{"id":2439,"date":"2023-05-10T15:24:43","date_gmt":"2023-05-10T15:24:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=2439"},"modified":"2024-10-18T20:55:16","modified_gmt":"2024-10-18T20:55:16","slug":"simple-and-compound-interest-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/simple-and-compound-interest-learn-it-2\/","title":{"raw":"Simple and Compound Interest: Learn It 2","rendered":"Simple and Compound Interest: Learn It 2"},"content":{"raw":"

Compounding<\/h2>\r\n

With simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding<\/strong>.<\/p>\r\n

\r\n
\r\n

compounding<\/h3>\r\n

Compounding <\/strong>refers to the process where the interest earned on an investment or the interest charged on a loan is added to the principal amount, so that interest can be earned or charged on a larger amount over time.<\/p>\r\n<\/div>\r\n<\/section>\r\n

Suppose that we deposit [latex]$1000[\/latex] in a bank account offering [latex]3\\%[\/latex] interest, compounded monthly. How will our money grow?<\/p>\r\n

The [latex]3\\%[\/latex] interest is an annual percentage rate (APR) \u2013 the total interest to be paid during the year. Since interest is being paid monthly, each month, we will earn [latex]\\frac{3\\%}{12}= 0.25\\%[\/latex] per month.<\/p>\r\n

In the first month,<\/p>\r\n

    \r\n\t
  • [latex]P_0 = $1000[\/latex]<\/li>\r\n\t
  • [latex]r = 0.0025 (0.25\\%)[\/latex]<\/li>\r\n\t
  • [latex]I= $1000 (0.0025) = $2.50[\/latex]<\/li>\r\n\t
  • [latex]A = $1000 + $2.50 = $1002.50[\/latex]<\/li>\r\n<\/ul>\r\n

    In the first month, we will earn [latex]$2.50[\/latex] in interest, raising our account balance to [latex]$1002.50[\/latex].<\/p>\r\n

    In the second month,<\/p>\r\n

      \r\n\t
    • [latex]P_0 = $1002.50[\/latex]<\/li>\r\n\t
    • [latex]I= $1002.50 (0.0025) = $2.51[\/latex] (rounded)<\/li>\r\n\t
    • [latex]A = $1002.50 + $2.51 = $1005.01[\/latex]<\/li>\r\n<\/ul>\r\n

      Notice that in the second month we earned more interest than we did in the first month. This is because we earned interest not only on the original [latex]$1000[\/latex] we deposited, but we also earned interest on the [latex]$2.50[\/latex] of interest we earned the first month. This is the key advantage that compounding\u00a0interest gives us.<\/p>\r\n

      Calculating out a few more months gives the following:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Month<\/strong><\/td>\r\nStarting balance<\/strong><\/td>\r\nInterest earned<\/strong><\/td>\r\nEnding Balance<\/strong><\/td>\r\n<\/tr>\r\n
      [latex]1[\/latex]<\/td>\r\n[latex]1000.00[\/latex]<\/td>\r\n[latex]2.50[\/latex]<\/td>\r\n[latex]1002.50[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]2[\/latex]<\/td>\r\n[latex]1002.50[\/latex]<\/td>\r\n[latex]2.51[\/latex]<\/td>\r\n[latex]1005.01[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]3[\/latex]<\/td>\r\n[latex]1005.01[\/latex]<\/td>\r\n[latex]2.51[\/latex]<\/td>\r\n[latex]1007.52[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]4[\/latex]<\/td>\r\n[latex]1007.52[\/latex]<\/td>\r\n[latex]2.52[\/latex]<\/td>\r\n[latex]1010.04[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]5[\/latex]<\/td>\r\n[latex]1010.04[\/latex]<\/td>\r\n[latex]2.53[\/latex]<\/td>\r\n[latex]1012.57[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]6[\/latex]<\/td>\r\n[latex]1012.57[\/latex]<\/td>\r\n[latex]2.53[\/latex]<\/td>\r\n[latex]1015.10[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]7[\/latex]<\/td>\r\n[latex]1015.10[\/latex]<\/td>\r\n[latex]2.54[\/latex]<\/td>\r\n[latex]1017.64[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]8[\/latex]<\/td>\r\n[latex]1017.64[\/latex]<\/td>\r\n[latex]2.54[\/latex]<\/td>\r\n[latex]1020.18[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]9[\/latex]<\/td>\r\n[latex]1020.18[\/latex]<\/td>\r\n[latex]2.55[\/latex]<\/td>\r\n[latex]1022.73[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]10[\/latex]<\/td>\r\n[latex]1022.73[\/latex]<\/td>\r\n[latex]2.56[\/latex]<\/td>\r\n[latex]1025.29[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]11[\/latex]<\/td>\r\n[latex]1025.29[\/latex]<\/td>\r\n[latex]2.56[\/latex]<\/td>\r\n[latex]1027.85[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]12[\/latex]<\/td>\r\n[latex]1027.85[\/latex]<\/td>\r\n[latex]2.57[\/latex]<\/td>\r\n[latex]1030.42[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"

      Compounding<\/h2>\n

      With simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding<\/strong>.<\/p>\n

      \n
      \n

      compounding<\/h3>\n

      Compounding <\/strong>refers to the process where the interest earned on an investment or the interest charged on a loan is added to the principal amount, so that interest can be earned or charged on a larger amount over time.<\/p>\n<\/div>\n<\/section>\n

      Suppose that we deposit [latex]$1000[\/latex] in a bank account offering [latex]3\\%[\/latex] interest, compounded monthly. How will our money grow?<\/p>\n

      The [latex]3\\%[\/latex] interest is an annual percentage rate (APR) \u2013 the total interest to be paid during the year. Since interest is being paid monthly, each month, we will earn [latex]\\frac{3\\%}{12}= 0.25\\%[\/latex] per month.<\/p>\n

      In the first month,<\/p>\n

        \n
      • [latex]P_0 = $1000[\/latex]<\/li>\n
      • [latex]r = 0.0025 (0.25\\%)[\/latex]<\/li>\n
      • [latex]I= $1000 (0.0025) = $2.50[\/latex]<\/li>\n
      • [latex]A = $1000 + $2.50 = $1002.50[\/latex]<\/li>\n<\/ul>\n

        In the first month, we will earn [latex]$2.50[\/latex] in interest, raising our account balance to [latex]$1002.50[\/latex].<\/p>\n

        In the second month,<\/p>\n

          \n
        • [latex]P_0 = $1002.50[\/latex]<\/li>\n
        • [latex]I= $1002.50 (0.0025) = $2.51[\/latex] (rounded)<\/li>\n
        • [latex]A = $1002.50 + $2.51 = $1005.01[\/latex]<\/li>\n<\/ul>\n

          Notice that in the second month we earned more interest than we did in the first month. This is because we earned interest not only on the original [latex]$1000[\/latex] we deposited, but we also earned interest on the [latex]$2.50[\/latex] of interest we earned the first month. This is the key advantage that compounding\u00a0interest gives us.<\/p>\n

          Calculating out a few more months gives the following:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
          Month<\/strong><\/td>\nStarting balance<\/strong><\/td>\nInterest earned<\/strong><\/td>\nEnding Balance<\/strong><\/td>\n<\/tr>\n
          [latex]1[\/latex]<\/td>\n[latex]1000.00[\/latex]<\/td>\n[latex]2.50[\/latex]<\/td>\n[latex]1002.50[\/latex]<\/td>\n<\/tr>\n
          [latex]2[\/latex]<\/td>\n[latex]1002.50[\/latex]<\/td>\n[latex]2.51[\/latex]<\/td>\n[latex]1005.01[\/latex]<\/td>\n<\/tr>\n
          [latex]3[\/latex]<\/td>\n[latex]1005.01[\/latex]<\/td>\n[latex]2.51[\/latex]<\/td>\n[latex]1007.52[\/latex]<\/td>\n<\/tr>\n
          [latex]4[\/latex]<\/td>\n[latex]1007.52[\/latex]<\/td>\n[latex]2.52[\/latex]<\/td>\n[latex]1010.04[\/latex]<\/td>\n<\/tr>\n
          [latex]5[\/latex]<\/td>\n[latex]1010.04[\/latex]<\/td>\n[latex]2.53[\/latex]<\/td>\n[latex]1012.57[\/latex]<\/td>\n<\/tr>\n
          [latex]6[\/latex]<\/td>\n[latex]1012.57[\/latex]<\/td>\n[latex]2.53[\/latex]<\/td>\n[latex]1015.10[\/latex]<\/td>\n<\/tr>\n
          [latex]7[\/latex]<\/td>\n[latex]1015.10[\/latex]<\/td>\n[latex]2.54[\/latex]<\/td>\n[latex]1017.64[\/latex]<\/td>\n<\/tr>\n
          [latex]8[\/latex]<\/td>\n[latex]1017.64[\/latex]<\/td>\n[latex]2.54[\/latex]<\/td>\n[latex]1020.18[\/latex]<\/td>\n<\/tr>\n
          [latex]9[\/latex]<\/td>\n[latex]1020.18[\/latex]<\/td>\n[latex]2.55[\/latex]<\/td>\n[latex]1022.73[\/latex]<\/td>\n<\/tr>\n
          [latex]10[\/latex]<\/td>\n[latex]1022.73[\/latex]<\/td>\n[latex]2.56[\/latex]<\/td>\n[latex]1025.29[\/latex]<\/td>\n<\/tr>\n
          [latex]11[\/latex]<\/td>\n[latex]1025.29[\/latex]<\/td>\n[latex]2.56[\/latex]<\/td>\n[latex]1027.85[\/latex]<\/td>\n<\/tr>\n
          [latex]12[\/latex]<\/td>\n[latex]1027.85[\/latex]<\/td>\n[latex]2.57[\/latex]<\/td>\n[latex]1030.42[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":15,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":89,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2439"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2439\/revisions"}],"predecessor-version":[{"id":13631,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2439\/revisions\/13631"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/89"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2439\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2439"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2439"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2439"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2439"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}