Here
is a fun activity that can be presented in a number of different
ways. The justification uses simple algebra, but the fun is in the oddity.
Consider this very unusual relationship.
Any two-digit number ending in 9 can be expressed as the sum of the product of the digits and the sum of the digits.
More
simply stated:
Any
two-digit number ending in 9
= [product
of digits] + [sum of digits]
One of
the real advantages of algebra is the facility with which, through its use, we
can justify many mathematical applications. Why is it possible to represent a
number ending in a 9 in the following way?
9 = (0 x 9) + (0 + 9)
19 = (1 x 9) + (1 + 9)
29 = (2 x 9) + (2 + 9)
39 = (3 x 9) + (3 + 9)
49 = (4 x 9) + (4 + 9)
59 = (5 x 9) + (5 + 9)
69 = (6 x 9) + (6 + 9)
79 = (7 x 9) + (7 + 9)
89 = (8 x 9) + (8 + 9)
99 = (9 x 9) + (9 + 9)
Why this
actually works?
Let’s use
algebra to clear up this very strange result, established above by example.
We
typically represent a two-digit number as 10t + u, where t represents
the tens digit and u represents the units digit.
Then the
sum of the digits is t + u and the
product of the digits is tu.
The
number meeting the above conditions = 10t + u = (tu) + (t + u)
10t = tu + t
9t = tu
u = 9
This
discussion should evoke a curiosity about numbers with more than two digits.
For example:
109 = (10 x 9) + (10 + 9)
119 = (11 x 9) + (11 + 9)
129 = (12 x 9) + (12 + 9)
Here the
digits to the left of the 9 are considered as a number and treated just as we
treated the tens digit above. The results are the same.
This can
be extended to any number of digits as long as the unit’s digit is a 9.
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