How to Do a Simple Rate Distance Time Problem with Neutral Operations

Two Parts:The TutorialHelpful Guidance

Neutral Operations has been shown thus far to be capable in the realm of computer graphics (see top Related wikiHow). Now apply it to a simple problem of Rate = Distance/Time = Distance - Time, as one begins to delve into its applicability in the realm of Space-Time problems.

Part 1
The Tutorial

  1. 1
    Open a new workbook and create 3 worksheets: Data, Chart and Saves. Save the workbook into a logical file folder.
  2. 2
    Set Preferences. The Goal Seeking Preferences will be important. Open Preferences in the Excel menu. Recommended Settings: Set General to R1C1 Off and Show the 10 Most Recent Documents; Edit - set all the Top options to checked except Automatically Convert Date System. Display number of decimal places = blank (for integers preferred), Preserve display of dates and set 30 for 21st century cutoff; View - show Formula Bar and Status Bar, hover for comments and all of Objects, Show grid lines and all boxes below that auto or checked; Chart - show chart names and data markers on hover. Leave rest unchecked for now; Calculation -- Automatically and calculate before save, Limit iteration checked, max iterations 100, max change .000,000,000,000,01 w/o commas as this problem involves precise goal seeking and save external link values and use 1904 date system; Error checking - check all; Save - save preview picture with new files and Save Autorecover after 5 minutes; Ribbon -- all checked except Hide group titles and Developer.
  3. 3
    To test the hypothesis that D-T = D/T = R, where D=Distance, T=Time and R=Rate, a chart based on data that includes the hypothesis data of about 60 miles (97 km) in 60 minutes will be created. Enter into cell A1 x and into cell B1 y. Select row 1 and Format Cells Alignment Center Font Underline Single.
  4. 4
    Enter into cell A2 -10 and enter into cell A30 60 and select cell range A2:A30 and Edit Fill Series Columns Linear accept step value of 2.5, OK. Select row 6 and Insert Rows. Select rows 8:10 and Insert Rows. Select cell A6 and enter -1; select cell A8 and enter .01; select cell A9 and enter .0.95445 and select cell A10 and enter 1.0172465, then select column A and Format Cells Number Number Decimal Places 2.
  5. 5
    Select cell range B2: B34 and with B2 the active cell, enter w/o quotes the formula "=A2/(1-(1/A2))" and Edit Fill Down. Select column B and Format Cells Number Number Decimal Places 15. There should be a #DIV/0! error in cell B7 since an attempt was made to divide by 0 there. Just ignore that. Double-click the right side of the column header to auto-expand the column to fit the overflow.
  6. 6
    Create the chart. Select cell range A2:B34 and go to Charts on the Ribbon or use Chart Wizard and do All or Other and scroll down and select Scatter Smoothed Line Scatter. Double-click on the series plot line and select Line Weights and Arrows, Weight 1.00. Do Chart Layout on the Ribbon by clicking in the chart first and select Axis Titles, Horizontal Axis Title, Title Below Chart and edit it to read Distance x. Select Axis Titles, Vertical Axis Title, Horizontal Axis Title and edit it to read Time y. Go to Chart Title and select Title Above Chart, and edit it to read "y = x/(1- 1/x) for: x/y = x-y or R = D/T = D-T", without quotes. Cut or copy the chart and paste it onto the Chart worksheet and then hold down the shift key and do command+c copy for Copy Picture and then activate the Saves worksheet and hold down the shift key and do command+v paste for Paste Picture onto the Saves worksheet.
  7. 7
    With Neutral Operations involving Division and Subtraction, the sub-steps to do are as follows:
    • D - T = D/T = Rate R where D = Distance and T = Time.
    • D-D - T = D/T -D as D is subtracted from both sides.
    • -T = D(1/T -1) as the left side is simplified and D is factored out on the right.
    • -T/(1/T -1) = D as both sides are divided by (1/T -1) and the right side is simplified since (1/T -1)/(1/T -1) = 1. D has become isolated and defined in terms of T and 1 and T may not = 1 or 0, lest division by 0 result.
    • T/(1 - 1/T) = D as the left sides is multiplied by 1 in the form of -1/-1 and the denominator is rearranged. From this is obtained y = x/(1 - 1/x) or in B2 as =A2/(1-(1/A2)), entered above.
    • Entering 60 for T gives 60/(1 - 1/60) or 60/(59/60) = 3600/59 = 61.0169491525424 = D.
    • Testing the hypothesis: D - T - D/T = R?
    • D - T = 61.0169491525424 - 60 = 1.0169491525424 = R
    • D/T = 61.0169491525424 / 60 = 1.0169491525424 = R; check.
    • So that if T = 60 minutes and D = 61.0169491525424 miles (98.197261016949 km), Rate R is 1.0169491525424 miles (1.6366210169492 km) per minute. Multiplying both numerator and denominator by 60 gives 61.0169491525424 mph (98.197261016949 km/h) = the Rate R. This is also true if one subtracts the time, 60 minutes, from the distance, 61.0169491525424 miles (98.197261016949 km), to obtain the Residual Rate of 1.0169491525424 miles (1.6366210169492 km) per minute.
    • This "Residual Rate" is positive for T>1 and otherwise is negative, infinite or non-existent. The chart clearly shows that Infinity will be asymptotically approached for values approaching closer and closer to 1, from either side of 1. Otherwise, slope y/x is somewhat greater than 1 for numbers greater than 1 and just less than 1 for values of y less than about -9.
  8. 8
    One can argue about negative distance and time. They are included here because the formula works even when they're negative and for no other reason.
  9. 9
    Select cell B35 and enter -60 into it. In cell B36, enter the formula by entering = and then clicking on cells B34 and B35. Copy the answer, 1.01694915254237, and Edit Paste Special, Values back into the cell A10. The answer in B10 is off from B34 by -1.0203393685515E-11 as may easily be confirmed by subtraction. Why it is off is less clear.
  10. 10
    Save the workbook. You're done!

Part 2
Helpful Guidance

  1. 1
    Make use of helper articles when proceeding through this tutorial:
    • See the Related wikiHows below and the article How to Do the Sub Steps of Neutral Operations for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation relating to Neutral Operations.
    • For more art charts and graphs, you might also want to click on Category:Algebra, Category:Mathematics, Category:Spreadsheets or Category:Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.


  • ERRORS: If you have errors or error values, either the sheet in incomplete and needs further input or Lookup Tables for critical variables or perhaps you've made a mistake somewhere along the line. If the instructions have been completed and there are still errors, select the cell that has the error value that is furthest left and topmost first. Look for a typo in a formula or unmatched parentheses. Possibly, a Defined Name is wrong -- they need to be input into the formulas exactly as they were defined. Do Insert Name Define to check. If you have #DIV/0!, I do too or not and will have mentioned it above, so look for a variable that somehow did not get filled in with a value perhaps. At any rate, what you want to do is select the cell with the error, and after checking all those typical errors, do Tools Auditing Trace Precedents and/or Trace Error. If fixing all the topmost leftmost errors does not fix the rest of your errors on your worksheet, you may need to do it the hard way, from the bottom right upwards then leftwards; that is the slow but sure way to fix all errors.
  • Also, errors in your chart data will most likely plot as zeroes. This may be acceptable or desirable even. However, if too many lines (or curves) are returning to 0, it may indicate a logical flaw in the data -- or too many tiny values and then perhaps rescaling the chart is needed by inspecting the horizontal and vertical axes and changing them to zero in on the problem. Hover over or click on a data marker on the series plot and then do a search in the proper column by value for that value, and identify its precedents.

Sources and Citations

  • The file used for this article is "x/y = x-y or R = D/T = D-T.xlsx"

Article Info

Categories: Mathematics | Algebra