{"id":3818,"date":"2025-09-09T17:00:14","date_gmt":"2025-09-09T17:00:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=3818"},"modified":"2026-03-17T17:55:38","modified_gmt":"2026-03-17T17:55:38","slug":"exponential-functions-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-functions-apply-it-2\/","title":{"raw":"Exponential Functions: Apply It 2","rendered":"Exponential Functions: Apply It 2"},"content":{"raw":"

Compound Interest<\/h2>\r\nSavings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest<\/strong>. The term compounding<\/em> refers to interest earned not only on the original value, but on the accumulated value of the account.\r\n\r\nThe annual percentage rate (APR)<\/strong> of an account, also called the nominal rate<\/strong>, is the yearly interest rate earned by an investment account. The term\u00a0nominal<\/em>\u00a0is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater<\/em> than the nominal rate! This is a powerful tool for investing.\r\n\r\n
\r\n

compound interest formula<\/h3>\r\nCompound interest<\/strong> can be calculated using the formula\r\n

[latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}[\/latex]<\/p>\r\nwhere\r\n

    \r\n \t
  • [latex]A(t)[\/latex] is the accumulated value of the account<\/li>\r\n \t
  • [latex]t[\/latex] is measured in years<\/li>\r\n \t
  • [latex]P[\/latex]\u00a0is the starting amount of the account, often called the principal, or more generally present value<\/li>\r\n \t
  • [latex]r[\/latex]\u00a0is the annual percentage rate (APR) expressed as a decimal<\/li>\r\n \t
  • [latex]n[\/latex]\u00a0is the number of times compounded in a year<\/li>\r\n<\/ul>\r\n<\/section>
    If we invest [latex]$3,000[\/latex] in an investment account paying [latex]3\\%[\/latex] interest compounded quarterly, how much will the account be worth in [latex]10[\/latex] years?[reveal-answer q=\"476919\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"476919\"]Let's break it down.\r\n
      \r\n \t
    • Because we are starting with [latex]$3,000[\/latex], [latex]P\u00a0= 3000[\/latex].<\/li>\r\n \t
    • Our interest rate is [latex]3\\%[\/latex], so [latex]r\u00a0=\u00a00.03[\/latex].<\/li>\r\n \t
    • Because we are compounding quarterly, we are compounding [latex]4[\/latex] times per year, so [latex]n\u00a0= 4[\/latex].<\/li>\r\n \t
    • We want to know the value of the account in [latex]10[\/latex] years, so we are looking for [latex]A(10)[\/latex], the value when [latex]t = 10[\/latex].<\/li>\r\n<\/ul>\r\n

      [latex]\\begin{array}{llllll}A\\left(t\\right)\\hfill & =P\\left(1+\\frac{r}{n}\\right)^{nt}\\hfill & \\text{Use the compound interest formula}. \\\\ A\\left(10\\right)\\hfill & =3000\\left(1+\\frac{0.03}{4}\\right)^{4\\cdot 10}\\hfill & \\text{Substitute using given values}. \\\\ \\text{ }\\hfill & \\approx 4045.05\\hfill & \\text{Round to two decimal places}.\\end{array}[\/latex]<\/p>\r\nThe account will be worth about [latex]$4,045.05[\/latex] in [latex]10[\/latex] years.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

      When calculating the value of an exponential function such as the compound interest formula, be careful when entering your calculation into a calculator. Use as many parentheses as needed to ensure your intent is clear. The calculator will apply order of operations as it is typed, which can cause an incorrect calculation for your equation.\r\n[latex]\\\\[\/latex]\r\nEx. To find the accumulated investment in the example above, you must calculate\r\n
      [latex]A\\left(10\\right)=3000\\left(1+\\frac{0.03}{4}\\right)^{4\\cdot 10}[\/latex].<\/center>[latex]\\\\[\/latex]This would be entered in most scientific or graphing calculators as
      [latex]3000\\left(1+(0.03\/4)\\right)\\wedge\\left(4\\star10\\right)[\/latex].<\/center><\/section>
      [ohm_question hide_question_numbers=1]321417[\/ohm_question]<\/section>
      A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child\u2019s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to [latex]$40,000[\/latex] over [latex]18[\/latex] years. She believes the account will earn [latex]6\\%[\/latex] compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?[reveal-answer q=\"550421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"550421\"]Let's break it down.\r\n
        \r\n \t
      • The nominal interest rate is [latex]6\\%[\/latex], so [latex]r\u00a0= 0.06[\/latex].<\/li>\r\n \t
      • Interest is compounded twice a year, so [latex]n = 2.[\/latex]<\/li>\r\n \t
      • We want to find the initial investment\u00a0[latex]P[\/latex] needed so that the value of the account will be worth [latex]$40,000[\/latex] in [latex]18[\/latex] years. So, [latex]A = $40,000[\/latex] and [latex]t = 18[\/latex]. <\/span><\/li>\r\n<\/ul>\r\nSubstitute the given values into the compound interest formula and solve for <\/span>[latex]P[\/latex].<\/span>\r\n

        [latex]\\begin{array}{c}A\\left(t\\right)\\hfill & =P{\\left(1+\\frac{r}{n}\\right)}^{nt}\\hfill & \\text{Use the compound interest formula}.\\hfill \\\\ 40,000\\hfill & =P{\\left(1+\\frac{0.06}{2}\\right)}^{2\\left(18\\right)}\\hfill & \\text{Substitute using given values }A\\text{, }r, n\\text{, and }t.\\hfill \\\\ 40,000\\hfill & =P{\\left(1.03\\right)}^{36}\\hfill & \\text{Simplify}.\\hfill \\\\ \\frac{40,000}{{\\left(1.03\\right)}^{36}}\\hfill & =P\\hfill & \\text{Isolate }P.\\hfill \\\\ P\\hfill & \\approx 13,801\\hfill & \\text{Divide and round to the nearest dollar}.\\hfill \\end{array}[\/latex]<\/p>\r\nLily will need to invest [latex]$13,801[\/latex] to have [latex]$40,000[\/latex] in [latex]18[\/latex] years.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

        [ohm_question hide_question_numbers=1]321419[\/ohm_question]<\/section>
        \r\n

        Annual Percentage Yield (APY)<\/h3>\r\nAPY<\/strong> stands for Annual Percentage Yield<\/strong>, and it represents the real rate of return earned on an investment, taking into account the effect of compounding interest.\r\n

        [latex]\\text{APY} = (1+\\frac{r}{n})^n - 1 [\/latex]<\/p>\r\nwhere\r\n

          \r\n \t
        • [latex]r[\/latex] is the nominal interest rate (or the APR.<\/li>\r\n \t
        • [latex]n[\/latex] is the number of compounding periods per year.<\/li>\r\n<\/ul>\r\n<\/section>
          Recall the example: We invest [latex]$3,000[\/latex] in an investment account paying [latex]3\\%[\/latex] interest compounded quarterly. What is the APY?[reveal-answer q=\"56101\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"56101\"]
          [latex]\\begin{align*} \\text{APY} &= \\left(1 + \\frac{r}{n}\\right)^n - 1 \\\\ r &= 0.03 \\quad (\\text{3% interest rate}) \\\\ n &= 4 \\quad (\\text{compounded quarterly}) \\\\ \\text{APY} &= \\left(1 + \\frac{0.03}{4}\\right)^4 - 1 \\\\ \\text{APY} &= \\left(1 + 0.0075\\right)^4 - 1 \\\\ \\text{APY} &= \\left(1.0075\\right)^4 - 1 \\\\ \\text{APY} &= 1.030339 - 1 \\\\ \\text{APY} &= 0.030339 \\\\ \\text{APY} &= 3.0339\\% \\end{align*}[\/latex]<\/center>Note: The APY is larger than the APR because it takes into account the effect of compounding, providing a more accurate measure of the actual return on the investment.[\/hidden-answer]<\/section>
          [ohm_question hide_question_numbers=1]321420[\/ohm_question]<\/section>","rendered":"

          Compound Interest<\/h2>\n

          Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest<\/strong>. The term compounding<\/em> refers to interest earned not only on the original value, but on the accumulated value of the account.<\/p>\n

          The annual percentage rate (APR)<\/strong> of an account, also called the nominal rate<\/strong>, is the yearly interest rate earned by an investment account. The term\u00a0nominal<\/em>\u00a0is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater<\/em> than the nominal rate! This is a powerful tool for investing.<\/p>\n

          \n

          compound interest formula<\/h3>\n

          Compound interest<\/strong> can be calculated using the formula<\/p>\n

          [latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}[\/latex]<\/p>\n

          where<\/p>\n

            \n
          • [latex]A(t)[\/latex] is the accumulated value of the account<\/li>\n
          • [latex]t[\/latex] is measured in years<\/li>\n
          • [latex]P[\/latex]\u00a0is the starting amount of the account, often called the principal, or more generally present value<\/li>\n
          • [latex]r[\/latex]\u00a0is the annual percentage rate (APR) expressed as a decimal<\/li>\n
          • [latex]n[\/latex]\u00a0is the number of times compounded in a year<\/li>\n<\/ul>\n<\/section>\n
            If we invest [latex]$3,000[\/latex] in an investment account paying [latex]3\\%[\/latex] interest compounded quarterly, how much will the account be worth in [latex]10[\/latex] years?<\/p>\n