The 500 Problem

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

1. Introduce the problem as a puzzle. Ensure that the students understand that the answer to both problems is the same. You may wish to get a child to guess a number for a and then substitute it in the problems as an example.
2. 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.
3. 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 think of alternative ways to come to the same answer.

Extension to the problem

This problem can be played with in many ways. First of all spend some time changing 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?

Then look at

What are a, b and c?

Then look at

Solution

Certainly the problem can be solved using guess and check and some other approaches. However, we think that the nicest way to do it is to just observe that in the left subtraction, we get a 5 in the units column. In the right subtraction we get a b in the units column. Hence b has to be 5 because the two answers are the same.

Now tackle the tens column. Since b = 5, the number that is to be found in the units column of the answer of the left subtraction is 4. So going to 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. Are you getting to see a pattern yet? (5 x 9 = 45; 2 x 9 = 18; …)

Incidentally this is a problem where a lot of knowledge is a dangerous thing. Many secondary students will attempt this using algebra. While algebra works it isn’t the slickest method in the initial stages.

Solution to the Extension:

But now the 5000 problem is very interesting. It doesn’t work! There are no values of a, b, c that will make the two subtractions equal. It’s not often you will have given your students something that won’t work. We tend not to do that in maths classes.

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 generally seem to find it fun to do.