The Newberry herald and news. (Newberry, S.C.) 1884-1903, 1886-03-03
TEACHERS DEPARTMENT.
c. W. WEIGH, A. NJ, EDITOR.
For Teachers Department.
A Problem and a Principle.

Mr.. EDITOR: The following prob
lem appears in Robinson's Progress
i've Practical Arithmetic, No. -,
p. l4S. A laborer receives Sl.4O per
day, and spends SIS for his sup
port, how much does he save in o
week? The special point to which l
desire to direct attention in this ex-
ample is this should the number
of days for the receipts and expen.
ditures be the same? If so, whether
six or seven. l find that students
invariably use seven. / would be
pleased to have you clear up another
point which apparently disturbs the
harmony in the process ofmultiplica-
ton of integers and multiplication OF
fractions. The multiplication Of in
tegers implies an increase; the mul-
tiplication of fractions decrease.
Again, in integers the product Of two
factors usually gives q greater result
than the sum of those same factors,
and in fractions the reverse is true.
In short, I would be pleased to
have the relation between whole num-
bers and fractions explained.
We (our school) are very much in
terested as we'll as gratitied at the
appearance Of your first article on
English grammar An effective plan
is a felt need in our school &. +.
[As Sunday is rest-day, the Ia
borer only receives pay for six day.
work. AS the laborer must be on
expense even while resting, he pays
for seven days support. Hence. 6
times $l.4O will produce SS.4O the
amount Of his wages. And 7 times
SIS will produce S5.25 the amount
of his expenses. Therefore, s8.4O-
1925 will leave S3.15 the amount of
his savings.
Multiplication is the process of
taking one number (the multiplicand)
as many times as there are units in
the other (the multiplicity. The null
tiplier is always an abstract number.
In whole numbers, the unit is one.
In fractions. the unit is one Of the
equal parts into which single thing
may be divided. The fractional unit
varies; the intregal unit is constant.
When seeking principle of fractions
therefore, always reduce the fractions
to q common denominator, that the
fractional unit may be the same to-
all of them.
The multiplication of integral num-
bers is the process OF taking one in
tegral number as many times as there
are integral units in the other. In
every instance but one the product
will be greater than the sum of the
factors, except when one of the fac
tors is unity.
The multiplication of fractional!
numbers is the process Of taking one
fractional number as many times as
there ere fractional units in the other.
In every instance, the sum is greater
than the product, except when both
factors are improper fractions. In
the exception, since the magnitude
of the product depends upon the rel
ativc values of the factors, it may be
either greater or less than the sum.
If we regard every improper fraction
as a mixed number-this is the more
philosophical and simple-the excep-
ton may be omitted; and. then it is
universally true that the sum Of two
fractions is greater than the product.
Multiplication is a short process OF
addition. Addition in arithmetic i'm
plies an increase; therefore, ,multiplica-
ton in arithmetic implies an increase.
But when two fractions-the improp-
er form being excluded-are multi
piled together decrease is usually
the result. The reason is this: Sup
pose we wish to multiply 8 by !.
To do so, we virtually take the ! fS.
Now, the ] Of thing is always less
than the thing itself. Here is plain
" the idea of decrease. But if you
add ! to there can be no decrease.
Hence, the sum Of ! and : is greater
than their product. Similarly for
any two proper fractions.
In arithmetic all numbers are CON-
prised between zero and ininity, zero
being the absence of value and in
6nity being greater than any conceiv-
able number. If2 and ? be the fac
tors of q number, their product is
equal to their sum. This is not true
of any other number or numbers. If
each of two numbers is not less than
3, their product is greater than their
sum. If each Of two numbers is less
than 3, their product is less than their
sum. If, of two numbers, the one is
greater than 2 and the other less
than ,, but greater than |, their pro-
duct is greater than their sum. If, OF
two numbers, the one is greater than
, and the other less than l, their pro-
duet is less than their sum. This re-