When a two digit number is reversed it decreases by 1.9 how many such numbers are possible

A transposition error describes an event where a bookkeeper accidentally reverses two adjacent digits, when recording transactional data. Although this error may seem small in scale, it often results in substantial financial incongruities that can have a great impact in other areas. Transpositional errors, which tend to occur in accounting firms, brokerages, and other financial services providers, fall under the broader category of transcription errors.

• A transposition error is a data entry snafu that occurs when two digits are accidentally reversed.
• These mistakes are caused by human error.
• Though seemingly small in scope, transposition errors can result in significant financial consequences.
• If a business’ accounting records show a discrepancy, the difference between the correct amount and the incorrectly-entered amount will be evenly divisible by 9.

Transposition errors are generally the result of human error. For example, in accounting, when a bookkeeper manually enters data into a ledger, they may, by mistake, incorrectly transfer information from an invoice into a balance sheet. Transposition errors may also occur when checks are filled out incorrectly, resulting in improper payment amounts that can cause overdrafts and other banking issues. Furthermore, transportation errors can result in incorrectly-recorded phone numbers, street addresses, or ZIP codes in customer profiles. And although the aforementioned mistakes are typically easily remedied, in some cases, transposition errors relating to medicinal dosing information may lead to tragic consequences.

If a company fails to catch and correct transposition errors, the incorrect value of assets may be perpetuated to outside agencies and individuals, such as corporate shareholders and the Internal Revenue Service. This could cause a chain effect of inaccuracies. For example, a business may be saddled with an increased tax liability if the transposition error is large enough to slingshot that company into a higher tax bracket. Of course, this largely depends on the degree of error in question. If a bookkeeper mistakenly writes $24.74 instead of $24.47, the resulting $0.27 discrepancy would hardly be consequential. On the other hand, if $1,823,000 were accidentally recorded as $1,283,000, the resulting $540,000 error is sure to have a profound financial ripple effect.

Transposition errors made in the trading world are sometimes called "fat-finger trades." In one famous example, a Japanese trader accidentally ordered 1.9 billion shares in Toyota. Luckily, that order never went through.

Interestingly, transposition errors may be revealed by a rather peculiar mathematical phenomenon. Pointedly: the difference between the incorrectly-recorded amount and the correct amount will always be evenly divisible by 9. For example, if a bookkeeper errantly writes 72 instead of 27, this would result in an error of 45, which may be evenly divided by 9, to give us 5. Likewise, if a bookkeeper mistakenly records 63 instead of 36, the difference between those two figures (27) may be evenly divided by 9 to give us 3. Bank tellers can use this rule to detect errors.

Transposition errors also describe scenarios where bookkeepers enter spreadsheet data into the wrong cells.

Here’s a cool mathematical magic trick. Write down a three-digit number whose digits are decreasing. Then reverse the digits to create a new number, and subtract this number from the original number. With the resulting number, add it to the reverse of itself. The number you will get is 1089!

For example, if you start with 532 (three digits, decreasing order), then the reverse is 235. Subtract 532-235 to get 297. Now add 297 and its reverse 792, and you will get 1089!

Presentation Suggestions:
You might ask your students to see if they can explain this magic trick using a little algebra.

The Math Behind the Fact:
If we let a, b, c denote the three digits of the original number, then the three-digit number is 100a+10b+c. The reverse is 100c+10b+a. Subtract: (100a+10b+c)-(100c+10b+a) to get 99(a-c). Since the digits were decreasing, (a-c) is at least 2 and no greater than 9, so the result must be one of 198, 297, 396, 495, 594, 693, 792, or 891. When you add any one of those numbers to the reverse of itself, you get 1089!

See the reference for more mathematical magic tricks.

How to Cite this Page: 
Su, Francis E., et al. “Magic 1089.” Math Fun Facts. <//www.math.hmc.edu/funfacts>.

References:
A. Benjamin and M. Shermer, Secrets of Mental Math, Three Rivers Press, 2006.

Fun Fact suggested by:
Francis Su

Tags: algebra, easy, magic trick, math tricks, mathematical magic, other

This problem also offers the opportunity for students to "play" with numbers. As well as practising addition the students are encouraged to look for patterns in their answers. This play encourages students to increase their understanding of numbers and how they relate to one another.  It also helps develop problem solving skill and creativity.

As numbers are 'reversed' they swap places. (eg. 41 to 14) It is therefore important to discuss what is happening to the place value of the numbers.

Required Resource Materials

Take any 2-digit number. Reverse the digits to make another 2-digit number. Add the two numbers together. How many answers do you get which are still 2-digit numbers?

What do the answers have in common?

Teaching Sequence

1. Introduce the problem – you could do this by writing 2 reversed 2-digit numbers eg 14 and 41. Ask the students what they can tell you about the 2 numbers. If they identify that the digits have swapped places then introduce the problem.
2. It is important that they are clear about how to reverse numbers and that they understand the difference between 2 and 3-digit numbers.
3. You may decide to do an example with the class.
4. Discuss with the students the strategies that they could use to add 2-digit numbers.
5. Let the students work on the problem individually before putting them in small groups. Some students are quicker than others when computing and it is important that all students have the opportunity to "play" with the problem before getting them to share their findings. If all students  have some work to bring to the group they are more likely to be involved in the solution.
6. As you circulate, encourage the students to explain how they are getting the answers.
7. Ask the students how they could organise their reversed numbers so that they could look for patterns in the answers. A good starting point would be to sort the 2 and 3-digit answers into lists or they may decide to identify the reversed numbers that give a 2-digit answer on the hundreds board.
8. Share patterns.

Extension to the problem

Is there a pattern in the numbers that give 3-digit sums?

Solution

There are many patterns that can be found in this problem.  Let's try a few numbers and see what we get: 13 + 31 = 44 26 + 62 = 88 47 + 74 = 121 54 + 45 = 99 68 + 86 = 154 Now we can see that if the sum of the digits in the 2-digit number is less than 10 then the sum of the reversed numbers is less than 100. 27 + 72 = 99 The sum of the digits in the 2-digit number determines the sum of the reversed numbers in the following way: If the sum is 6 the answer is 66 (24 + 42 = 66; 15 + 51 = 66 etc)

If the sum is 8 then the sum of the reversed numbers is 88.

You might support your students to notice that the sum in every case above is a multiple of 11.

Solution to the Extension:

Once again the 3-digit sums are all multiples of 11.  To see this notive that 68 + 86 gives the same answer as 66 + 88.  Now both 66 and 88 are multiples of 11, so the sum is too.