{"id":3966,"date":"2024-10-29T14:33:04","date_gmt":"2024-10-29T18:33:04","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&p=3966"},"modified":"2025-06-04T17:01:34","modified_gmt":"2025-06-04T21:01:34","slug":"chap-6-knowledge-check-solutions","status":"web-only","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/chap-6-knowledge-check-solutions\/","title":{"raw":"Chap 6 Knowledge Check Solutions","rendered":"Chap 6 Knowledge Check Solutions"},"content":{"raw":"

Knowledge Check 6.1<\/h2>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Time<\/td>\r\nInterest<\/td>\r\nDeposit<\/td>\r\nWithdrawal<\/td>\r\nBalance<\/td>\r\n<\/tr>\r\n
0<\/td>\r\n<\/td>\r\n$200,000<\/td>\r\n<\/td>\r\n$200,000<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n$40,000<\/td>\r\n40,000<\/td>\r\n<\/td>\r\n280,000<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n56,000<\/td>\r\n<\/td>\r\n50,000<\/td>\r\n286,00<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n57,200<\/td>\r\n<\/td>\r\n50,000<\/td>\r\n293,200<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n58,640<\/td>\r\n<\/td>\r\n50,000<\/td>\r\n301,840<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n60,368<\/td>\r\n<\/td>\r\n300,000<\/td>\r\n62,208<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote that $62,208 is the future value of $25,000!\r\n

Knowledge Check 6.2<\/h2>\r\nAt 15% effective:\r\n\r\nWe have the outflows:\r\n
    \r\n \t
  • Year 0: [latex]PV($200,000)= $200,000[\/latex]<\/li>\r\n \t
  • Year 1: [latex]PV($40,000) = \\frac{$40,000}{1.15^1} = $34,782.61[\/latex]<\/li>\r\n \t
  • Total PV(Outflows) = $234,782.61<\/li>\r\n<\/ul>\r\nAnd the Inflows:\r\n
      \r\n \t
    • Year 2: [latex]PV($50,000) = \\frac{$50,000}{1.15^2} = $37,807.18[\/latex]<\/li>\r\n \t
    • Year 3: [latex]PV($50,000) = \\frac{$50,000}{1.15^3} = $32,875.81[\/latex]<\/li>\r\n \t
    • Year 4: [latex]PV($50,000) = \\frac{$50,000}{1.15^4} = $28,587.66[\/latex]<\/li>\r\n \t
    • Year 5: [latex]PV($300,000) = \\frac{$300,000}{1.15^5} = $149,153.02[\/latex]<\/li>\r\n \t
    • Total PV Inflows = $248,423.68<\/li>\r\n<\/ul>\r\nSo that\r\n\r\n[latex]\r\n\\begin{align*}\r\nNPV &=PV_{Inflows} - PV_{Outflows}\\\\\r\n&=$234,782.61-248,423.68\\\\\r\n&=$13,641.68\r\n\\end{align*}\r\n[\/latex]\r\n\r\nSince the NPV is positive, the investment would be acceptable at 15% effective.\r\n

      Knowledge Check 6.3<\/h2>\r\nOption A\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Time (yr)<\/strong><\/span><\/td>\r\nCash Flow<\/strong><\/span><\/td>\r\nPV<\/strong><\/span><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
      0<\/td>\r\n-$30,000<\/td>\r\n\\(-$30,000\\)<\/td>\r\n<\/tr>\r\n
      1<\/td>\r\n12,000<\/td>\r\n\\(\\frac{12,000}{1.20^1}= 10,000\\)<\/td>\r\n<\/tr>\r\n
      2<\/td>\r\n12,000<\/td>\r\n\\(\\frac{12,000}{1.20^2}= 8,333.33\\)<\/td>\r\n<\/tr>\r\n
      3<\/td>\r\n12,000<\/td>\r\n\\(\\frac{12,000}{1.20^3}= 6,944.44\\)<\/td>\r\n<\/tr>\r\n
      4<\/td>\r\n26,000<\/td>\r\n\\(\\frac{26,000}{1.20^4}= 12,538.58\\)<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      <\/td>\r\nNPV =<\/strong><\/td>\r\n\\($7,816.36\\)<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOption B\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
      Time (yr)<\/strong><\/span><\/td>\r\nCash Flow<\/strong><\/span><\/td>\r\nPV<\/strong><\/span><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
      0<\/td>\r\n-$92,000<\/td>\r\n\\(-$92,000\\)<\/td>\r\n<\/tr>\r\n
      1<\/td>\r\n21,000<\/td>\r\n\\(\\frac{21,000}{1.20^1}= 17,500\\)<\/td>\r\n<\/tr>\r\n
      2<\/td>\r\n21,000<\/td>\r\n\\(\\frac{21,000}{1.20^2}= 14,583.33\\)<\/td>\r\n<\/tr>\r\n
      3<\/td>\r\n21,000<\/td>\r\n\\(\\frac{21,000}{1.20^3}= 12,152.78\\)<\/td>\r\n<\/tr>\r\n
      4<\/td>\r\n21,000<\/td>\r\n\\(\\frac{21,000}{1.20^4}= 10,127.31\\)<\/td>\r\n<\/tr>\r\n
      5<\/td>\r\n21,000+95,000<\/td>\r\n\\(\\frac{116,000}{1.20^5}= 46,617.80\\)<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      <\/td>\r\nNPV =<\/strong><\/td>\r\n\\($8,981.22\\)<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

      Knowledge Check 6.4<\/h2>\r\nOption A:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Time (yr)<\/strong><\/span><\/td>\r\nCash Flow<\/strong><\/span><\/td>\r\nBALANCE<\/strong><\/span><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
      0<\/td>\r\n[latex]-$30,000[\/latex]<\/td>\r\n\\(-$30,000\\)<\/td>\r\n<\/tr>\r\n
      1<\/td>\r\n[latex]12,000[\/latex]<\/td>\r\n\\(-30,000+12,000 = -18,000\\)<\/td>\r\n<\/tr>\r\n
      2<\/td>\r\n[latex]12,000[\/latex]<\/td>\r\n\\(-18,000+12,000=-6,000\\)<\/td>\r\n<\/tr>\r\n
      3<\/td>\r\n[latex]12,000[\/latex]<\/td>\r\n\\(-6,000+12,000 = 6,000\\)<\/td>\r\n<\/tr>\r\n
      4<\/td>\r\n[latex]26,000[\/latex]<\/td>\r\n\\(6,000+26,000=32,000\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPayback is reached halfway between years 2 and 3, so we use a value of 2.5\r\n\r\nOption B\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Time (yr)<\/strong><\/span><\/td>\r\nCash Flow<\/strong><\/span><\/td>\r\nBALANCE<\/strong><\/span><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
      0<\/td>\r\n[latex]-$92,000[\/latex]<\/td>\r\n[latex]\u00a0-$92,000[\/latex]\r\n<\/td>\r\n<\/tr>\r\n
      1<\/td>\r\n[latex]21,000[\/latex]<\/td>\r\n\\(-92,000 + 21,000= -71,000\\)<\/td>\r\n<\/tr>\r\n
      2<\/td>\r\n[latex]21,000[\/latex]<\/td>\r\n\\(-71,000+21,000 = -50,000\\)<\/td>\r\n<\/tr>\r\n
      3<\/td>\r\n[latex]21,000[\/latex]<\/td>\r\n\\(-50,000+21,000 = -29,000\\)<\/td>\r\n<\/tr>\r\n
      4<\/td>\r\n[latex]21,000[\/latex]<\/td>\r\n\\(-29,000+21,000 = -8,000\\)<\/td>\r\n<\/tr>\r\n
      5<\/td>\r\n[latex]21,000+95,000[\/latex]<\/td>\r\n\\(-8,000+116,000=108,000\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHere we are in between years 4 and 5.\u00a0 To approximate we use:\r\n\r\n\\[Payback = 4 + \\frac{$8,000}{$116,000} = 4.07 \\text{ years}\\]","rendered":"

      Knowledge Check 6.1<\/h2>\n\n\n\n\n\n\n\n\n\n
      Time<\/td>\nInterest<\/td>\nDeposit<\/td>\nWithdrawal<\/td>\nBalance<\/td>\n<\/tr>\n
      0<\/td>\n<\/td>\n$200,000<\/td>\n<\/td>\n$200,000<\/td>\n<\/tr>\n
      1<\/td>\n$40,000<\/td>\n40,000<\/td>\n<\/td>\n280,000<\/td>\n<\/tr>\n
      2<\/td>\n56,000<\/td>\n<\/td>\n50,000<\/td>\n286,00<\/td>\n<\/tr>\n
      3<\/td>\n57,200<\/td>\n<\/td>\n50,000<\/td>\n293,200<\/td>\n<\/tr>\n
      4<\/td>\n58,640<\/td>\n<\/td>\n50,000<\/td>\n301,840<\/td>\n<\/tr>\n
      5<\/td>\n60,368<\/td>\n<\/td>\n300,000<\/td>\n62,208<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Note that $62,208 is the future value of $25,000!<\/p>\n

      Knowledge Check 6.2<\/h2>\n

      At 15% effective:<\/p>\n

      We have the outflows:<\/p>\n

        \n
      • Year 0: [latex]PV($200,000)= $200,000[\/latex]<\/li>\n
      • Year 1: [latex]PV($40,000) = \\frac{$40,000}{1.15^1} = $34,782.61[\/latex]<\/li>\n
      • Total PV(Outflows) = $234,782.61<\/li>\n<\/ul>\n

        And the Inflows:<\/p>\n

          \n
        • Year 2: [latex]PV($50,000) = \\frac{$50,000}{1.15^2} = $37,807.18[\/latex]<\/li>\n
        • Year 3: [latex]PV($50,000) = \\frac{$50,000}{1.15^3} = $32,875.81[\/latex]<\/li>\n
        • Year 4: [latex]PV($50,000) = \\frac{$50,000}{1.15^4} = $28,587.66[\/latex]<\/li>\n
        • Year 5: [latex]PV($300,000) = \\frac{$300,000}{1.15^5} = $149,153.02[\/latex]<\/li>\n
        • Total PV Inflows = $248,423.68<\/li>\n<\/ul>\n

          So that<\/p>\n

          [latex]\\begin{align*} NPV &=PV_{Inflows} - PV_{Outflows}\\\\ &=$234,782.61-248,423.68\\\\ &=$13,641.68 \\end{align*}[\/latex]<\/p>\n

          Since the NPV is positive, the investment would be acceptable at 15% effective.<\/p>\n

          Knowledge Check 6.3<\/h2>\n

          Option A<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
          Time (yr)<\/strong><\/span><\/td>\nCash Flow<\/strong><\/span><\/td>\nPV<\/strong><\/span><\/td>\n<\/tr>\n<\/thead>\n
          0<\/td>\n-$30,000<\/td>\n\\(-$30,000\\)<\/td>\n<\/tr>\n
          1<\/td>\n12,000<\/td>\n\\(\\frac{12,000}{1.20^1}= 10,000\\)<\/td>\n<\/tr>\n
          2<\/td>\n12,000<\/td>\n\\(\\frac{12,000}{1.20^2}= 8,333.33\\)<\/td>\n<\/tr>\n
          3<\/td>\n12,000<\/td>\n\\(\\frac{12,000}{1.20^3}= 6,944.44\\)<\/td>\n<\/tr>\n
          4<\/td>\n26,000<\/td>\n\\(\\frac{26,000}{1.20^4}= 12,538.58\\)<\/td>\n<\/tr>\n
          <\/td>\n<\/td>\n<\/td>\n<\/tr>\n
          <\/td>\nNPV =<\/strong><\/td>\n\\($7,816.36\\)<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          Option B<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
          Time (yr)<\/strong><\/span><\/td>\nCash Flow<\/strong><\/span><\/td>\nPV<\/strong><\/span><\/td>\n<\/tr>\n<\/thead>\n
          0<\/td>\n-$92,000<\/td>\n\\(-$92,000\\)<\/td>\n<\/tr>\n
          1<\/td>\n21,000<\/td>\n\\(\\frac{21,000}{1.20^1}= 17,500\\)<\/td>\n<\/tr>\n
          2<\/td>\n21,000<\/td>\n\\(\\frac{21,000}{1.20^2}= 14,583.33\\)<\/td>\n<\/tr>\n
          3<\/td>\n21,000<\/td>\n\\(\\frac{21,000}{1.20^3}= 12,152.78\\)<\/td>\n<\/tr>\n
          4<\/td>\n21,000<\/td>\n\\(\\frac{21,000}{1.20^4}= 10,127.31\\)<\/td>\n<\/tr>\n
          5<\/td>\n21,000+95,000<\/td>\n\\(\\frac{116,000}{1.20^5}= 46,617.80\\)<\/td>\n<\/tr>\n
          <\/td>\n<\/td>\n<\/td>\n<\/tr>\n
          <\/td>\nNPV =<\/strong><\/td>\n\\($8,981.22\\)<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          Knowledge Check 6.4<\/h2>\n

          Option A:<\/p>\n\n\n\n\n\n\n\n\n\n
          Time (yr)<\/strong><\/span><\/td>\nCash Flow<\/strong><\/span><\/td>\nBALANCE<\/strong><\/span><\/td>\n<\/tr>\n<\/thead>\n
          0<\/td>\n[latex]-$30,000[\/latex]<\/td>\n\\(-$30,000\\)<\/td>\n<\/tr>\n
          1<\/td>\n[latex]12,000[\/latex]<\/td>\n\\(-30,000+12,000 = -18,000\\)<\/td>\n<\/tr>\n
          2<\/td>\n[latex]12,000[\/latex]<\/td>\n\\(-18,000+12,000=-6,000\\)<\/td>\n<\/tr>\n
          3<\/td>\n[latex]12,000[\/latex]<\/td>\n\\(-6,000+12,000 = 6,000\\)<\/td>\n<\/tr>\n
          4<\/td>\n[latex]26,000[\/latex]<\/td>\n\\(6,000+26,000=32,000\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          Payback is reached halfway between years 2 and 3, so we use a value of 2.5<\/p>\n

          Option B<\/p>\n\n\n\n\n\n\n\n\n\n\n
          Time (yr)<\/strong><\/span><\/td>\nCash Flow<\/strong><\/span><\/td>\nBALANCE<\/strong><\/span><\/td>\n<\/tr>\n<\/thead>\n
          0<\/td>\n[latex]-$92,000[\/latex]<\/td>\n<\/td>\n<\/tr>\n
          1<\/td>\n[latex]21,000[\/latex]<\/td>\n\\(-92,000 + 21,000= -71,000\\)<\/td>\n<\/tr>\n
          2<\/td>\n[latex]21,000[\/latex]<\/td>\n\\(-71,000+21,000 = -50,000\\)<\/td>\n<\/tr>\n
          3<\/td>\n[latex]21,000[\/latex]<\/td>\n\\(-50,000+21,000 = -29,000\\)<\/td>\n<\/tr>\n
          4<\/td>\n[latex]21,000[\/latex]<\/td>\n\\(-29,000+21,000 = -8,000\\)<\/td>\n<\/tr>\n
          5<\/td>\n[latex]21,000+95,000[\/latex]<\/td>\n\\(-8,000+116,000=108,000\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          Here we are in between years 4 and 5.\u00a0 To approximate we use:<\/p>\n

          \\[Payback = 4 + \\frac{$8,000}{$116,000} = 4.07 \\text{ years}\\]<\/p>\n","protected":false},"author":883,"menu_order":10,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3966","chapter","type-chapter","status-web-only","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/3966","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/users\/883"}],"version-history":[{"count":13,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/3966\/revisions"}],"predecessor-version":[{"id":4130,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/3966\/revisions\/4130"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/3966\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/media?parent=3966"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapter-type?post=3966"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/contributor?post=3966"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/license?post=3966"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}