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2011-02-01T00:00:01
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Take a look at the two multiplications below. What do you notice?
<br></br>
<br></br>
$32 \times 46 = 1472$ <br></br>
$23 \times 64 = 1472$
<p> </p>
<p>The digits in this multiplication have been reversed, and the
answer has stayed the same! </p>
<p>Is this surprising? Can you find other examples where this
happens?</p>
<p>What do you notice about the pairs of two digit numbers that
produce this special result?</p>
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<p><span class="editorial">Here is a very interesting observation from <strong>Abigail</strong> (Chelmondiston Primary School).</span><br></br>
"32 doubled is 64 and 46 halved is 23."</p>
<p>Think about it! Can you find more examples of multiplications that work the same way?</p>
<p> </p>
<p><span class="editorial">Jasmine took a look at</span></p>
<p><span class="editorial">32 × 46 = 1472<br></br>
23 × 64 = 1472</span></p>
<p><span class="editorial">and sent us her findings:</span></p>
<p>I have thought about these numbers and something catches my eye:</p>
<p>If you times the 3 from the 30 with the 4 from the 40, then you get 12.</p>
<p>If you times the units digit numbers (2 and 6) together, then you get 12 again.</p>
<p>The same thing if you times the 2 from the 20 and the 6 from the 60 and so on.</p>
<p>To prove my theory right, here is another example:<br></br>
<br></br>
48 x 42 = 2016<br></br>
84 x 24 = 2016</p>
<p>As you can see, the same thing happens here, but the number I get is 16.</p>
<p> </p>
<p><span class="editorial"><strong>Daniel</strong> (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:</span></p>
<p> </p>
<p>If ab x cd = ba x dc</p>
<p>(10a + b) (10c + d) = (10b +a) (10d + c)</p>
<p>100ac + 10ad + 10bc + bd = 100bd + 10bc + 10ad + ac</p>
<p>99ac = 99 bd</p>
<p>ac = bd</p>
<p> </p>
<p>So 32 x 46 = 23 x 64</p>
<p>because 3x4 = 2x6</p>
<p><span class="editorial">Daniel gave two more examples:</span></p>
<p>36 x 21 = 63 x 12 = 756</p>
<p>13 x 62 = 31 x 26 = 806</p>
<p> </p>
<p><span class="editorial"><strong>Does Abagail's doubling and halving idea work with these examples?</strong></span></p>
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Please Explain
Visitors to Earth from the distant planet of Zub-Zorna were amazed
when they found out that when the digits in this multiplication
were reversed, the answer was the same! Find a way to explain why
this works.
Numbers and the Number System
Number - generally
Calculations and Numerical Methods
Multiplication & division
Algebra
Algebra - generally
Numbers and the Number System
Properties of numbers