# The 500 problem

*Keywords:*

Here’s a subtraction problem.

The numbers a and b stand for digits.

If the two subtraction sums give the same answer, what digits do a and b stand for?

AO elaboration and other teaching resources

This problem demands a sound knowledge of place value. It is essential for students to understand the grouping and place value basis of our number system. They must be able to:

- Understand that the same digit is used to represent different amounts
- Interpret the value of each digit according to its position
- Express amounts by using digits

### Problem

Here’s a subtraction problem. The numbers a and b stand for digits. If the two subtraction sums give the same answer, what digits do a and b stand for?

### Teaching Sequence

- Introduce the problem as a puzzle. Ensure that the students understand that the answer to both problems is the same. You may ask a student to guess a number for
**a**and then substitute it in the problems as an example. - Give the students time to work on the problem individually before sharing their work with a partner. Encourage the students to discuss the reasons for their guesses. When the students are working on extensions prompt them to look for patterns.
- As the answer may be "guessed" by some students quickly ask them to write their answer and method. If they have guessed the answer encourage them to find alternative solutions.

#### Extension to the problem

This problem can be adjusted in many ways.

For example:

Change the number 5. Replace it in all the subtraction sums by 2 or 8 or ? What happens to a and b? Is there a pattern?

Pose these problems:

5000 | 5abc |

-abc5 | -5000 |

Or

50000 | -5abcd |

-5abcd | -5000 |

### Solution

Several strategies can be used to solve this problem, including guess and check. One way is to just observe that in the left subtraction, there must be 5 ones. In the right subtraction there are b ones. Therefore, b has to be 5 because the two answers are the same.

Consider the tens. Since b = 5, the number that is to be found in the units column of the answer of the left subtraction is 4. In the right subtraction we see that

a = 4. Just check that there is nothing wrong with a = 4 and b = 5.

Changing 5 to 2 will give a = 1 and b = 8; changing 5 to 8 will give a = 7 and b = 2. Perhaps you see that pattern: (5 x 9 = 45; 2 x 9 = 18; …)

#### Solution to the Extension:

5000 problem is very interesting. It doesn’t work! There are no values of a, b, c that will make the two subtractions equal. Posing your students a problem that doesn't work is a worthwhile and legitimate challenge.

The surprising thing is that the 50000 problem works again! And the pattern here looks a lot like the 500 problem!

This problem might make a good investigation. There’s a lot in it and students find it fun to do.

Attachment | Size |
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500Problem.pdf | 97.97 KB |

HeTangohanga.pdf | 152.67 KB |