Slide 1 / 128 Slide 2 / 128 Algebra II Exponential Growth and Decay 2015-11-19 www.njctl.org Slide 3 / 128 Slide 4 / 128 Table of Contents Click on topic to go to that section. Simple Annual Interest Simple Compound Interest Annual Interest The Constant, e Population Growth Half-Lives & Decay Applications Return to Table of PARCC Sample Questions Contents Standards Slide 5 / 128 Slide 6 / 128 Simple Interest One important reason to invest your money is the opportunity to earn interest; which means your bank pays you money for keeping it in one of their accounts. The money you earn depends on the percentage interest you are paid per time period and how long your money is in the account. There are a few different ways interest can be calculated, but simple interest is earned based on the initial investment amount only.
Slide 7 / 128 Slide 8 / 128 Simple Interest In general, this becomes Where A is the accrued amount P is the principal (initial investment) r is the interest rate for that time period t is the time invested Slide 9 / 128 Slide 10 / 128 Simple Interest Simple Interest With simple interest, your interest is always calculated based on your Continuing with our example... initial investment, or starting principal. If you are paid 10% simple interest per year on your initial You can see that the $100 remains the same each year because the investment of $1000, what would be your account balance after 3 initial investment was $1000. years? Year Account Balance Interest 0 $1000 1 $1100 $100 2 $1200 $100 3 $1300 $100 4 $1400 $100 Slide 11 / 128 Slide 12 / 128 1 Which equation describes your ending bank balance if 2 Which equation describes your ending bank balance if $1000 earns 5% simple annual interest for 7 years? $500 earns 6% simple annual interest for 3 years? A A B B C C D D E None of these E None of these
Slide 13 / 128 Slide 14 / 128 3 What will be your bank balance if you put $600 in your 4 What will be your bank balance if you put $1800 in your account and earn 5% simple annual interest for seven account and earn 4% simple annual interest for six years? years? Slide 15 / 128 Slide 16 / 128 5 What will be your bank balance if you put $3000 in your 6 If you are earning 7% simple annual interest and your account and earn 2% simple annual interest for ten goal is to have $3000 in your account after six years, how years? much will you have to initially deposit? Slide 17 / 128 Slide 18 / 128 7 If you are earning 10% simple annual interest and your 8 If you are earning 2% interest and your goal is to have goal is to have $3000 in your account after six years, how $3000 in your account after six years, how much will you much will you have to initially deposit? have to initially deposit?
Slide 19 / 128 Slide 20 / 128 Compound Interest Compound interest can be thought of as "making interest on Compound Interest interest." Every time the interest is calculated, the current account balance is used to calculate the new interest. This means you are earning slightly more each time period (assuming the other factors are constant) compared to simple interest. Return to Table of Contents Slide 21 / 128 Slide 22 / 128 Compound Interest Compound Interest Recalling our example from the first section, if you are paid 10% Earning 10% compound interest, yield the table below. Notice, the simple interest per year on your balance of $1000, you would be paid $100 at the end of one year so your balance at the end of one interest is calculated based on the previous year's ending balance. year is $1100. With compound interest, the following years you will earn interest not Year Balance Interest only on your original $1000, but also the interest you've earned in 0 $1000 $100 prior years. 1 $1100 $110 This is called the compounding effect of interest. 2 $1210 $121 In the real world, it is better to be earning compounding interest than 3 $1331 $133.1 to be paying it...it grows very fast. That's why saving and investing 4 $1464.1 $146.41 early is so important. 5 $1610.51 At the same time, this is why it can be hard to get out of debt, when you're on the wrong side of compounding interest. Slide 23 / 128 Slide 24 / 128 Compound Interest Compound Interest Why does the amount of interest earned increase each year? Algebraically, Math Practice Instead of total interest of $500 (with simple interest), you earn After two years, the amount you earn would be given by $610.51. Why? Year Balance Interest But we can rewrite this expression to yield: 0 $1000 $100 1 $1100 $110 2 $1210 $121 3 $1331 $133.1 What do you think your account balance will be after three years? 4 $1464.1 $146.41 5 $1610.51
Slide 25 / 128 Slide 26 / 128 Compound Interest Compound Interest Practice: Calculate the total account balance after investing $750 at Therefore, in general, your account balance with compound interest 5% interested compounded yearly for 8 years. will be given by where A(t) is the amount of money after t time periods P is the principal, or initial investment t is the number of time periods (usually years) r is the interest rate per time period Slide 27 / 128 Slide 28 / 128 Quarterly Compounding Compound Interest If n = 4, that means that we calculate and pay interest four times. With annual interest, you receive your interest at the end of the time period, in this case the year. It also means that only 1/4 of a year will have passed between each interest calculation. But, it's also possible for interest to compound within the year. So, we have to divide the annual interest rate by 4 to get the For instance, your interest rate could be compounded quarterly. interest rate for one calendar quarter: 10% divided by 4 = 2.5% In this case, the interest is paid four time each year. Then we calculate the interest 4 times. The number of times per year that interest is compounded is called n. So, in this case, n = 4. The power of 4 reflects that the interest is calculated four times a year, each time at the annual rate divided by 4. Slide 29 / 128 Slide 30 / 128 Quarterly Compounding Compounding In general, the result of compounding more frequently is given by the formula: So, even though the annual interest rate is the same: 10% where In this case, you earn an extra $3.81 by quarterly compounding A is the total account balance as compared to annual interest. P is the principal, or starting balance You end with $1103.81 rather than $1100.00 r is the annual interest rate t is the number of years n is the number of times per year that the interest is compounded
Slide 31 / 128 Slide 32 / 128 Weekly Compounding Weekly Compounding What if we compounded weekly? When we compounded 4 times, we gained $3.08 more than if we What would the formula look like for that? had used simple interest Discuss and write a formula for that case. Compounding 52 times earns a bit more: $5.06 is $1.98 more than the $3.08 we earned by compounding 4 times. Then, determine your bank balance after one year, starting with Let's see what happens if we keep increasing our compounding $1000 and compounding weekly with 10% interest. Slide 33 / 128 Slide 34 / 128 Compound Interest Compound Interest Answer Fill in this chart for compounding: Math Practice Daily (365.25 times) Compounding Interest Balance Each second (31,557,600 times) Annually $100 $1100 Quarterly $103.81 $1103.81 Interest Balance Weekly $5.06 $1105.06 Annual $100 $1100 Daily Quarterly $103.81 $1103.81 Every second Weekly $5.06 $1105.06 Daily Every second Slide 35 / 128 Slide 36 / 128 9 Which equation describes your bank balance if $5250 10 Which equation describes your bank balance if $1000 earns 4% annual interest, compounded annually for 9 earns 6% annual interest, compounded quarterly, for 7 years? years? A A B B C C D D E None of these E None of these
Slide 37 / 128 Slide 38 / 128 11 Which equation describes your ending bank balance if 12 What will your bank balance be if you put $600 in your $500 earns 9% annual interest, compounded monthly, for account and earn 5% interest, compounded weekly, for 3 years? seven years? A B C D E None of these Slide 39 / 128 Slide 40 / 128 13 What will be your bank balance if you put $1800 in your 14 What will be your bank balance if you put $3000 in your account and earn 4% interest, compounded daily, for six account and earn 2% interest, compounded weekly, for years? ten years? Slide 41 / 128 Slide 42 / 128 15 If you are earning 7% interest, compounded daily, and 16 If you are earning 10% interest, compounded weekly, and your goal is to have $3000 in your account after six your goal is to have $3000 in your account after six years, how much will you have to initially deposit? years, how much will you have to initially deposit?
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