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 9 (Answer) / 128 Simple Interest Simple Interest Continuing with our example... Continuing with our example... If you are paid 10% simple interest per year on your initial If you are paid 10% simple interest per year on your initial investment of $1000, what would be your account balance after 3 investment of $1000, what would be your account balance after 3 years? years? Answer [This object is a pull tab] Slide 10 / 128 Slide 11 / 128 Simple Interest 1 Which equation describes your ending bank balance if $1000 earns 5% simple annual interest for 7 years? With simple interest, your interest is always calculated based on your initial investment, or starting principal. A You can see that the $100 remains the same each year because the initial investment was $1000. B Year Account Balance Interest C 0 $1000 1 $1100 $100 D 2 $1200 $100 E None of these 3 $1300 $100 4 $1400 $100
Slide 11 (Answer) / 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 Answer B B B C C D D [This object is a pull tab] E None of these E None of these Slide 12 (Answer) / 128 Slide 13 / 128 2 Which equation describes your ending bank balance if 3 What will be your bank balance if you put $600 in your $500 earns 6% simple annual interest for 3 years? account and earn 5% simple annual interest for seven years? A Answer A B C D [This object is a pull tab] E None of these Slide 13 (Answer) / 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? Answer $810 [This object is a pull tab]
Slide 14 (Answer) / 128 Slide 15 / 128 4 What will be your bank balance if you put $1800 in your 5 What will be your bank balance if you put $3000 in your account and earn 4% simple annual interest for six account and earn 2% simple annual interest for ten years? years? Answer $2,232 [This object is a pull tab] Slide 15 (Answer) / 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? Answer $3,600 [This object is a pull tab] Slide 16 (Answer) / 128 Slide 17 / 128 6 If you are earning 7% simple annual interest and your 7 If you are earning 10% simple annual interest and your goal is to have $3000 in your account after six years, how goal is to have $3000 in your account after six years, how much will you have to initially deposit? much will you have to initially deposit? Answer $2112.68 [This object is a pull tab]
Slide 17 (Answer) / 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? Answer $1,875 [This object is a pull tab] Slide 18 (Answer) / 128 Slide 19 / 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? Compound Interest Answer $2,678.57 [This object is a pull tab] Return to Table of Contents Slide 20 / 128 Slide 21 / 128 Compound Interest 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 Compound interest can be thought of as "making interest on year is $1100. interest." Every time the interest is calculated, the current account balance is used to calculate the new interest. This means you are With compound interest, the following years you will earn interest not earning slightly more each time period (assuming the other factors only on your original $1000, but also the interest you've earned in are constant) compared to simple interest. 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 Slide 23 / 128 Compound Interest Compound Interest Why does the amount of interest earned increase each year? Earning 10% compound interest, yield the table below. Notice, the Math Practice interest is calculated based on the previous year's ending balance. Instead of total interest of $500 (with simple interest), you earn $610.51. Why? Year Balance Interest Year Balance Interest 0 $1000 $100 0 $1000 $100 1 $1100 $110 1 $1100 $110 2 $1210 $121 2 $1210 $121 3 $1331 $133.1 3 $1331 $133.1 4 $1464.1 $146.41 4 $1464.1 $146.41 5 $1610.51 5 $1610.51 Slide 24 / 128 Slide 24 (Answer) / 128 Compound Interest Compound Interest Algebraically, Algebraically, The question on this slide addresses After two years, the amount you earn would be given by 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: 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? What do you think your account balance will be after three years? 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 26 (Answer) / 128 Slide 27 / 128 Compound Interest Compound Interest With annual interest, you receive your interest at the end of the Practice: Calculate the total account balance after investing $750 at time period, in this case the year. 5% interested compounded yearly for 8 years. But, it's also possible for interest to compound within the year. Answer 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. [This object is a pull tab] So, in this case, n = 4. Slide 28 / 128 Slide 29 / 128 Quarterly Compounding 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 So, even though the annual interest rate is the same: 10% interest rate for one calendar quarter: 10% divided by 4 = 2.5% In this case, you earn an extra $3.81 by quarterly compounding Then we calculate the interest 4 times. as compared to annual interest. You end with $1103.81 rather than $1100.00 The power of 4 reflects that the interest is calculated four times a year, each time at the annual rate divided by 4. Slide 30 / 128 Slide 31 / 128 Weekly Compounding Compounding In general, the result of compounding more frequently is given by What if we compounded weekly? the formula: What would the formula look like for that? Discuss and write a formula for that case. where A is the total account balance Then, determine your bank balance after one year, starting with $1000 and compounding weekly with 10% interest. 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
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