Hey there!
Today I'm going to do something a little bit
old-school and I'm going to be reviewing an
exam from 1866.
So, this is an arithmetic exam and it's from
152 years ago if my arithmetic is any good.
Now, this is called the Regents Questions
and I believe it's from an exam from the state
of New York.
The Regents exams that are designed for I
think graduating high school students to be
able to graduate and get their diploma.
So, to be able to prove you know, everything
that you've sort of learned during your course
is there.
I believe the Regents exams still sort of
exists today in some form but it's not the
same as it was back then.
This is an arithmetic exam and I think it's
since been replaced by things like geometry,
algebra, calculus and many other different
disciplines.
So regardless of what form the Regents exams
look like these days, we're going to have
a look through these very old questions and
just see what life was like back then for
a budding mathematician.
I'm going to go out on a hunch and assume
that there are no calculators involved here,
so you're going to have to be doing all this
by hand and it kind of makes sense that they
want to train you up to be good at arithmetic,
to be good at things like bookkeeping or just
general day-to-day sums.
Now, I got these questions from a book which
was published in 1880.
It's publicly available online and I'll link
to it and it's just a database of all these
past problems going all the way back to the
start.
So, follow along with that if you would like
to and in my next video which will be uploaded
shortly after this one, I will actually work
through in detail one of the more tricky questions
from this exam because in this video, we're
not going to really work them out, just have
a look at what they're like.
So, let's get started.
So, here is page 1.
We have The Regents Questions: Arithmetic
Examination 1.
November 8th, 1866.
Let's go through the problems.
Number 1: Write in figures each of the following
numbers, add them and express in words their
sum.
So, we've got some numbers written down in
words.
It would seem kind of easy to just write them
down in digits but this is an 1866 exam after
all.
So, it's a little bit trickier than that.
They've given us fifty-six thousand and fourteen
thousandths.
So, I guess they're talking about the places
after the decimal point here, so you would
have like your tenths, hundredths, thousandths
and tens of thousands digits.
So, the numbers are a little bit trickier.
We've got nineteen and nineteen hundredths,
fifty-seven and forty-eight ten-thousandths,
twenty three thousand five, and four tenths
and fourteen millionths.
So, a little bit trickier but I think it's
still all right.
Number two; What is the difference between
3 8/4 plus 7 5/8 and 4 plus 2 8/4?
So this is a fractions problem.
We've got some mixed fractions here.
I think the rule is just going to be you need
to add some fractions essentially and to do
that you'll need to, I guess, convert them
to have a common denominator, so getting out
of mixed fractions into improper ones.
Question three: In multiplying by more than
one figure, where is the first figure in each
partial product written, and why is it so
written?
Now, this one seems a little bit confusing.
I guess I don't really know what they're saying.
As far as I can tell, they're talking about
multiplying by more than one figure and to
me that means when you have say two numbers
being multiplied together and each number
is like more than one digit in size and you
do that thing where you do it column by column.
So you'd go 2 by 8, 16 carry 1, 1 by 8 plus
that.
9 and then you do like the next one, the 10s,
so that would be 4 and that would be 2 and
then you add these up.
That would be 3 3 6.
So I think that's what they're talking about
but I guess I don't really know how to answer
the question; where is each partial product
written.
Are these are the partial products?
Are they talking about carrying the one, uh,
something like that.
Number four: If the divisor is 19, the quotient
37, and the remainder 11, what is the dividend?
So, the dividend is the number to be divided
by something.
So, the dividend is divided by 19, the answer
is 37 with a remainder of 11.
So, to work backwards and find what the dividend
is, we do 19 times 37 plus 11.
Number five: What is the quotient of 65 bu.
1 pk and 3 qt. divided by 12?
So, these are a bunch of I guess units that
I'm not familiar with.
I did a quick Google of them and I think it's
called Peck and Bushel but yeah, I guess I
don't really know how many of each of these
is in this one and likewise but you'd have
to convert them down to I guess the Quart
divided by 12 and then maybe convert it back
into these types of units again.
Number six: Which of the fundamental operations
(or ground rules) of arithmetic is employed
in reduction descending?
So, initially I had no idea what reduction
descending is but I gave this one a quick
Google as well and I think it refers to when
you're like going down in denomination.
Like so, if you have dollars and you want
to convert it into cents.
If I had a dollar and I want to tell how many
cents was in it, I would times it by a hundred.
So, I believe this one would be multiplication.
Number seven: In exchanging gold dust for
cotton, by what weight would each be weighed?
This one seems a bit out of my range.
I guess I don't know exactly what they're
saying but I feel like maybe they're on some
old-school scale and they're like needing
to weigh gold dust and cotton against some
standard weight.
I'm not really sure.
Here's one that I think we'll be able to get.
What is the only even prime number?
The answer to that would be 2.
Number nine: How many weeks in (this many
minutes)?
It about eight million.
I think this would be a process of dividing
and probably a series of long division problems.
So you'd want to convert from minutes divide
it by 60 to find out how many hours there
are, divided by 24 to get how many days, divided
by seven days into weeks.
So, a series of division.
Hope you'll sharpen up your long division
skills because mine are very rusty.
Number ten: To what term and division does
the value of a common fraction correspond?
I don't really know what the terms in division
are except that it could be like the dividend
or the divisor, the quotient or the remainder
or something.
I'm gonna have to pass on that one though.
Number eleven: What is the product of a fraction
multiplied by its denominator?
Well, if we had the fraction one-half and
multiplied it by two, you just end up with
one or you just end up with whatever you started
with on the top, so that would be ending up
with the numerator.
Number twelve: What is the rule for the multiplication
of decimals?
Now, this one hasn't been outdated.
This rule still applies.
The rule is that the decimal point is placed
in the product so that the number of decimal
places in the product is the sum of decimal
places and what you're multiplying together.
So, for example if you're multiplying a number
with two decimal places by a number with one
decimal place, your answer will have to have
three decimal places in it.
That's still something that I remember doing
in school.
Number thirteen: How is a common fraction
reduced to a decimal?
I guess to convert a fraction to a decimal,
what you're doing is you're dividing the numerator
by the denominator.
So, I guess that's all you can do, maybe do
it by long division.
Number fourteen: What is ratio and how may
it be expressed?
This one's a little difficult to put into
words but I guess I would say that a ratio
is the number of times one value is like contained
within the other.
15: If this amount of coal costs this amount
of dollars, what will this amount of coal
costs?
I think this is another case of like converting
this whole value into like the lowest denominator.
So like pounds is that?
I don't know, and then working out how much
each pound costs.
Think converting this one into pounds too
and working out its total cost.
Number 16: Find the cost of the several articles
and the amount of the following bill.
This one is a little bit awkward.
I think it might got something to do with
the rent or something.
So we've got A.P Jewett paying to Dr. Samuel
Palmer for some amount feet of board at some
amount per M.
So, I guess this is literally square feet
of space or that you're boarding and this
is either the cost per meter, like square
meter per month.
I'm not sure.
I guess in essence, you'd have to understand
the language here and what's common in terms
of these types of things but it probably comes
down to just adding up these different articles.
17: What is the length of the side of a cubical
box which contains 389,017 solid inches?
Excuse me, but this one seems pretty difficult
to do without a calculator, like so much more
than the other ones.
I guess doing a cube root without a calculator
is not something that I am well-versed in
but you would have to do it through either
like long division and type sums or through
some sort of estimation and checking method.
Number eighteen: What is the present worth
of the following note discounted at a bank,
and when will it become due?
So, we've got a nice little letter here; ninety
days from date, for value received.
I promised to pay to the order of John Smith
one hundred dollars at the Albany City National
Bank.
John Brown.
Forgive me if I'm mistaken, this one seems
to just be asking how much it's worth and
it seems to be a hundred dollars paid 90 days
from the date and this is the date here.
So, maybe this is some basic bookkeeping type
skills although I would say these types of
word problems kind of do age this exam a little.
You really wouldn't see something written
in the same way today.
Something that has not aged itself though,
in fact has aged pretty well is things like
this where it's just using the numbers and
the most basic sentences.
Maybe this one here involve 5/8 to the seventh
power.
That's pretty much the same as what you'd
see today except maybe the word involved would
be changed to find 5/8 or like so, 5/8 or
something like that.
For this question though, it seems easy in
essence.
You just need to take 5/8 times it by itself
seven times, although when you do do it, it
turns out to get a little bit difficult on
the algebra.
For example, I worked out what?
Five eighths to the sixth power would be...
So this would represent what you're going
to have to do in your final sum to get the
seventh power, but to the sixth power, you're
already dealing with fraction 15625 over 262144.
So, already you're getting quite high and
you still need to take this and times it again
by 5 over 8.
So, I don't think that's necessarily like
a walk in the park because even though you
know what to do, you do have to actually go
into the details every time and it would be
a little bit of time to solve it.
Number 20: What is the square root of 0.0043046721?
Now, this question has me a little bit shook
because just like the cube root question on
the previous page, this seems sort of absurd
to be asking I think without the calculator,
but actually I guess it can be done and it's
interesting to me to actually learn more about
how I would have done this.
So, in the next video which I'm going to upload
shortly after this one, I'll take this problem
here and we'll see if we can solve it by hand.
Maybe we'll have a little bit of fun there,
but I'm really interested to actually work
through how you would do that without the
luxury of the calculator.
Maybe I'm too used to the luxury.
Number 21: We have an amount of sugar at a
certain price.
It looks like it's like lost 12% in price,
so how much was the whole cost.
For something like this, I guess in essence
it's the same as what we still do these days.
If you want to find out how much it was worth
before it depreciated, you would take the
total amount now divide it by 0.88 and that
would tell you how much it was worth to start
with.
Number 22: A person owned five eighths of
a mine and sold eight quarters of his interest
for this amount, what was the value of the
entire mine?
I feel like this would have made sense to
me if he was selling some sensible portion
of his share of the mine but it seems like
he's selling... well, two of his share.
So, I don't know how you can actually sell
more than what you have, that sort of seems
like to me.
I guess this must be like a multiplication
and fractions somewhere but the wording is
so awkward I'd probably be a little bit tricked
here.
Nearly at the end.
Number 23.
It's talking about, at a certain time, at
a certain longitude, use that to work out
what time it will be at a different longitude.
This one actually seems interesting to me
because I guess we take for granted like time
zones and longitude and I guess the relationship
between.
But if you really do think about it, every
hour of time zone difference would actually
correspond to a change of 15 degrees in the
longitude.
So, I guess you can use that effect to work
out here how the time is changing.
And finally another question using several
different units here which are not [?] questions.
I mean, that's why it's hard to start with.
I guess converting it down into some unit
that you choose and then dividing this price
by their number.
I don't know what potash is but okay.
I guess [?] this exam spooks me a little because
I've gone through a lot of mathematics but
this - this paper is not really about mathematics.
Arithmetic is what it says.
It's really testing your long division, your
multiplication, your basic skills like that
which can actually fall out of practice the
further you go into maths.
It's sort of funny to talk with my friends
about how bad we are sometimes at arithmetic
considering our qualifications in some of
these really high level ideas.
You know, if we like go out to dinner together,
it will take several of us to just work out
how to split the bill.
It's not like these are the skills that you
actually really practice every day the further
you go, it gets more and more abstract from
here.
But it makes sense.
This is 1866 and I guess even when I was in
high school, our teachers in hindsight's sort
of lied to us and said "you've got to learn
these skills because you're not going to just
have a calculator in your pocket everywhere
you go".
It turns out that's - it turned out to be
sort of true with smartphones and everything
but back in 1866 I guess that sentiment actually
did ring true, that you didn't have a calculator
in your pocket.
I guess it raises the question of nowadays
is arithmetic important.
Like eventually some of these skills have
been phased out even in the Regents exam since
this first one and it's kind of my opinion
that arithmetic is less and less important
to learn and to know and to have so much practice
in because whilst I think it's still valuable
to know the foundations of what you're doing
and to call yourself a mathematician without
understanding how to do long division, it's
a little bit strange I admit but also there's
so much to learn and so much to do that really
we only have time to focus on doing the important
things that are new.
But in having said that, probably around the
world there are still some exams like this.
Some countries I think would focus more on
making sure that someone who wants to call
themselves a mathematician can do these like
square roots by hand.
I never had to prove that but I'm sure somewhere
in the world, this is still the kind of thing
you're having to prove and I guess I'm not
the judge of whether that's good or bad or
not, but respect to the people that could
solve all of this exam no worries because
it's already what they've been practicing.
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