Reading a Cumulative-Change Table
A table of changes can describe how a quantity develops over several consecutive intervals. For example, an inventory might increase during one interval, decrease during the next, and increase again afterward. A cumulative-change column keeps a running record of the net change from the beginning to each point in the table.
The key is to distinguish three related quantities: the change during one interval, the cumulative change up to a point, and the actual quantity at that point. They are connected, but they are not interchangeable. In particular, a cumulative change is measured relative to a starting point; it is not automatically the quantity itself.
Changes are signed. An increase is positive and a decrease is negative. Thus a negative interval change reduces the cumulative change, even if the cumulative change remains positive overall. The cumulative change records the net effect of all the interval changes so far, not the total amount of movement in both directions.
Building the Cumulative-Change Column
To build a table from interval changes, start with cumulative change \(0\) at \(t_0\). Then add each interval’s signed change to the cumulative value at the preceding time. Equivalently, each new cumulative value is the previous cumulative value plus the change over the interval just completed.
If the initial quantity \(Q(t_0)\) is known, the actual quantity at any later time is the initial quantity plus the cumulative change to that time. The initial value is not itself an interval change; it supplies the baseline from which changes are measured.
Worked Example: Tracking an Inventory
Worked Example: Tracking an Inventory
A storeroom begins with \(40\) boxes. During three consecutive days, its inventory changes by \(+12\) boxes, \(-5\) boxes, and \(+8\) boxes. Find the cumulative change and the inventory at the end of each day.
At the starting time, the cumulative change is \(0\) boxes, and the inventory is \(40\) boxes.
After day 1, the cumulative change is \(0+12=12\) boxes. The inventory is \(40+12=52\) boxes.
After day 2, the cumulative change is \(12+(-5)=7\) boxes. The inventory is \(40+7=47\) boxes.
After day 3, the cumulative change is \(7+8=15\) boxes. The inventory is \(40+15=55\) boxes.
| Time | Interval change during the day | Cumulative change | Inventory |
|---|---|---|---|
| Start | — | 0 boxes | 40 boxes |
| End of day 1 | +12 boxes | 12 boxes | 52 boxes |
| End of day 2 | −5 boxes | 7 boxes | 47 boxes |
| End of day 3 | +8 boxes | 15 boxes | 55 boxes |
Check the result by comparing consecutive inventory values: \(52-40=12\), \(47-52=-5\), and \(55-47=8\) boxes. Each difference agrees with the corresponding interval change. The final inventory also agrees with the initial inventory plus the cumulative change: \(40+15=55\) boxes.
Recovering an Interval Change
A table may provide cumulative values instead of individual interval changes. To recover the change on an interval, subtract the cumulative value at the start of that interval from the cumulative value at its end. The cumulative value at the end already includes all earlier changes, so subtracting the preceding cumulative value removes those earlier contributions.
Why Subtracting Consecutive Cumulative Values Works
Here is the reasoning in full. Let the successive interval changes be \(d_1,d_2,\ldots,d_n\). By definition, the cumulative change \(C_i\) is obtained by adding the first \(i\) interval changes. For an interval after the beginning, the two cumulative values have all earlier changes in common:
Subtract the second equality from the first. Each earlier change appears once with a positive sign and once with a negative sign, so those terms cancel, leaving only the change on the new interval:
For the first interval, the same rule works because \(C_0=0\): \(C_1-C_0=C_1=d_1\). Therefore, subtracting consecutive cumulative values recovers the interval change, including the first one.
Worked Example: Recovering Changes from a Cumulative Table
Worked Example: Recovering Changes from a Cumulative Table
A tank’s cumulative net volume change from the start of an observation is recorded at three successive times as \(4.5\), \(3.0\), and \(9.2\) cubic metres. Find the net volume change during each interval. If the tank initially contains \(20\) cubic metres, also find its volume at each recorded time.
At the start, the cumulative change is \(C_0=0\) cubic metres. The first interval change is \(4.5-0=4.5\) cubic metres.
The second interval change is \(3.0-4.5=-1.5\) cubic metres. The negative sign means the tank’s volume decreased over that interval.
The third interval change is \(9.2-3.0=6.2\) cubic metres.
Add the initial volume to each cumulative change to find the volume at the recorded times: \(20+4.5=24.5\), \(20+3.0=23.0\), and \(20+9.2=29.2\) cubic metres.
| Time | Cumulative change | Change during the interval just completed | Tank volume |
|---|---|---|---|
| Start | 0 m³ | — | 20 m³ |
| First recorded time | 4.5 m³ | 4.5 − 0 = 4.5 m³ | 24.5 m³ |
| Second recorded time | 3.0 m³ | 3.0 − 4.5 = −1.5 m³ | 23.0 m³ |
| Third recorded time | 9.2 m³ | 9.2 − 3.0 = 6.2 m³ | 29.2 m³ |
Check by rebuilding the cumulative values: \(0+4.5=4.5\), \(4.5+(-1.5)=3.0\), and \(3.0+6.2=9.2\) cubic metres. The final tank volume is also \(20+9.2=29.2\) cubic metres, as shown in the table.
Tables of Actual Quantities
Sometimes a table lists the value of the quantity at each time rather than cumulative changes from a baseline. The same subtraction principle applies, but the entries have a different meaning: subtract consecutive quantity values to obtain the interval change. You do not add the initial value again if the table already gives actual quantities.
Worked Example: Finding Changes from Position Values
An object’s position, measured in metres relative to a fixed origin, is \(-2\) metres at the start, \(5\) metres at the next time, \(1\) metre at the following time, and \(4.5\) metres at the final time. Find each interval’s displacement and the cumulative displacement from the start.
First interval: \(5-(-2)=7\) metres. Second interval: \(1-5=-4\) metres. Third interval: \(4.5-1=3.5\) metres.
The cumulative displacement is \(0\) metres at the start, \(7\) metres after the first interval, \(7+(-4)=3\) metres after the second, and \(3+3.5=6.5\) metres after the third.
| Time | Position | Interval displacement | Cumulative displacement |
|---|---|---|---|
| Start | −2 m | — | 0 m |
| Next time | 5 m | 5 − (−2) = 7 m | 7 m |
| Following time | 1 m | 1 − 5 = −4 m | 3 m |
| Final time | 4.5 m | 4.5 − 1 = 3.5 m | 6.5 m |
Check each displacement by subtracting consecutive positions: \(5-(-2)=7\), \(1-5=-4\), and \(4.5-1=3.5\) metres. Also, the final position minus the initial position is \(4.5-(-2)=6.5\) metres, matching the final cumulative displacement.
A Common Mistake: Subtracting in the Wrong Order
Suppose cumulative changes at two consecutive times are \(6\) kilograms and \(10\) kilograms. A common error is to calculate \(6-10=-4\) kilograms for the later interval. This reverses the order and gives the change from the later cumulative value back to the earlier one.
The correct calculation is later cumulative change minus earlier cumulative change: \(10-6=4\) kilograms. The cumulative amount has risen by \(4\) kilograms across that interval. A useful check is to add the proposed interval change to the earlier cumulative value: \(6+4=10\) kilograms. The incorrect value fails this check, because \(6+(-4)=2\), not \(10\).
Another possible confusion is treating a cumulative change as the actual quantity. If a quantity begins at \(20\) units and its cumulative change is \(7\) units, the quantity is \(20+7=27\) units, not \(7\) units. Always check whether a table column is labelled as a change from the start or as the quantity itself.
Using Cumulative Tables Reliably
When reading or completing a table, identify its baseline and interpret each column before calculating. Check that all changes and cumulative values use compatible units. A signed change can be negative, while the cumulative change may be positive or negative depending on the net effect so far. When the starting quantity is given, combine it with cumulative change only when finding the actual quantity.
Determine whether the table gives interval changes, cumulative changes from the start, or actual quantity values.
Add an interval change to the preceding cumulative change; subtract consecutive cumulative values to recover an interval change; add the initial quantity to a cumulative change to find the actual quantity.
Verify that consecutive cumulative values differ by the interval change, and that initial quantity plus final cumulative change equals the final quantity.
These tables make the relationship between local changes and an overall net change visible. Their central arithmetic is simple, but careful attention to the starting point, subtraction order, sign, and units prevents most errors.
Check Your Understanding
Answer each question, showing the subtraction or addition used.
- A cumulative-change column reads \(0\), \(5\), \(2\), and \(8\) metres. What are the changes on the three successive intervals?
- A quantity begins at \(14\) litres and has cumulative change \(-3\) litres at a later time. What is its value then?
- Why is the change on an interval found by subtracting the earlier cumulative value from the later cumulative value?
- A table gives actual values \(9\), \(6\), and \(11\) units at consecutive times. What are the changes on the two intervals?
- A cumulative change increases from \(-2\) kilograms to \(1\) kilogram. What is the interval change, and what does its sign indicate?
