Challenge Level

Here is a very interesting observation from **Abigail** (Chelmondiston Primary School).

"32 doubled is 64 and 46 halved is 23."

Think about it! Can you find more examples of multiplications that work the same way?

Jasmine took a look at

32 Ã— 46 = 1472

23 Ã— 64 = 1472

and sent us her findings:

I have thought about these numbers and something catches my eye:

If you times the 3 from the 30 with the 4 from the 40, then you get 12.

If you times the units digit numbers (2 and 6) together, then you get 12 again.

The same thing if you times the 2 from the 20 and the 6 from the 60 and so on.

To prove my theory right, here is another example:

48 x 42 = 2016

84 x 24 = 2016

As you can see, the same thing happens here, but the number I get is 16.

**Daniel** (Anglo-Chinese Primary School) used some algebra to look at how the numbers relate to each other and came to the same conclusion about the 'tens' digits and the 'units' digits:

If ab x cd = ba x dc

(10a + b) (10c + d) = (10b +a) (10d + c)

100ac + 10ad + 10bc + bd = 100bd + 10bc + 10ad + ac

99ac = 99 bd

ac = bd

So 32 x 46 = 23 x 64

because 3x4 = 2x6

Daniel gave two more examples:

36 x 21 = 63 x 12 = 756

13 x 62 = 31 x 26 = 806

**Does Abagail's doubling and halving idea work with these examples?**