How to use the transposition error detector
- Enter the total of debits and the total of credits.
- Optionally paste the posted entries, one amount per line, if you want the page to point at likely culprits.
- Read whether the difference is divisible by 9 and what the digits of the error look like.
- Check each candidate entry against its source document.
Formula
difference D = |debits - credits| in cents
a swap of neighboring digits changes a number by 9 x (digit difference) x 10^m
a decimal point slid one place changes it by 9 x the smaller amount
If D is not divisible by 9 then one swap or one slide cannot be the whole story. If it is, the page checks each entry for a swap that would produce exactly D, or an amount equal to D/9 or 10D/9.
Books off by $36.00
Debits are $5,481.50 and credits are $5,445.50, so the difference is $36.00. Divided by 9 it is $4.00, which is 4 x 10^2 cents. That fits two neighboring digits that differ by 4 in the place worth $1.00. Among the entries, 904.20 becomes 940.20 when its 0 and 4 swap, a change of $36.00, so it is flagged with the fix to try.
A difference of $90.00 with an entry of 10.00 fits a decimal point slid one place: the entry may be one place too small and should be 100.00. A difference of $900.00 fits an entry of 1,000.00 that should be 100.00.
A hint, not proof
About one in nine unrelated differences is divisible by 9, so a match is a reason to look and not an answer. Check each candidate against the receipt, invoice or bank line before you change a record.
If the difference is not divisible by 9, look for a missed entry, an entry posted twice or on the wrong side, or several errors that add up.
Reading the digits of the error
When the difference divided by 9 is a single digit times a power of ten, the page tells you the size of the digit gap and the place value. A gap of 4 in the dollars place means the two neighboring digits differ by 4 and sit in the tens and units of dollars. Leading zeros and equal digits cannot produce a difference.
For many entries that could add up to the total, use the Subset Sum Reconciliation Finder.
Assumptions and limits
- Totals and entries have at most two decimal places.
- Only one swap of neighboring digits or one slid decimal point is considered at a time.
- The page does not know which side of the books an entry is on beyond the sign of the difference.
- A swap that would create a leading zero is not considered.
- Up to 500 entries are searched.
Frequently asked questions
Why does divisibility by 9 work?
Swapping two neighboring digits a and b changes the number by 9 times (a minus b) times a power of ten, because moving a digit one place multiplies its value by ten. So the change is always a multiple of 9.
What if the swapped digits are not neighbors?
Swaps of digits two places apart change the number by a multiple of 99, which is also divisible by 9, but this page only searches for neighboring swaps in the entries.
Which side of the books has the error?
The page shows which total is larger. The error is on the side with the larger total if an amount was entered too high, or on the other side if one was entered too low.
Can a decimal slide give a different difference?
A slide by one place changes the amount by 9 times the smaller value, so the difference is again divisible by 9. A slide by two places changes it by 99 times the smaller value, which this page does not search for.