Algebra II Exponential Growth and Decay 2015-11-19 www.njctl.org - PDF document

Slide 1 / 128 Slide 2 / 128 Algebra II Exponential Growth and Decay 2015-11-19 www.njctl.org Slide 3 / 128 Table of Contents Click on topic to go to that section. Simple Annual Interest Compound Interest The Constant, e Population Growth Half-Lives & Decay Applications PARCC Sample Questions Standards

Slide 4 / 128 Simple Annual Interest Return to Table of Contents Slide 5 / 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 6 / 128

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 Simple Interest Continuing with our example... If you are paid 10% simple interest per year on your initial investment of $1000, what would be your account balance after 3 years?

Slide 9 (Answer) / 128 Simple Interest Continuing with our example... If you are paid 10% simple interest per year on your initial investment of $1000, what would be your account balance after 3 years? Answer [This object is a pull tab] Slide 10 / 128 Simple Interest With simple interest, your interest is always calculated based on your initial investment, or starting principal. You can see that the $100 remains the same each year because the initial investment was $1000. Year Account Balance Interest 0 $1000 1 $1100 $100 2 $1200 $100 3 $1300 $100 4 $1400 $100 Slide 11 / 128 1 Which equation describes your ending bank balance if $1000 earns 5% simple annual interest for 7 years? A B C D E None of these

Slide 11 (Answer) / 128 1 Which equation describes your ending bank balance if $1000 earns 5% simple annual interest for 7 years? A Answer B B C D [This object is a pull tab] E None of these Slide 12 / 128 2 Which equation describes your ending bank balance if $500 earns 6% simple annual interest for 3 years? A B C D E None of these Slide 12 (Answer) / 128 2 Which equation describes your ending bank balance if $500 earns 6% simple annual interest for 3 years? A Answer A B C D [This object is a pull tab] E None of these

Slide 13 / 128 3 What will be your bank balance if you put $600 in your account and earn 5% simple annual interest for seven years? Slide 13 (Answer) / 128 3 What will be your bank balance if you put $600 in your account and earn 5% simple annual interest for seven years? Answer $810 [This object is a pull tab] Slide 14 / 128 4 What will be your bank balance if you put $1800 in your account and earn 4% simple annual interest for six years?

Slide 14 (Answer) / 128 4 What will be your bank balance if you put $1800 in your account and earn 4% simple annual interest for six years? Answer $2,232 [This object is a pull tab] Slide 15 / 128 5 What will be your bank balance if you put $3000 in your account and earn 2% simple annual interest for ten years? Slide 15 (Answer) / 128 5 What will be your bank balance if you put $3000 in your account and earn 2% simple annual interest for ten years? Answer $3,600 [This object is a pull tab]

Slide 16 / 128 6 If you are earning 7% simple annual interest and your goal is to have $3000 in your account after six years, how much will you have to initially deposit? Slide 16 (Answer) / 128 6 If you are earning 7% simple annual interest and your goal is to have $3000 in your account after six years, how much will you have to initially deposit? Answer $2112.68 [This object is a pull tab] Slide 17 / 128 7 If you are earning 10% simple annual interest and your goal is to have $3000 in your account after six years, how much will you have to initially deposit?

Slide 17 (Answer) / 128 7 If you are earning 10% simple annual interest and your goal is to have $3000 in your account after six years, how much will you have to initially deposit? Answer $1,875 [This object is a pull tab] Slide 18 / 128 8 If you are earning 2% interest and your goal is to have $3000 in your account after six years, how much will you have to initially deposit? Slide 18 (Answer) / 128 8 If you are earning 2% interest and your goal is to have $3000 in your account after six years, how much will you have to initially deposit? Answer $2,678.57 [This object is a pull tab]

Slide 19 / 128 Compound Interest Return to Table of Contents Slide 20 / 128 Compound Interest Compound interest can be thought of as "making interest on 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. Slide 21 / 128 Compound Interest Recalling our example from the first section, if you are paid 10% 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 year is $1100. With compound interest, the following years you will earn interest not only on your original $1000, but also the interest you've earned in prior years. This is called the compounding effect of interest. In the real world, it is better to be earning compounding interest than to be paying it...it grows very fast. That's why saving and investing early is so important. 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 22 / 128 Compound Interest Earning 10% compound interest, yield the table below. Notice, the interest is calculated based on the previous year's ending balance. Year Balance Interest 0 $1000 $100 1 $1100 $110 2 $1210 $121 3 $1331 $133.1 4 $1464.1 $146.41 5 $1610.51 Slide 23 / 128 Compound Interest Why does the amount of interest earned increase each year? Math Practice Instead of total interest of $500 (with simple interest), you earn $610.51. Why? Year Balance Interest 0 $1000 $100 1 $1100 $110 2 $1210 $121 3 $1331 $133.1 4 $1464.1 $146.41 5 $1610.51 Slide 24 / 128 Compound Interest Algebraically, After two years, the amount you earn would be given by But we can rewrite this expression to yield: What do you think your account balance will be after three years?

Slide 24 (Answer) / 128 Compound Interest Algebraically, The question on this slide addresses After two years, the amount you earn would be given by Math Practice MP.8 Additional Question to address MPs: What generalization can you make? But we can rewrite this expression to yield: (MP.8) [This object is a pull tab] What do you think your account balance will be after three years? Slide 25 / 128 Compound Interest Therefore, in general, your account balance with compound interest 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 26 / 128 Compound Interest Practice: Calculate the total account balance after investing $750 at 5% interested compounded yearly for 8 years.

Slide 26 (Answer) / 128 Compound Interest Practice: Calculate the total account balance after investing $750 at 5% interested compounded yearly for 8 years. Answer [This object is a pull tab] Slide 27 / 128 Compound Interest With annual interest, you receive your interest at the end of the time period, in this case the year. But, it's also possible for interest to compound within the year. For instance, your interest rate could be compounded quarterly. In this case, the interest is paid four time each year. The number of times per year that interest is compounded is called n. So, in this case, n = 4. Slide 28 / 128 Quarterly Compounding If n = 4, that means that we calculate and pay interest four times. It also means that only 1/4 of a year will have passed between each interest calculation. So, we have to divide the annual interest rate by 4 to get the interest rate for one calendar quarter: 10% divided by 4 = 2.5% Then we calculate the interest 4 times. 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 Quarterly Compounding So, even though the annual interest rate is the same: 10% In this case, you earn an extra $3.81 by quarterly compounding as compared to annual interest. You end with $1103.81 rather than $1100.00 Slide 30 / 128 Compounding In general, the result of compounding more frequently is given by the formula: where A is the total account balance P is the principal, or starting balance 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 Weekly Compounding What if we compounded weekly? What would the formula look like for that? Discuss and write a formula for that case. Then, determine your bank balance after one year, starting with $1000 and compounding weekly with 10% interest.