Special Numbers
Lots of you sent in particular numbers that
you spotted.
Kristian noticed that $19$
works:
$1+9=10$
$1 \times 9=9$
$10+9=19$
Daniel found that $29$ works:
$(2+9)+(2 \times 9) = 11+18=29$
Bill and Ben discovered that $39$
works:
Because $3 + 9 = 12$ and $3 \times 9 = 27$ and $27 + 12 = 39$
Class 5AA at Raglan Junior sent us
something interesting:
Any two digit number which ends in $9$ will give you the solution.
However you can't use $99$ as the digits are the same.
But why does this work? Daniel, from Bacons
College, sent us his explanation:
The $2$ digit number has $a$ tens and $b$ units, so I can write the
equation for this question like this:
$a + b + a \times b = 10 \times a + b$
so: $a + a \times b = 10 \times a$
so: $ a \times (1 + b) = 10 \times a$
so: $b = 9$
and it turns out that $a$ can be anything!
I think that what Daniel means is that a
can be any of $1, 2, 3, 4, 5, 6, 7, 8,$ because it can't be $9$ (as
Class 5AA pointed out, $99$ isn't allowed). But the idea of using
algebra is a good one, if you've met it. Otherwise, you could just
try all $8$ numbers.