{"id":60,"date":"2020-04-17T15:41:09","date_gmt":"2020-04-17T19:41:09","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&p=60"},"modified":"2021-07-12T14:12:28","modified_gmt":"2021-07-12T18:12:28","slug":"topic","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/topic\/","title":{"raw":"3.2 Exact Time\u00a0\u00a0","rendered":"3.2 Exact Time\u00a0\u00a0"},"content":{"raw":"If the two dates <\/strong>that determine the interest period are known, it is important to calculate the exact <\/strong>number of days between those two dates. To do this, we count the beginning day but not the ending day of the period. We will always use a 365 day year, essentially ignoring leap year.\r\n\r\n
\r\n

Example 3.2.1<\/h2>\r\n<\/header>\r\n
If the interest period ranges from May 12 to May 27,<\/div>\r\n
<\/div>\r\n
\u00a0[latex]t = \\frac{27 - 12}{365} =\\frac{15}{365}\\; years[\/latex]<\/div>\r\nBe sure to satisfy yourself that this simple subtraction of dates actually includes the first date but not the last.\r\n\r\nYou should also know that April, June, September and November have 30 days, February has 28 days (29 days during a leap year) and the remaining months have 31 days. Leap years occur every four years and are determined by getting an integer (a whole number) answer when dividing the last two digits of the year by four.\r\n\r\n \r\n

Example 3.2.2<\/h2>\r\nHow many days between April 16 and August 12?\r\n\r\nStart Date:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 April 16 (counted)\r\n\r\nEnd Date: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 August 12 (not counted)\r\n\r\nNumber of days:\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
April<\/td>\r\n=<\/td>\r\n(30 - 15)<\/td>\r\n=<\/td>\r\n15<\/td>\r\n<\/tr>\r\n
May<\/td>\r\n=<\/td>\r\n<\/td>\r\n=<\/td>\r\n31<\/td>\r\n<\/tr>\r\n
June<\/td>\r\n=<\/td>\r\n<\/td>\r\n=<\/td>\r\n30<\/td>\r\n<\/tr>\r\n
July<\/td>\r\n=<\/td>\r\n<\/td>\r\n=<\/td>\r\n31<\/td>\r\n<\/tr>\r\n
August<\/td>\r\n=<\/td>\r\n(12-1)<\/td>\r\n=<\/td>\r\n11<\/td>\r\n<\/tr>\r\n
TOTAL<\/td>\r\n=<\/td>\r\n<\/td>\r\n=<\/td>\r\n118 days<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nTherefore,\r\n

[latex]t = \\frac{118}{365}\\;days[\/latex]<\/p>\r\nSome calculators and computer software will calculate the days for the user.\u00a0 In this text we present two methods of date counting: using your calculator, and using a table.\u00a0 We also highly encourage students to take the time to learn how to use the date functionality in Microsoft Excel, as it is well worth the effort.\r\n

Using the BA II Plus to Find Exact Dates<\/h2>\r\n[embed]https:\/\/youtu.be\/fBaU7Lxl43E[\/embed]\r\n\r\nWe will use Question 16 from the Practice Problems to illustrate the exact date feature of the BA II plus calculator.\r\n\r\n
\r\n

Example 3.2.3<\/h2>\r\n<\/header>\r\n
AW borrowed $9,000 on January 30, 2002 and agreed to pay 14% simple interest on the balance outstanding at any time. He paid $5,000 on March 9, 2002 and $2,500 on May 25, 2002. How much did he have to pay on June 30, 2002 in order to pay off the debt?<\/div>\r\n
    \r\n \t
  • First, find the number of days between the debt and the focal date.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n\r\n\r\n
    2nd Date<\/td>\r\nDTl =<\/td>\r\n1.3002 Enter<\/td>\r\nEnter the date of the loan (month.day year)<\/td>\r\n<\/tr>\r\n
    <\/td>\r\nDT2=<\/td>\r\n6.3002 Enter<\/td>\r\nEnter the focal date<\/td>\r\n<\/tr>\r\n
    <\/td>\r\nDBD=<\/td>\r\nCPT<\/td>\r\n151 <\/strong>from debt to focal date<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
      \r\n \t
    • Next, find the number of days from the $5,000 payment to the focal date.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n\r\n\r\n
      2nd Date<\/td>\r\nDTl =<\/td>\r\n3.0902 Enter<\/td>\r\nEnter the date of the first payment<\/td>\r\n<\/tr>\r\n
      <\/td>\r\nDT2=<\/td>\r\n6.3002 Enter<\/td>\r\nThis is the focal date - you do not need to re-enter<\/td>\r\n<\/tr>\r\n
      <\/td>\r\nDBD=<\/td>\r\nCPT<\/td>\r\n113 <\/strong>days from the payment to the focal date<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
        \r\n \t
      • Find the number of days from the second payment to the focal date.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n\r\n\r\n
        2nd Date<\/td>\r\nDTl =<\/td>\r\n5.2502 Enter<\/td>\r\nEnter the date of the second payment<\/td>\r\n<\/tr>\r\n
        <\/td>\r\nDT2=<\/td>\r\n6.300 Enter<\/td>\r\nThis is the focal date -\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 you do not need to re-enter<\/td>\r\n<\/tr>\r\n
        <\/td>\r\nDBD=<\/td>\r\nCPT<\/td>\r\n36 <\/strong>days from the payment to the focal date<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe will finish solving this problem later on.\r\n

        Dates using a Table<\/h2>\r\nIf you prefer to use tables, you can use the following:\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\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\r\n\r\n\r\n\r\n
        Day of Month<\/strong><\/td>\r\nJan<\/strong><\/td>\r\nFeb<\/strong><\/td>\r\nMar<\/strong><\/td>\r\nApr<\/strong><\/td>\r\nMay<\/strong><\/td>\r\nJun<\/strong><\/td>\r\nJul<\/strong><\/td>\r\nAug<\/strong><\/td>\r\nSep<\/strong><\/td>\r\nOct<\/strong><\/td>\r\nNov<\/strong><\/td>\r\nDec<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
        1<\/strong><\/td>\r\n1<\/td>\r\n32<\/td>\r\n60<\/td>\r\n91<\/td>\r\n121<\/td>\r\n152<\/td>\r\n182<\/td>\r\n213<\/td>\r\n244<\/td>\r\n274<\/td>\r\n305<\/td>\r\n335<\/td>\r\n<\/tr>\r\n
        2<\/strong><\/td>\r\n2<\/td>\r\n33<\/td>\r\n61<\/td>\r\n92<\/td>\r\n122<\/td>\r\n153<\/td>\r\n183<\/td>\r\n214<\/td>\r\n245<\/td>\r\n275<\/td>\r\n306<\/td>\r\n336<\/td>\r\n<\/tr>\r\n
        3<\/strong><\/td>\r\n3<\/td>\r\n34<\/td>\r\n62<\/td>\r\n93<\/td>\r\n123<\/td>\r\n154<\/td>\r\n184<\/td>\r\n215<\/td>\r\n246<\/td>\r\n276<\/td>\r\n307<\/td>\r\n337<\/td>\r\n<\/tr>\r\n
        4<\/strong><\/td>\r\n4<\/td>\r\n35<\/td>\r\n63<\/td>\r\n94<\/td>\r\n124<\/td>\r\n155<\/td>\r\n185<\/td>\r\n216<\/td>\r\n247<\/td>\r\n277<\/td>\r\n308<\/td>\r\n338<\/td>\r\n<\/tr>\r\n
        5<\/strong><\/td>\r\n5<\/td>\r\n36<\/td>\r\n64<\/td>\r\n95<\/td>\r\n125<\/td>\r\n156<\/td>\r\n186<\/td>\r\n217<\/td>\r\n248<\/td>\r\n278<\/td>\r\n309<\/td>\r\n339<\/td>\r\n<\/tr>\r\n
        6<\/strong><\/td>\r\n6<\/td>\r\n37<\/td>\r\n65<\/td>\r\n96<\/td>\r\n126<\/td>\r\n157<\/td>\r\n187<\/td>\r\n218<\/td>\r\n249<\/td>\r\n279<\/td>\r\n310<\/td>\r\n340<\/td>\r\n<\/tr>\r\n
        7<\/strong><\/td>\r\n7<\/td>\r\n38<\/td>\r\n66<\/td>\r\n97<\/td>\r\n127<\/td>\r\n158<\/td>\r\n188<\/td>\r\n219<\/td>\r\n250<\/td>\r\n280<\/td>\r\n311<\/td>\r\n341<\/td>\r\n<\/tr>\r\n
        8<\/strong><\/td>\r\n8<\/td>\r\n39<\/td>\r\n67<\/td>\r\n98<\/td>\r\n128<\/td>\r\n159<\/td>\r\n189<\/td>\r\n220<\/td>\r\n251<\/td>\r\n281<\/td>\r\n312<\/td>\r\n342<\/td>\r\n<\/tr>\r\n
        9<\/strong><\/td>\r\n9<\/td>\r\n40<\/td>\r\n68<\/td>\r\n99<\/td>\r\n129<\/td>\r\n160<\/td>\r\n190<\/td>\r\n221<\/td>\r\n252<\/td>\r\n282<\/td>\r\n313<\/td>\r\n343<\/td>\r\n<\/tr>\r\n
        10<\/strong><\/td>\r\n10<\/td>\r\n41<\/td>\r\n69<\/td>\r\n100<\/td>\r\n130<\/td>\r\n161<\/td>\r\n191<\/td>\r\n222<\/td>\r\n253<\/td>\r\n283<\/td>\r\n314<\/td>\r\n344<\/td>\r\n<\/tr>\r\n
        11<\/strong><\/td>\r\n11<\/td>\r\n42<\/td>\r\n70<\/td>\r\n101<\/td>\r\n131<\/td>\r\n162<\/td>\r\n192<\/td>\r\n223<\/td>\r\n254<\/td>\r\n284<\/td>\r\n315<\/td>\r\n345<\/td>\r\n<\/tr>\r\n
        12<\/strong><\/td>\r\n12<\/td>\r\n43<\/td>\r\n71<\/td>\r\n102<\/td>\r\n132<\/td>\r\n163<\/td>\r\n193<\/td>\r\n224<\/td>\r\n255<\/td>\r\n285<\/td>\r\n316<\/td>\r\n346<\/td>\r\n<\/tr>\r\n
        13<\/strong><\/td>\r\n13<\/td>\r\n44<\/td>\r\n72<\/td>\r\n103<\/td>\r\n133<\/td>\r\n164<\/td>\r\n194<\/td>\r\n225<\/td>\r\n256<\/td>\r\n286<\/td>\r\n317<\/td>\r\n347<\/td>\r\n<\/tr>\r\n
        14<\/strong><\/td>\r\n14<\/td>\r\n45<\/td>\r\n73<\/td>\r\n104<\/td>\r\n134<\/td>\r\n165<\/td>\r\n195<\/td>\r\n226<\/td>\r\n257<\/td>\r\n287<\/td>\r\n318<\/td>\r\n348<\/td>\r\n<\/tr>\r\n
        15<\/strong><\/td>\r\n15<\/td>\r\n46<\/td>\r\n74<\/td>\r\n105<\/td>\r\n135<\/td>\r\n166<\/td>\r\n196<\/td>\r\n227<\/td>\r\n258<\/td>\r\n288<\/td>\r\n319<\/td>\r\n349<\/td>\r\n<\/tr>\r\n
        16<\/strong><\/td>\r\n16<\/td>\r\n47<\/td>\r\n75<\/td>\r\n106<\/td>\r\n136<\/td>\r\n167<\/td>\r\n197<\/td>\r\n228<\/td>\r\n259<\/td>\r\n289<\/td>\r\n320<\/td>\r\n350<\/td>\r\n<\/tr>\r\n
        17<\/strong><\/td>\r\n17<\/td>\r\n48<\/td>\r\n76<\/td>\r\n107<\/td>\r\n137<\/td>\r\n168<\/td>\r\n198<\/td>\r\n229<\/td>\r\n260<\/td>\r\n290<\/td>\r\n321<\/td>\r\n351<\/td>\r\n<\/tr>\r\n
        18<\/strong><\/td>\r\n18<\/td>\r\n49<\/td>\r\n77<\/td>\r\n108<\/td>\r\n138<\/td>\r\n169<\/td>\r\n199<\/td>\r\n230<\/td>\r\n261<\/td>\r\n291<\/td>\r\n322<\/td>\r\n352<\/td>\r\n<\/tr>\r\n
        19<\/strong><\/td>\r\n19<\/td>\r\n50<\/td>\r\n78<\/td>\r\n109<\/td>\r\n139<\/td>\r\n170<\/td>\r\n200<\/td>\r\n231<\/td>\r\n262<\/td>\r\n292<\/td>\r\n323<\/td>\r\n353<\/td>\r\n<\/tr>\r\n
        20<\/strong><\/td>\r\n20<\/td>\r\n51<\/td>\r\n79<\/td>\r\n110<\/td>\r\n140<\/td>\r\n171<\/td>\r\n201<\/td>\r\n232<\/td>\r\n263<\/td>\r\n293<\/td>\r\n324<\/td>\r\n354<\/td>\r\n<\/tr>\r\n
        21<\/strong><\/td>\r\n21<\/td>\r\n52<\/td>\r\n80<\/td>\r\n111<\/td>\r\n141<\/td>\r\n172<\/td>\r\n202<\/td>\r\n233<\/td>\r\n264<\/td>\r\n294<\/td>\r\n325<\/td>\r\n355<\/td>\r\n<\/tr>\r\n
        22<\/strong><\/td>\r\n22<\/td>\r\n53<\/td>\r\n81<\/td>\r\n112<\/td>\r\n142<\/td>\r\n173<\/td>\r\n203<\/td>\r\n234<\/td>\r\n265<\/td>\r\n295<\/td>\r\n326<\/td>\r\n356<\/td>\r\n<\/tr>\r\n
        23<\/strong><\/td>\r\n23<\/td>\r\n54<\/td>\r\n82<\/td>\r\n113<\/td>\r\n143<\/td>\r\n174<\/td>\r\n204<\/td>\r\n235<\/td>\r\n266<\/td>\r\n296<\/td>\r\n327<\/td>\r\n357<\/td>\r\n<\/tr>\r\n
        24<\/strong><\/td>\r\n24<\/td>\r\n55<\/td>\r\n83<\/td>\r\n114<\/td>\r\n144<\/td>\r\n175<\/td>\r\n205<\/td>\r\n236<\/td>\r\n267<\/td>\r\n297<\/td>\r\n328<\/td>\r\n358<\/td>\r\n<\/tr>\r\n
        25<\/strong><\/td>\r\n25<\/td>\r\n56<\/td>\r\n84<\/td>\r\n115<\/td>\r\n145<\/td>\r\n176<\/td>\r\n206<\/td>\r\n237<\/td>\r\n268<\/td>\r\n298<\/td>\r\n329<\/td>\r\n359<\/td>\r\n<\/tr>\r\n
        26<\/strong><\/td>\r\n26<\/td>\r\n57<\/td>\r\n85<\/td>\r\n116<\/td>\r\n146<\/td>\r\n177<\/td>\r\n207<\/td>\r\n238<\/td>\r\n269<\/td>\r\n299<\/td>\r\n330<\/td>\r\n360<\/td>\r\n<\/tr>\r\n
        27<\/strong><\/td>\r\n27<\/td>\r\n58<\/td>\r\n86<\/td>\r\n117<\/td>\r\n147<\/td>\r\n178<\/td>\r\n208<\/td>\r\n239<\/td>\r\n270<\/td>\r\n300<\/td>\r\n331<\/td>\r\n361<\/td>\r\n<\/tr>\r\n
        28<\/strong><\/td>\r\n28<\/td>\r\n59<\/td>\r\n87<\/td>\r\n118<\/td>\r\n148<\/td>\r\n179<\/td>\r\n209<\/td>\r\n240<\/td>\r\n271<\/td>\r\n301<\/td>\r\n332<\/td>\r\n362<\/td>\r\n<\/tr>\r\n
        29<\/strong><\/td>\r\n29<\/td>\r\n<\/td>\r\n88<\/td>\r\n119<\/td>\r\n149<\/td>\r\n180<\/td>\r\n210<\/td>\r\n241<\/td>\r\n272<\/td>\r\n302<\/td>\r\n333<\/td>\r\n363<\/td>\r\n<\/tr>\r\n
        30<\/strong><\/td>\r\n30<\/td>\r\n<\/td>\r\n89<\/td>\r\n120<\/td>\r\n150<\/td>\r\n181<\/td>\r\n211<\/td>\r\n242<\/td>\r\n273<\/td>\r\n303<\/td>\r\n334<\/td>\r\n364<\/td>\r\n<\/tr>\r\n
        31<\/strong><\/td>\r\n31<\/td>\r\n<\/td>\r\n90<\/td>\r\n<\/td>\r\n151<\/td>\r\n<\/td>\r\n212<\/td>\r\n243<\/td>\r\n<\/td>\r\n304<\/td>\r\n<\/td>\r\n365<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTable 3-1: <\/strong>Days of the Year\r\n\r\n \r\n\r\nNote carefully: <\/strong>For leap years (those years where the last two digits are divisible by four with an integer result), February 29 must be included and becomes day 60. One day must then be added to all dates following. (Examples of leap years: 2016, 2020, 2024, 2028, 2032 )\r\n\r\nThere are two occasions when extra care must be taken when using Table 3-1:\r\n

        Leap Year<\/h1>\r\nIn a leap year, if the time period includes the end of February, every date after February 28 increases its count by one.\r\n\r\nBe careful when your calculations include a leap year or a period that covers the end of a year.\r\n\r\nWrite the day count on a diagram.\r\n\r\n \r\n\r\n
        \r\n

        Example 3.2.4<\/h2>\r\n<\/header>\r\n
        \r\n\r\nHow many days between January 15 and May 28, 2024? (2024 is a leap year.)\r\n\r\nJanuary 15 is the 15th day of the year from Table 3-1.\r\n\r\nMay 28 is the 148th day of the year from Table 3-1. However, since the time period includes February 29, you must add one day to the value shown in the table.\r\n\r\nTherefore, the exact number of days = (148 + 1) \u2212 15 = 134.\r\n\r\n<\/div>\r\n \r\n

        When the Time Period Includes the End of the Year<\/h1>\r\n

        Example 3.2.5<\/h2>\r\nHow many days between October 15, 2021 and February 13, 2022? The technique is to subtract the first day value from 365 (to get to the end of the year) and then to add the second day value to that difference (which moves you into a second year).\r\n\r\nOctober 15 is the 288th day of the year from Table 3-1. Therefore, the number of days to the end of the year is (365 - 288) or 77.\r\n\r\nFebruary 13 is the 44th day of the year from Table 3-1. Therefore, the exact number of days is 77 + 44 = 121. You can draw a time line with the exact dates shown. To assist in calculating the exact time, get into the habit of also writing the day count on the diagram.\r\n\r\n\"Empty\r\n\r\nThis will automatically lead you to an exact count of the number of days between any two dates. To illustrate this, note:\r\n\r\n178 - 44 = 134 days between the first two dates,\r\n\r\n288 - 178 = 110 days between the second two dates, and\r\n\r\n288 - 44 = 244 days between the first and last dates.\r\n\r\n \r\n\r\n \r\n\r\n \r\n

        Your Own Notes<\/h2>\r\n
          \r\n \t
        • Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t
        • These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t
        • Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"1\"]","rendered":"

          If the two dates <\/strong>that determine the interest period are known, it is important to calculate the exact <\/strong>number of days between those two dates. To do this, we count the beginning day but not the ending day of the period. We will always use a 365 day year, essentially ignoring leap year.<\/p>\n

          \n

          Example 3.2.1<\/h2>\n<\/header>\n
          If the interest period ranges from May 12 to May 27,<\/div>\n
          <\/div>\n
          \u00a0[latex]t = \\frac{27 - 12}{365} =\\frac{15}{365}\\; years[\/latex]<\/div>\n

          Be sure to satisfy yourself that this simple subtraction of dates actually includes the first date but not the last.<\/p>\n

          You should also know that April, June, September and November have 30 days, February has 28 days (29 days during a leap year) and the remaining months have 31 days. Leap years occur every four years and are determined by getting an integer (a whole number) answer when dividing the last two digits of the year by four.<\/p>\n

           <\/p>\n

          Example 3.2.2<\/h2>\n

          How many days between April 16 and August 12?<\/p>\n

          Start Date:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 April 16 (counted)<\/p>\n

          End Date: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 August 12 (not counted)<\/p>\n

          Number of days:<\/p>\n

          \n\n\n\n\n\n\n\n\n
          April<\/td>\n=<\/td>\n(30 – 15)<\/td>\n=<\/td>\n15<\/td>\n<\/tr>\n
          May<\/td>\n=<\/td>\n<\/td>\n=<\/td>\n31<\/td>\n<\/tr>\n
          June<\/td>\n=<\/td>\n<\/td>\n=<\/td>\n30<\/td>\n<\/tr>\n
          July<\/td>\n=<\/td>\n<\/td>\n=<\/td>\n31<\/td>\n<\/tr>\n
          August<\/td>\n=<\/td>\n(12-1)<\/td>\n=<\/td>\n11<\/td>\n<\/tr>\n
          TOTAL<\/td>\n=<\/td>\n<\/td>\n=<\/td>\n118 days<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

          Therefore,<\/p>\n

          [latex]t = \\frac{118}{365}\\;days[\/latex]<\/p>\n

          Some calculators and computer software will calculate the days for the user.\u00a0 In this text we present two methods of date counting: using your calculator, and using a table.\u00a0 We also highly encourage students to take the time to learn how to use the date functionality in Microsoft Excel, as it is well worth the effort.<\/p>\n

          Using the BA II Plus to Find Exact Dates<\/h2>\n