## Thursday, February 14, 2008

### 16/64 = 1/4...How to Reduce Fractions the 'Easy Way'!

Totally Clueless sent me an email reminding me of some famous 'fractured fraction' examples like the one in the title. Can you think of a couple of other two-digit examples of the same type that 'reduce' this way? Note that 10/30 = 1/3 doesn't qualify (the digits have to 'cancel' diagonally!).

Here is TC's version:
Note that the product 16 x 4 can be obtained by deleting the '1' and the 'x'!

(a) Find the other two instances of this 'weird' multiplication. The two factors have to be of the same type as in the example, i.e., a 2-digit number by a 1-digit number and the tens' digit of the 2-digit number must be 1.

(b) Most would find the other instances by guess-test. Here's a more significant challenge. Verify algebraically that there are exactly three such solutions.

(c) Is this problem equivalent to the 'easy way' to reduce fractions mentioned in the title of this post? Why or why not?

Eric Jablow said...

Do you mean you have never heard of the freshmen's universal laws of commutativity, associativity, and distributivity?

Dave Marain said...

Eric--
Probably the same laws as for freshmen, sophs, juniors and seniors in high school. My favorite is the fallacy that one can perform any random operation on both sides of an equation with impunity. For example, a^n = b^n iff a = b. This and the belief that √ symbol implies both positive and negative values. Now you've done it, I may have to start a post of the TEN most common student errors/fallacies in math, even though this has been done numerous times on other web sites. Maybe I should start another poll! I'll bet you have your own Top ten list in mind...

Totally_clueless said...

Hi Dave,

In (a), it may be unnecessary to specify that the 10's digit is a 1. Wouldn't you get the same 3 solutions nevertheless.

TC

Eric Jablow said...

1. √(a²+b²) = a+b

2. (-x)ⁿ = - xⁿ

3. (fg)ʹ = fʹgʹ

Matt Doar said...

I make it 4 cases:

16/64 = 1/4
19/95 = 1/5
26/65 = 2/5
49/98 = 4/8

Anyone generalized it to more than 2 digits?

~Matt

Anonymous said...

199/995 = 1/5
742/424 = 7/4
654/545 = 6/5
484/847 = 4/7
499/998 = 4/8
166/664 = 1/4
266/665 = 2/5

Ohel Moed said...

187 / 880 = 17/80

Krishnanunni said...

what are these numbers called 98/49, 64/16 95/19 etc ???

Lulu said...

These kind of fractions are countable infinite
1/4 = 16/64 = 166/664 = 1666/6664 =
...
1/9 = 19/95 = 199/995 = 1999/9995 = ...
So are the rest