{"id":383,"date":"2016-10-11T22:56:31","date_gmt":"2016-10-11T22:56:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/math4libarts\/?post_type=chapter&#038;p=383"},"modified":"2021-02-05T23:58:37","modified_gmt":"2021-02-05T23:58:37","slug":"annuities","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/annuities\/","title":{"raw":"Savings Annuities","rendered":"Savings Annuities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Calculate the balance on an annuity after a specific amount of time<\/li>\r\n \t<li>Calculate interest earned and amount deposited in an annuity problem<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Savings Annuity<\/h2>\r\nFor most of us, we aren\u2019t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. This idea is called a <strong>savings annuity<\/strong>. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/02171735\/7027606047_cac49c3b79_z.jpg\"><img class=\"aligncenter size-full wp-image-737\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/02171735\/7027606047_cac49c3b79_z.jpg\" alt=\"Glass jar labeled &quot;Retirement.&quot; Inside are crumpled $100 bills\" width=\"640\" height=\"560\" \/><\/a>\r\n\r\nAn annuity can be described recursively in a fairly simple way. Recall that basic compound interest follows from the relationship\r\n<p style=\"text-align: center\">[latex]{{P}_{m}}=\\left(1+\\frac{r}{k}\\right){{P}_{m-1}}[\/latex]<\/p>\r\nFor a savings annuity, we simply need to add a deposit, <em>d<\/em>, to the account with each compounding period:\r\n<p style=\"text-align: center\">[latex]{{P}_{m}}=\\left(1+\\frac{r}{k}\\right){{P}_{m-1}}+d[\/latex]<\/p>\r\nTaking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general.\r\n<div class=\"textbox examples\">\r\n<h3>reading examples: the paper-and-pencil approach<\/h3>\r\nIn mathematics, we say <em>the best way to read a math text is with a paper and pencil.<\/em>The example below is challenging. Resist the temptation to gloss over it and cut straight to the formula given at the end. Instead, work through the example, line by line, with a pencil and paper. Do the math in each line to see how each subsequent, equivalent equation is formed. Ask questions if you can't see how one line was rewritten algebraically into the next. Reading math with pencil in hand helps make future math less challenging, increases your ability to apply logic in the real world by forming new thought patterns, and it really pays off at test time!\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall Algebraic Skills<\/h3>\r\nFor this example, you'll need to recall these skills in particular.\r\n\r\nThe distributive property: [latex]a\\left(b+c\\right)=ab+ac[\/latex]\r\n\r\nFactoring out a greatest common factor: [latex]m\\left(a+b\\right) + n\\left(a+b\\right)=\\left(a+b\\right)\\left(m+n\\right)[\/latex]\r\n\r\n<span style=\"font-size: 1rem;text-align: initial\">How to multiply like bases with exponents: [latex]a^{m-1}\\cdot a=a^{m-1+1}=a^{m}[\/latex]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose we will deposit $100 each month into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. Write an explicit formula that represents this scenario.\r\n[reveal-answer q=\"747493\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"747493\"]\r\n\r\nIn this example:\r\n<ul>\r\n \t<li><em>r<\/em> = 0.06 (6%)<\/li>\r\n \t<li><em>k<\/em> = 12 (12 compounds\/deposits per year)<\/li>\r\n \t<li><em>d<\/em> = $100 (our deposit per month)<\/li>\r\n<\/ul>\r\nWriting out the recursive equation gives\r\n\r\n[latex]{{P}_{m}}=\\left(1+\\frac{0.06}{12}\\right){{P}_{m-1}}+100=\\left(1.005\\right){{P}_{m-1}}+100[\/latex]\r\n\r\nAssuming we start with an empty account, we can begin using this relationship:\r\n\r\n[latex]P_0=0[\/latex]\r\n\r\n[latex]P_1=(1.005)P_0+100=100[\/latex]\r\n\r\n[latex]P_2=(1.005)P_1+100=(1.005)(100)+100=100(1.005)+100[\/latex]\r\n\r\n[latex]P_3=(1.005)P_2+100=(1.005)(100(1.005)+100)+100=100(1.005)^2+100(1.005)+100[\/latex]\r\n\r\nContinuing this pattern, after <em>m<\/em> deposits, we\u2019d have saved:\r\n\r\n[latex]P_m=100(1.005)^{m-1}+100(1.005)^{m-2} +L+100(1.005)+100[\/latex]\r\n\r\nIn other words, after <em>m<\/em> months, the first deposit will have earned compound interest for <em>m-<\/em>1 months. The second deposit will have earned interest for <em>m\u00ad<\/em>-2 months. The last month's deposit (L) would have earned only one month's worth of interest. The most recent deposit will have earned no interest yet.\r\n\r\nThis equation leaves a lot to be desired, though \u2013 it doesn\u2019t make calculating the ending balance any easier! To simplify things, multiply both sides of the equation by 1.005:\r\n\r\n[latex]1.005{{P}_{m}}=1.005\\left(100{{\\left(1.005\\right)}^{m-1}}+100{{\\left(1.005\\right)}^{m-2}}+\\cdots+100(1.005)+100\\right)[\/latex]\r\n\r\nDistributing on the right side of the equation gives\r\n\r\n[latex]1.005{{P}_{m}}=100{{\\left(1.005\\right)}^{m}}+100{{\\left(1.005\\right)}^{m-1}}+\\cdots+100{{(1.005)}^{2}}+100(1.005)[\/latex]\r\n\r\nNow we\u2019ll line this up with like terms from our original equation, and subtract each side\r\n\r\n[latex]\\begin{align}&amp;\\begin{matrix}1.005{{P}_{m}}&amp;=&amp;100{{\\left(1.005\\right)}^{m}}+&amp;100{{\\left(1.005\\right)}^{m-1}}+\\cdots+&amp;100(1.005)&amp;{}\\\\{{P}_{m}}&amp;=&amp;{}&amp;100{{\\left(1.005\\right)}^{m-1}}+\\cdots+&amp;100(1.005)&amp;+100\\\\\\end{matrix}\\\\&amp;\\\\\\end{align}[\/latex]\r\n\r\nAlmost all the terms cancel on the right hand side when we subtract, leaving\r\n\r\n[latex]1.005{{P}_{m}}-{{P}_{m}}=100{{\\left(1.005\\right)}^{m}}-100[\/latex]\r\n\r\nFactor [latex]P_m[\/latex] out of the terms on the left side.\r\n\r\n[latex]\\begin{array}{c}P_m(1.005-1)=100{{\\left(1.005\\right)}^{m}}-100\\\\(0.005)P_m=100{{\\left(1.005\\right)}^{m}}-100\\end{array}[\/latex]\r\n\r\nSolve for <em>P<sub>m<\/sub><\/em>\r\n\r\n[latex]\\begin{align}&amp;0.005{{P}_{m}}=100\\left({{\\left(1.005\\right)}^{m}}-1\\right)\\\\&amp;\\\\&amp;{{P}_{m}}=\\frac{100\\left({{\\left(1.005\\right)}^{m}}-1\\right)}{0.005}\\\\\\end{align}[\/latex]\r\n\r\nReplacing <em>m<\/em> months with 12<em>N<\/em>, where <em>N<\/em> is measured in years, gives\r\n\r\n[latex]{{P}_{N}}=\\frac{100\\left({{\\left(1.005\\right)}^{12N}}-1\\right)}{0.005}[\/latex]\r\n\r\nRecall 0.005 was <em>r\/k<\/em> and 100 was the deposit <em>d. <\/em>12 was <em>k<\/em>, the number of deposit each year.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nGeneralizing this result, we get the savings annuity formula.\r\n<div class=\"textbox\">\r\n<h3>Annuity Formula<\/h3>\r\n[latex]P_{N}=\\frac{d\\left(\\left(1+\\frac{r}{k}\\right)^{Nk}-1\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]\r\n<ul>\r\n \t<li><em>P<sub>N<\/sub><\/em> is the balance in the account after <em>N<\/em> years.<\/li>\r\n \t<li><em>d<\/em> is the regular deposit (the amount you deposit each year, each month, etc.)<\/li>\r\n \t<li><em>r <\/em> is the annual interest rate in decimal form.<\/li>\r\n \t<li><em>k <\/em>is the number of compounding periods in one year.<\/li>\r\n<\/ul>\r\nIf the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.\r\n\r\n<\/div>\r\nFor example, if the compounding frequency isn\u2019t stated:\r\n<ul>\r\n \t<li>If you make your deposits every month, use monthly compounding, <em>k<\/em> = 12.<\/li>\r\n \t<li>If you make your deposits every year, use yearly compounding, <em>k<\/em> = 1.<\/li>\r\n \t<li>If you make your deposits every quarter, use quarterly compounding, <em>k<\/em> = 4.<\/li>\r\n \t<li>Etc.<\/li>\r\n<\/ul>\r\n<div class=\"textbox\">\r\n<h3>When do you use this?<\/h3>\r\nAnnuities assume that you put money in the account <strong>on a regular schedule<\/strong> (every month, year, quarter, etc.) and let it sit there earning interest.\r\n\r\nCompound interest assumes that you put money in the account <strong>once<\/strong> and let it sit there earning interest.\r\n<ul>\r\n \t<li>Compound interest: One deposit<\/li>\r\n \t<li>Annuity: Many deposits.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall order of operations<\/h3>\r\nUsing the order of operations correctly is essential when using complicated formulas like the annuity formula.\r\n\r\nPEMDAS: First simplify like terms inside parentheses then handle exponents before multiplying or dividing. Do addition and subtraction outside of parentheses last.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nA traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?\r\n[reveal-answer q=\"261481\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"261481\"]\r\n\r\nIn this example,\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>d<\/em> = $100<\/td>\r\n<td>the monthly deposit<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>r<\/em> = 0.06<\/td>\r\n<td>6% annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>k<\/em> = 12<\/td>\r\n<td>since we\u2019re doing monthly deposits, we\u2019ll compound monthly<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>N<\/em> = 20<\/td>\r\n<td>\u00a0we want the amount after 20 years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPutting this into the equation:\r\n\r\n[latex]\\begin{align}&amp;{{P}_{20}}=\\frac{100\\left({{\\left(1+\\frac{0.06}{12}\\right)}^{20(12)}}-1\\right)}{\\left(\\frac{0.06}{12}\\right)}\\\\&amp;{{P}_{20}}=\\frac{100\\left({{\\left(1.005\\right)}^{240}}-1\\right)}{\\left(0.005\\right)}\\\\&amp;{{P}_{20}}=\\frac{100\\left(3.310-1\\right)}{\\left(0.005\\right)}\\\\&amp;{{P}_{20}}=\\frac{100\\left(2.310\\right)}{\\left(0.005\\right)}=\\$46200 \\\\\\end{align}[\/latex]\r\n\r\nThe account will grow to $46,200 after 20 years.\r\n\r\nNotice that you deposited into the account a total of $24,000 ($100 a month for 240 months). The difference between what you end up with and how much you put in is the interest earned. In this case it is $46,200 - $24,000 = $22,200.\r\n\r\n[\/hidden-answer]\r\n\r\nThis example is explained in detail here.\r\n\r\nhttps:\/\/youtu.be\/quLg4bRpxPA\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA conservative investment account pays 3% interest. If you deposit $5 a day into this account, how much will you have after 10 years? How much is from interest?\r\n[reveal-answer q=\"160692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"160692\"]\r\n<div>\r\n\r\n<em>d<\/em> = $5\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 the daily deposit\r\n\r\n<em>r<\/em> = 0.03 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3% annual rate\r\n\r\n<em>k<\/em> = 365 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 since we\u2019re doing daily deposits, we\u2019ll compound daily\r\n\r\n<em>N<\/em> = 10 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 we want the amount after 10 years\r\n\r\n[latex]P_{10}=\\frac{5\\left(\\left(1+\\frac{0.03}{365}\\right)^{365*10}-1\\right)}{\\frac{0.03}{365}}=21,282.07[\/latex]\r\n\r\nThe account will be worth $21,282.07 after 10 years. How much of that is from interest earned?\r\n\r\nYou deposited $5 per day for 10 years. That's [latex]5\\text{ dollars }\\ast 365\\text{ days } \\ast 10\\text{ years }=18250\\text{ dollars}[\/latex].\r\n\r\nSubtract the amount you deposited, $18,250, from the account balance, $21,282.07. You earned $3,32.07 from interest.\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]6691[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>\r\n<h2>Solving For The Deposit Amount<\/h2>\r\nFinancial planners typically recommend that you have a certain amount of savings upon retirement. \u00a0If you know the future value of the account, you can solve for the monthly contribution amount that will give you the desired result. In the next example, we will show you how this works.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nYou want to have $200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest. How much do you need to deposit each month to meet your retirement goal?\r\n[reveal-answer q=\"897790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"897790\"]\r\n\r\nIn this example, we\u2019re looking for <em>d<\/em>.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>r<\/em> = 0.08<\/td>\r\n<td>8% annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>k<\/em> = 12<\/td>\r\n<td>since we\u2019re depositing monthly<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>N<\/em> = 30<\/td>\r\n<td>30 years<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>P30<\/em> = $200,000<\/td>\r\n<td>The amount we want to have in 30 years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn this case, we\u2019re going to have to set up the equation, and solve for <em>d<\/em>.\r\n\r\n[latex]\\begin{align}&amp;200,000=\\frac{d\\left({{\\left(1+\\frac{0.08}{12}\\right)}^{30(12)}}-1\\right)}{\\left(\\frac{0.08}{12}\\right)}\\\\&amp;200,000=\\frac{d\\left({{\\left(1.00667\\right)}^{360}}-1\\right)}{\\left(0.00667\\right)}\\\\&amp;200,000=d(1491.57)\\\\&amp;d=\\frac{200,000}{1491.57}=\\$134.09 \\\\\\end{align}[\/latex]\r\n\r\nSo you would need to deposit $134.09 each month to have $200,000 in 30 years if your account earns 8% interest.\r\n\r\n[\/hidden-answer]\r\n\r\nView the solving of this problem\u00a0in the following video.\r\n\r\nhttps:\/\/youtu.be\/LB6pl7o0REc\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]6688[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<h2>Solving For Time<\/h2>\r\nWe can solve the annuities formula for time, like we did the compounding interest formula, by using logarithms. In the next example we will work through how this is done.\r\n<div class=\"textbox examples\">\r\n<h3>recall using a logarithm to solve for an exponent<\/h3>\r\nIn the following example, you'll need to recall that you can solve for a variable contained in an exponent by taking the log of both sides of the equation.\r\n\r\nEx. Solve for x in the following equation\r\n\r\n[latex]a = b^{mx}[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 we are solving for x, in the exponent\r\n\r\n[latex]log(a) = log\\left(b^{mx}\\right)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 take the log of both sides\r\n\r\n[latex]log(a)=mx\\ast log\\left(b\\right)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0use the exponent property\r\n\r\n[latex]\\frac{log(a)}{mb}=x[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0divide away all non-x terms to isolate x\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIf you invest $100 each month into an account earning 3% compounded monthly, how long will it take the account to grow to $10,000?\r\n[reveal-answer q=\"181207\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"181207\"]\r\n\r\nThis is a savings annuity problem since we are making regular deposits into the account.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>d<\/em> = $100<\/td>\r\n<td>the monthly deposit<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>r<\/em> = 0.03<\/td>\r\n<td>3% annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>k<\/em> = 12<\/td>\r\n<td>since we\u2019re doing monthly deposits, we\u2019ll compound monthly<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe don\u2019t know <em>N<\/em>, but we want <em>P<sub>N<\/sub><\/em> to be $10,000.\r\n\r\nPutting this into the equation:\r\n\r\n[latex]10,000=\\frac{100\\left({{\\left(1+\\frac{0.03}{12}\\right)}^{N(12)}}-1\\right)}{\\left(\\frac{0.03}{12}\\right)}[\/latex] \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Simplifying the fractions a bit\r\n\r\n[latex]10,000=\\frac{100\\left({{\\left(1.0025\\right)}^{12N}}-1\\right)}{0.0025}[\/latex]\r\n\r\nWe want to isolate the exponential term, 1.002512<em>N<\/em>, so multiply both sides by 0.0025\r\n\r\n[latex]25=100\\left({{\\left(1.0025\\right)}^{12N}}-1\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Divide both sides by 100\r\n\r\n[latex]0.25={{\\left(1.0025\\right)}^{12N}}-1[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Add 1 to both sides\r\n\r\n[latex]1.25={{\\left(1.0025\\right)}^{12N}}[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Now take the log of both sides\r\n\r\n[latex]\\log\\left(1.25\\right)=\\log\\left({{\\left(1.0025\\right)}^{12N}}\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Use the exponent property of logs\r\n\r\n[latex]\\log\\left(1.25\\right)=12N\\log\\left(1.0025\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Divide by 12log(1.0025)\r\n\r\n[latex]\\frac{\\log\\left(1.25\\right)}{12\\log\\left(1.0025\\right)}=N[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Approximating to a decimal\r\n\r\n<em>N<\/em> = 7.447 years\r\n\r\nIt will take about 7.447 years to grow the account to $10,000.\r\n\r\n[\/hidden-answer]\r\n\r\nThis example is demonstrated here:\r\n\r\nhttps:\/\/youtu.be\/F3QVyswCzRo\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Calculate the balance on an annuity after a specific amount of time<\/li>\n<li>Calculate interest earned and amount deposited in an annuity problem<\/li>\n<\/ul>\n<\/div>\n<h2>Savings Annuity<\/h2>\n<p>For most of us, we aren\u2019t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. This idea is called a <strong>savings annuity<\/strong>. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/02171735\/7027606047_cac49c3b79_z.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-737\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/02171735\/7027606047_cac49c3b79_z.jpg\" alt=\"Glass jar labeled &quot;Retirement.&quot; Inside are crumpled $100 bills\" width=\"640\" height=\"560\" \/><\/a><\/p>\n<p>An annuity can be described recursively in a fairly simple way. Recall that basic compound interest follows from the relationship<\/p>\n<p style=\"text-align: center\">[latex]{{P}_{m}}=\\left(1+\\frac{r}{k}\\right){{P}_{m-1}}[\/latex]<\/p>\n<p>For a savings annuity, we simply need to add a deposit, <em>d<\/em>, to the account with each compounding period:<\/p>\n<p style=\"text-align: center\">[latex]{{P}_{m}}=\\left(1+\\frac{r}{k}\\right){{P}_{m-1}}+d[\/latex]<\/p>\n<p>Taking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general.<\/p>\n<div class=\"textbox examples\">\n<h3>reading examples: the paper-and-pencil approach<\/h3>\n<p>In mathematics, we say <em>the best way to read a math text is with a paper and pencil.<\/em>The example below is challenging. Resist the temptation to gloss over it and cut straight to the formula given at the end. Instead, work through the example, line by line, with a pencil and paper. Do the math in each line to see how each subsequent, equivalent equation is formed. Ask questions if you can&#8217;t see how one line was rewritten algebraically into the next. Reading math with pencil in hand helps make future math less challenging, increases your ability to apply logic in the real world by forming new thought patterns, and it really pays off at test time!<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall Algebraic Skills<\/h3>\n<p>For this example, you&#8217;ll need to recall these skills in particular.<\/p>\n<p>The distributive property: [latex]a\\left(b+c\\right)=ab+ac[\/latex]<\/p>\n<p>Factoring out a greatest common factor: [latex]m\\left(a+b\\right) + n\\left(a+b\\right)=\\left(a+b\\right)\\left(m+n\\right)[\/latex]<\/p>\n<p><span style=\"font-size: 1rem;text-align: initial\">How to multiply like bases with exponents: [latex]a^{m-1}\\cdot a=a^{m-1+1}=a^{m}[\/latex]<\/span><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. Write an explicit formula that represents this scenario.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q747493\">Show Solution<\/span><\/p>\n<div id=\"q747493\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this example:<\/p>\n<ul>\n<li><em>r<\/em> = 0.06 (6%)<\/li>\n<li><em>k<\/em> = 12 (12 compounds\/deposits per year)<\/li>\n<li><em>d<\/em> = $100 (our deposit per month)<\/li>\n<\/ul>\n<p>Writing out the recursive equation gives<\/p>\n<p>[latex]{{P}_{m}}=\\left(1+\\frac{0.06}{12}\\right){{P}_{m-1}}+100=\\left(1.005\\right){{P}_{m-1}}+100[\/latex]<\/p>\n<p>Assuming we start with an empty account, we can begin using this relationship:<\/p>\n<p>[latex]P_0=0[\/latex]<\/p>\n<p>[latex]P_1=(1.005)P_0+100=100[\/latex]<\/p>\n<p>[latex]P_2=(1.005)P_1+100=(1.005)(100)+100=100(1.005)+100[\/latex]<\/p>\n<p>[latex]P_3=(1.005)P_2+100=(1.005)(100(1.005)+100)+100=100(1.005)^2+100(1.005)+100[\/latex]<\/p>\n<p>Continuing this pattern, after <em>m<\/em> deposits, we\u2019d have saved:<\/p>\n<p>[latex]P_m=100(1.005)^{m-1}+100(1.005)^{m-2} +L+100(1.005)+100[\/latex]<\/p>\n<p>In other words, after <em>m<\/em> months, the first deposit will have earned compound interest for <em>m-<\/em>1 months. The second deposit will have earned interest for <em>m\u00ad<\/em>-2 months. The last month&#8217;s deposit (L) would have earned only one month&#8217;s worth of interest. The most recent deposit will have earned no interest yet.<\/p>\n<p>This equation leaves a lot to be desired, though \u2013 it doesn\u2019t make calculating the ending balance any easier! To simplify things, multiply both sides of the equation by 1.005:<\/p>\n<p>[latex]1.005{{P}_{m}}=1.005\\left(100{{\\left(1.005\\right)}^{m-1}}+100{{\\left(1.005\\right)}^{m-2}}+\\cdots+100(1.005)+100\\right)[\/latex]<\/p>\n<p>Distributing on the right side of the equation gives<\/p>\n<p>[latex]1.005{{P}_{m}}=100{{\\left(1.005\\right)}^{m}}+100{{\\left(1.005\\right)}^{m-1}}+\\cdots+100{{(1.005)}^{2}}+100(1.005)[\/latex]<\/p>\n<p>Now we\u2019ll line this up with like terms from our original equation, and subtract each side<\/p>\n<p>[latex]\\begin{align}&\\begin{matrix}1.005{{P}_{m}}&=&100{{\\left(1.005\\right)}^{m}}+&100{{\\left(1.005\\right)}^{m-1}}+\\cdots+&100(1.005)&{}\\\\{{P}_{m}}&=&{}&100{{\\left(1.005\\right)}^{m-1}}+\\cdots+&100(1.005)&+100\\\\\\end{matrix}\\\\&\\\\\\end{align}[\/latex]<\/p>\n<p>Almost all the terms cancel on the right hand side when we subtract, leaving<\/p>\n<p>[latex]1.005{{P}_{m}}-{{P}_{m}}=100{{\\left(1.005\\right)}^{m}}-100[\/latex]<\/p>\n<p>Factor [latex]P_m[\/latex] out of the terms on the left side.<\/p>\n<p>[latex]\\begin{array}{c}P_m(1.005-1)=100{{\\left(1.005\\right)}^{m}}-100\\\\(0.005)P_m=100{{\\left(1.005\\right)}^{m}}-100\\end{array}[\/latex]<\/p>\n<p>Solve for <em>P<sub>m<\/sub><\/em><\/p>\n<p>[latex]\\begin{align}&0.005{{P}_{m}}=100\\left({{\\left(1.005\\right)}^{m}}-1\\right)\\\\&\\\\&{{P}_{m}}=\\frac{100\\left({{\\left(1.005\\right)}^{m}}-1\\right)}{0.005}\\\\\\end{align}[\/latex]<\/p>\n<p>Replacing <em>m<\/em> months with 12<em>N<\/em>, where <em>N<\/em> is measured in years, gives<\/p>\n<p>[latex]{{P}_{N}}=\\frac{100\\left({{\\left(1.005\\right)}^{12N}}-1\\right)}{0.005}[\/latex]<\/p>\n<p>Recall 0.005 was <em>r\/k<\/em> and 100 was the deposit <em>d. <\/em>12 was <em>k<\/em>, the number of deposit each year.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Generalizing this result, we get the savings annuity formula.<\/p>\n<div class=\"textbox\">\n<h3>Annuity Formula<\/h3>\n<p>[latex]P_{N}=\\frac{d\\left(\\left(1+\\frac{r}{k}\\right)^{Nk}-1\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]<\/p>\n<ul>\n<li><em>P<sub>N<\/sub><\/em> is the balance in the account after <em>N<\/em> years.<\/li>\n<li><em>d<\/em> is the regular deposit (the amount you deposit each year, each month, etc.)<\/li>\n<li><em>r <\/em> is the annual interest rate in decimal form.<\/li>\n<li><em>k <\/em>is the number of compounding periods in one year.<\/li>\n<\/ul>\n<p>If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.<\/p>\n<\/div>\n<p>For example, if the compounding frequency isn\u2019t stated:<\/p>\n<ul>\n<li>If you make your deposits every month, use monthly compounding, <em>k<\/em> = 12.<\/li>\n<li>If you make your deposits every year, use yearly compounding, <em>k<\/em> = 1.<\/li>\n<li>If you make your deposits every quarter, use quarterly compounding, <em>k<\/em> = 4.<\/li>\n<li>Etc.<\/li>\n<\/ul>\n<div class=\"textbox\">\n<h3>When do you use this?<\/h3>\n<p>Annuities assume that you put money in the account <strong>on a regular schedule<\/strong> (every month, year, quarter, etc.) and let it sit there earning interest.<\/p>\n<p>Compound interest assumes that you put money in the account <strong>once<\/strong> and let it sit there earning interest.<\/p>\n<ul>\n<li>Compound interest: One deposit<\/li>\n<li>Annuity: Many deposits.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall order of operations<\/h3>\n<p>Using the order of operations correctly is essential when using complicated formulas like the annuity formula.<\/p>\n<p>PEMDAS: First simplify like terms inside parentheses then handle exponents before multiplying or dividing. Do addition and subtraction outside of parentheses last.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q261481\">Show Solution<\/span><\/p>\n<div id=\"q261481\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this example,<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>d<\/em> = $100<\/td>\n<td>the monthly deposit<\/td>\n<\/tr>\n<tr>\n<td><em>r<\/em> = 0.06<\/td>\n<td>6% annual rate<\/td>\n<\/tr>\n<tr>\n<td><em>k<\/em> = 12<\/td>\n<td>since we\u2019re doing monthly deposits, we\u2019ll compound monthly<\/td>\n<\/tr>\n<tr>\n<td><em>N<\/em> = 20<\/td>\n<td>\u00a0we want the amount after 20 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Putting this into the equation:<\/p>\n<p>[latex]\\begin{align}&{{P}_{20}}=\\frac{100\\left({{\\left(1+\\frac{0.06}{12}\\right)}^{20(12)}}-1\\right)}{\\left(\\frac{0.06}{12}\\right)}\\\\&{{P}_{20}}=\\frac{100\\left({{\\left(1.005\\right)}^{240}}-1\\right)}{\\left(0.005\\right)}\\\\&{{P}_{20}}=\\frac{100\\left(3.310-1\\right)}{\\left(0.005\\right)}\\\\&{{P}_{20}}=\\frac{100\\left(2.310\\right)}{\\left(0.005\\right)}=\\$46200 \\\\\\end{align}[\/latex]<\/p>\n<p>The account will grow to $46,200 after 20 years.<\/p>\n<p>Notice that you deposited into the account a total of $24,000 ($100 a month for 240 months). The difference between what you end up with and how much you put in is the interest earned. In this case it is $46,200 &#8211; $24,000 = $22,200.<\/p>\n<\/div>\n<\/div>\n<p>This example is explained in detail here.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Saving Annuities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/quLg4bRpxPA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A conservative investment account pays 3% interest. If you deposit $5 a day into this account, how much will you have after 10 years? How much is from interest?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q160692\">Show Solution<\/span><\/p>\n<div id=\"q160692\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p><em>d<\/em> = $5\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 the daily deposit<\/p>\n<p><em>r<\/em> = 0.03 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3% annual rate<\/p>\n<p><em>k<\/em> = 365 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 since we\u2019re doing daily deposits, we\u2019ll compound daily<\/p>\n<p><em>N<\/em> = 10 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 we want the amount after 10 years<\/p>\n<p>[latex]P_{10}=\\frac{5\\left(\\left(1+\\frac{0.03}{365}\\right)^{365*10}-1\\right)}{\\frac{0.03}{365}}=21,282.07[\/latex]<\/p>\n<p>The account will be worth $21,282.07 after 10 years. How much of that is from interest earned?<\/p>\n<p>You deposited $5 per day for 10 years. That&#8217;s [latex]5\\text{ dollars }\\ast 365\\text{ days } \\ast 10\\text{ years }=18250\\text{ dollars}[\/latex].<\/p>\n<p>Subtract the amount you deposited, $18,250, from the account balance, $21,282.07. You earned $3,32.07 from interest.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6691\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6691&theme=oea&iframe_resize_id=ohm6691&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div>\n<h2>Solving For The Deposit Amount<\/h2>\n<p>Financial planners typically recommend that you have a certain amount of savings upon retirement. \u00a0If you know the future value of the account, you can solve for the monthly contribution amount that will give you the desired result. In the next example, we will show you how this works.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>You want to have $200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest. How much do you need to deposit each month to meet your retirement goal?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q897790\">Show Solution<\/span><\/p>\n<div id=\"q897790\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this example, we\u2019re looking for <em>d<\/em>.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>r<\/em> = 0.08<\/td>\n<td>8% annual rate<\/td>\n<\/tr>\n<tr>\n<td><em>k<\/em> = 12<\/td>\n<td>since we\u2019re depositing monthly<\/td>\n<\/tr>\n<tr>\n<td><em>N<\/em> = 30<\/td>\n<td>30 years<\/td>\n<\/tr>\n<tr>\n<td><em>P30<\/em> = $200,000<\/td>\n<td>The amount we want to have in 30 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In this case, we\u2019re going to have to set up the equation, and solve for <em>d<\/em>.<\/p>\n<p>[latex]\\begin{align}&200,000=\\frac{d\\left({{\\left(1+\\frac{0.08}{12}\\right)}^{30(12)}}-1\\right)}{\\left(\\frac{0.08}{12}\\right)}\\\\&200,000=\\frac{d\\left({{\\left(1.00667\\right)}^{360}}-1\\right)}{\\left(0.00667\\right)}\\\\&200,000=d(1491.57)\\\\&d=\\frac{200,000}{1491.57}=\\$134.09 \\\\\\end{align}[\/latex]<\/p>\n<p>So you would need to deposit $134.09 each month to have $200,000 in 30 years if your account earns 8% interest.<\/p>\n<\/div>\n<\/div>\n<p>View the solving of this problem\u00a0in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Savings annuities - solving for the deposit\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LB6pl7o0REc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6688\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6688&theme=oea&iframe_resize_id=ohm6688&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<h2>Solving For Time<\/h2>\n<p>We can solve the annuities formula for time, like we did the compounding interest formula, by using logarithms. In the next example we will work through how this is done.<\/p>\n<div class=\"textbox examples\">\n<h3>recall using a logarithm to solve for an exponent<\/h3>\n<p>In the following example, you&#8217;ll need to recall that you can solve for a variable contained in an exponent by taking the log of both sides of the equation.<\/p>\n<p>Ex. Solve for x in the following equation<\/p>\n<p>[latex]a = b^{mx}[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 we are solving for x, in the exponent<\/p>\n<p>[latex]log(a) = log\\left(b^{mx}\\right)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 take the log of both sides<\/p>\n<p>[latex]log(a)=mx\\ast log\\left(b\\right)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0use the exponent property<\/p>\n<p>[latex]\\frac{log(a)}{mb}=x[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0divide away all non-x terms to isolate x<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>If you invest $100 each month into an account earning 3% compounded monthly, how long will it take the account to grow to $10,000?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q181207\">Show Solution<\/span><\/p>\n<div id=\"q181207\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is a savings annuity problem since we are making regular deposits into the account.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>d<\/em> = $100<\/td>\n<td>the monthly deposit<\/td>\n<\/tr>\n<tr>\n<td><em>r<\/em> = 0.03<\/td>\n<td>3% annual rate<\/td>\n<\/tr>\n<tr>\n<td><em>k<\/em> = 12<\/td>\n<td>since we\u2019re doing monthly deposits, we\u2019ll compound monthly<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We don\u2019t know <em>N<\/em>, but we want <em>P<sub>N<\/sub><\/em> to be $10,000.<\/p>\n<p>Putting this into the equation:<\/p>\n<p>[latex]10,000=\\frac{100\\left({{\\left(1+\\frac{0.03}{12}\\right)}^{N(12)}}-1\\right)}{\\left(\\frac{0.03}{12}\\right)}[\/latex] \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Simplifying the fractions a bit<\/p>\n<p>[latex]10,000=\\frac{100\\left({{\\left(1.0025\\right)}^{12N}}-1\\right)}{0.0025}[\/latex]<\/p>\n<p>We want to isolate the exponential term, 1.002512<em>N<\/em>, so multiply both sides by 0.0025<\/p>\n<p>[latex]25=100\\left({{\\left(1.0025\\right)}^{12N}}-1\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Divide both sides by 100<\/p>\n<p>[latex]0.25={{\\left(1.0025\\right)}^{12N}}-1[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Add 1 to both sides<\/p>\n<p>[latex]1.25={{\\left(1.0025\\right)}^{12N}}[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Now take the log of both sides<\/p>\n<p>[latex]\\log\\left(1.25\\right)=\\log\\left({{\\left(1.0025\\right)}^{12N}}\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Use the exponent property of logs<\/p>\n<p>[latex]\\log\\left(1.25\\right)=12N\\log\\left(1.0025\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Divide by 12log(1.0025)<\/p>\n<p>[latex]\\frac{\\log\\left(1.25\\right)}{12\\log\\left(1.0025\\right)}=N[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Approximating to a decimal<\/p>\n<p><em>N<\/em> = 7.447 years<\/p>\n<p>It will take about 7.447 years to grow the account to $10,000.<\/p>\n<\/div>\n<\/div>\n<p>This example is demonstrated here:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Use logs to find the time it takes an annuity to grow\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/F3QVyswCzRo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-383\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Annuities. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>Project<\/strong>: Math in Society. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Retirement. <strong>Authored by<\/strong>: Tax Credits. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/flic.kr\/p\/bH1jrv\">https:\/\/flic.kr\/p\/bH1jrv<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Savings Annuities. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/quLg4bRpxPA\">https:\/\/youtu.be\/quLg4bRpxPA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Savings annuities - solving for the deposit. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/LB6pl7o0REc\">https:\/\/youtu.be\/LB6pl7o0REc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 6691, 6688. <strong>Authored by<\/strong>: Lippman,David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li> Determining The Value of an Annuity. <strong>Authored by<\/strong>: Sousa, James (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/DWFezRwYp0I\">https:\/\/youtu.be\/DWFezRwYp0I<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Determining The Value of an Annuity on the TI84. <strong>Authored by<\/strong>: Sousa, James (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/OKCZRZ-hWH8\">https:\/\/youtu.be\/OKCZRZ-hWH8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":20,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Annuities\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Retirement\",\"author\":\"Tax Credits\",\"organization\":\"\",\"url\":\"https:\/\/flic.kr\/p\/bH1jrv\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Savings Annuities\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/quLg4bRpxPA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Savings annuities - solving for the deposit\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/LB6pl7o0REc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 6691, 6688\",\"author\":\"Lippman,David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\" Determining The Value of an Annuity\",\"author\":\"Sousa, James 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