Please Explain
Take a look at the two multiplications below. What do you notice?
$32 \times 46 = 1472$
$23 \times 64 = 1472$
The digits in this multiplication have been reversed, and the answer has stayed the same!
Is this surprising? Can you find other examples where this happens?
What do you notice about the pairs of two digit numbers that produce this special result?
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?