New Math
AS MATh PERSoNS, we often are called upon to help with the next
generation’s homework. I’m sure you’ve all had a sibling, a cousin, or
your college roommate from sophomore year call you up out of the
blue with a math question. You might not have spoken in months—
possibly even years—and suddenly there’s a voice on the phone (or an
email, or a Facebook post) asking you to recite the quadratic formula
off the top of your head. It happens. You can take it as an excuse to feel
put upon, as an opportunity to show off, or as a chance to be of service.
The situation is inevitable. how you handle it is up to you.
Because I spent some years teaching before coming over to the actuarial
field, I enjoy helping people with math
problems. It’s harder to do by phone,
of course, but not impossible. I’ve even
helped people with algebra homework
by text message, and it was fine. Learning occurred.
The other day, I was helping my
youngest niece, Zoe, with a bit of arithmetic. Zoe’s a bright kid, but stubborn.
When she goes off track, it’s difficult to
get her to return to the “proper” course
of action (an endearing trait when it
comes to the choice of silverware at a
formal dinner but dangerous when it
comes to mathematics). In this case, we
were working with rational numbers—
fractions—a perilous area that can cause
lifelong math anxiety.
My concern, as a result, was that she
correct any mistakes as soon as she made
them and that she not practice wrong
division. Where we got tripped up (
naturally) was on a problem that Zoe got
right.
The subject was proper fractions—
strictly between zero and one—and
reducing them to lowest terms. The first
solutions may be emailed
to the author at cont.puzzles@
gmail.com. in order to make the
solver list, your solutions must
by received by
sept. 30, 2011.
problem was 16/64. Since you’re a math
person, you immediately broke 64 into 16
× 4, and reduced it to 1/4. Zoe is not yet
a math person, just a normal 7-year-old.
She crossed out the 6s and got 1/4.
“Hold up there, Zoe,” I said. “That’s
the right answer, but that’s not how you
get it.”
She looked hurt for a moment.
“You said cross off things that were
in both the top and the bottom,” she
countered.
“I did say that,” I explained, and then
froze. You know how it is when you say
something and think you’re making perfect sense—only to discover that you
didn’t make nearly as much sense as you
thought.
“It’s a bit more complicated,” I con-
tinued. “What you did wouldn’t work in
general. What if you were trying to re-
duce 6/9?”
“You can’t reduce six over nine be-
cause there’s only one thing in each
part,” Zoe said triumphantly, as if she’d
seen through a trick I was trying to pull.
I clearly needed to get her to go back to
factoring.
“Well, let’s just see about that,” I said
patiently. “Let’s try the next problem on
your worksheet. I’ll show you how my
way of doing it gets a different answer
than what you’re doing.”
The next problem was 26/65.
As it turns out, there are a few examples of two-digit over two-digit
proper fractions that, when “reduced”
by crossing out a common digit from the
numerator and from the denominator,
result in a true answer. Only one other, I
believe, gives you a completely reduced
fraction. You’ll want to find that.
But that’s not the puzzle—that’d
be too easy. I’m looking for a list of all
three-digit over three-digit proper fractions that can be reduced completely
by Zoe’s method. For example, 106/265
reducing to 10/25 by eliminating the
6s doesn’t count because 10/25 is further reducible according to the normal
rules of arithmetic. On the other hand,
187/880 is one of the ones you will be
looking for since 17/80 doesn’t reduce
any further and 187/880 = 17/80. At the
risk of unnecessarily complicating this,
trailing zeros also don’t count since that’s
normally considered correct. So, for instance, 100/730 reducing to 10/73 doesn’t
count. For any combative types who’d argue for eliminating the middle zero from
the numerator and the ending zero from
the denominator, the answer is still no.
To restate the puzzle, how many
proper fractions of three digits over
three digits can be incorrectly completely
reduced to the correct answer by the removal of a single non-zero digit from both
the numerator and the denominator?
