periods.
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Term: Population Growth
Definition:
The increase in the number of individuals in a population.
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Term: Depreciation
Definition:
Reduction in the value of an asset over time, due to wear and tear or obsolescence.
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Term: Exponential Growth
Definition:
A growth pattern where the quantity increases by a constant percentage over equal intervals.