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Please Explain

Age 11 to 14 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 × 46 = 1472
23 × 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?