Instead of dividing the number formed by the outside digits (9, 36, 54) by 9, we might divide the full amount of the difference (999, 3,996, 5,994) by 999; this would give the same result.
The reason for the divisibility of this difference by 999 in an error of this kind is apparent when a number is given algebraic notation instead of Arabic. The Arabic number 2,197 expressed algebraically would be 2,000 + 100 + 90 + 7. Generalizing, we may formulate any number of four figures by 1,000a + 100b + 10c + d, in which a, b, c, and d may have values from 0 to 9 inclusive. A transposition between the thousands and units digits, the “a” and the “d,” would result in the following number: 1,000d + 100b + 10c + a. The error would therefore be:
| Original number | 1,000a | + 100b + 10c + d |
| Transposed number | a | + 100b + 10c + 1,000d |
| Difference | 999a | - 999d |
This error is plainly divisible by 999, and the resulting quotient (a-d) is the difference between the two transposed digits.
It may be shown similarly why 99 is a divisor of the error cited under case (c) above.
Transplacements.—A transplacement or slide occurs when some or all of the digits of a number are moved one or more places to the right or left without change in the order of the figures; for instance, 736 written as 73.60, as 7.36, or as 700.36. The first is called a one-column slide, the second and third two-column slides. The error caused by a one-column slide is always divisible by 9, a two-column by 99, a three-column by 999, etc. The division by 9, 99, 999, etc., disregarding decimals, always gives the figures whose transplacement has caused the error. Thus the error caused by writing 736 as 73.60 is 662.40, which divided by 9 is 736; or 736 written as 7.36 produces an error of 728.64, which divided by 99 gives 736; or 736 written as 700.36 causes an error of 35.64, which divided by 99 gives 36, the part transplaced. The reason is similar to that given above for the transposition.
When a whole number of dollars is written as cents, the resulting error is divisible by 9 and moreover the cents added to the dollars gives 99 in each case. For instance in writing:
| .73 | instead of | 73.00, | the resulting error is | 72.27 |
| .58 | ” ” | 58.00, | ””” ” | 57.42 |
| .16 | ” ” | 16.00, | ””” ” | 15.84 |
When the error in the trial balance is of this kind, the amount transplaced may be found by subtracting the cents of the error from 100. In the above examples this difference would be 100-27, 100 -42, 100-84, or 73, 58, and 16 respectively, which are in each case the figures of the transplaced amount as seen in the example. Having determined this, the trial balance and ledger accounts should be gone over to look for a slide of the given number.
Checking the Postings.—From the above discussion, the impossibility of determining in all cases the nature of the error is quite evident—particularly as to whether it is one caused by a transposition or a slide. Unless the kind of error is readily discernible, it is usually advisable to employ the method of checking, i.e., going over all the work of posting to determine its correctness—or other methods to be discussed in [Chapter LI]. After all, careful work in making the record with legible figures and in proving additions and subtractions, wherever possible, more than pays for itself in the time saved hunting for errors caused by slovenly and inaccurate work.