Algebra word problems from two 1850's U.S. texts
Dave L. Renfro
had me looking at a lot of old algebra texts -- an overview of
a method of completing the square called the "Hindu Method" and
how prevalent it was in old textbooks (scheduled to appear in
the July 2007 issue of The Mathematical Gazette). Back when
I was doing this, I made note of anything interesting that I
came across. I may post some of these things from time to time.
Today I thought I'd post some algebra word problems from a
couple of books published during the 1850's. Each problem
is given exactly as it appeared, including the use or non-use
of commas in numbers and the spellings of words. A fraction 1/4
that appears in vertical form in the text is written as \{1}{4}.
Italized "math type" letters, such as 'a', are written as $a$,
unless they appear in a vertical fraction form such as \{a}{b}.
Ordinary text italized for emphasis is underscored, as one might
do for Pauli's quote "... what you said is not even _wrong_".
Disclaimer: The following problems were specifically chosen
for issues that would be considered inappropriate or strange
from our present vantage point, and do not necessarily
represent the typical problem in either text.
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William Smyth, "A Treatise on Algebra for the Use of
Schools and Colleges", Sanborn, Carter, Bazin & Co.,
1852, 336 pages.
http://www.famousamericans.net/williamsmyth/
http://www.tscholars.com/encyclopedia/William_Smyth
p. 8 (#10): "A man leaves by will his property, amounting to
$14000, to his wife, two sons and three daughters; each son
is to receive twice as much as a daughter, and the wife as
much as all the children together. What will each receive?"
p. 15 (#4): "A General having lost a battle found that he had
only 3600 men more than half his army left, fit for action;
600 more than one-eighth of his men being wounded, and the rest,
which were one-fifth of the whole army, either slain, taken
prisoners or missing. Of how many men did his army consist?
Ans. 24,000."
p. 17 (#13): "On an approaching war 594 men are to be raised
from three towns A, B, C, in proportion to their population.
Now the population of A is to that of B as 3 to 5; whilst the
population of B is to that of C as 8 to 7. How many men must
each town furnish?
Ans. A 144, B 240, C 210."
pp. 68-69 (#5): "Three men, A, B, C, driving their sheep to
market, says A to B and C, if each of you will give me 5 of
your sheep, I shall have just half as many as both of you
will have left. Says B to A and C, if each of you will give
me 5 of yours I shall have just as many as both of you will
have left. Says C to A and B, if each of you will give me
5 of yours I shall have just twice as many as both of you
will have left. How many had each?
Ans. 10, 20, and 30 respectively."
p. 149 (#9): "Two messengers, A and B, were dispatched at the
same time to a place 90 miles distant; the former of whom
riding one mile an hour more than the other, arrived at the
end of his journey an hour before him. At what rate did each
travel per hour?
Ans. A 10 miles, B 9."
p. 225 (#4): "A gentleman divided £210 among three servants
in geometrical progression; the first had £90 more than the
last. How much had each?"
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Horatio Nelson Robinson, "A Theoretical and Practical
Treatise on Algebra", 38'th edition, Ivison & Phinney,
1859, viii + 9-360 pages.
http://library.indstate.edu/level1.dir/cml/rbsc/floyd/math.html
p. 65 (#17): "In the composition of a quantity of gunpowder
The nitre was 10 lbs. more than \{2}{3} of the whole,
The sulphur 4\{1}{2} lbs. less than \{1}{6} of the whole,
The charcoal 2 lbs. less than \{1}{7} of the nitre.
What was the amount of gunpowder?
Ans. 69 lbs"
p. 66 (#21): "A man driving a flock of sheep to market, was
met by a party of soldiers, who plundered him of \{1}{3} of
his flock and 6 more. Afterwards he was met by another company,
who took \{1}{2} what he then had and 10 more: after that he
had but 2 left. How many had he at first?
Ans. 45."
p. 66 (#24): "A person engaged to work $a$ days on these conditions:
For each day he worked he was to receive $b$ cents, for each day
he was idle he was to forfeit $c$ cents. At the end of $a$ days
he received $d$ cents. How many days was he idle?
Ans. \{ab-d}{b+c} days."
p. 68 (#34): "My horse and saddle are together worth $a$ dollars,
and my horse is worth $n$ times my saddle. What is the value
of each?
Ans. Saddle \{a}{n+1}. Horse \{na}{n+1}."
p. 96 (#4): "A man when he was married was 30 years old, and his
wife 15. How many years must elapse before his age will be three
times the age of his wife?
Ans. The question is incorrectly enunciated; 7\{1}{2} years
_before_ the marriage, _not_ after, their ages bore the
specified relation."
p. 186 (#7): "A person dies, leaving children and a fortune of
$46,800, which, by the will, is to be divided equally among
them; but it happens that immediately after the death of the
father, two of the children also die; and if, in consequence
of this, each remaining child receive $1950 more than he or
she was entitled to by the will, how many children were there?
Ans. 8 children."
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Dave L. Renfro