More old word problems
These word problems were taken from John Stoddard's American Intellectual Arithmetic, published in New York in 1860 by Sheldon & Co.
Solving word problems with algebra is a three-step process:
(1) Define variables and constants, and write equations.
(2) Choose an algorithm for solving the equations, and implement it on a calculator if it is complex one.
(3) Execute the algorithm, and formulate the answer
1. If a hogshead of molasses containing 84 gallons cost $30; how much must it be sold a gallon, to gain 40 percent?
Variables, constants, and equations.
c = 30/84 unit price, in dollars per gallon, paid for molasses;
x unknown unit price for which molasses must be sold;
x = 1.4*c the selling price per gallon must be 40% = .4 higher than
the unit price for which it was bought.
Algorithm.
Compute 1.4*30/84
Execution.
This computation can be done mentally, on paper, or with a calculator.
The answer: The selling price of molasses should be 50 cents per gallon.
2. A merchant bought broadcloth for $1.20 a yard and sold it for 33 1/3 percent more than he gave for it; which, however, was 33 1/3 percent less than his marked price for it. How much was his marked price per yard?
Variables, constants, and equations.
b = 1.20 price, in dollars per yard, for which a merchant bought cloth;
p = 33+1/3 percentage of profit, and "mark-down" from marked price;
s = b*(1+p/100) price for which the merchant sold the cloth, which was 33 1/3% more than he bought it for;
x unknown marked price for cloth,
s = x*(1-p/100) selling price was 33 1/3% less than marked price.
Algorithm.
Compute: x = b*(1+p/100)/(1-p/100) for b = 1.20 and p = 33+1/3.
Execution.
Recommended method: use a calculator.
The answer: The marked price of cloth was $2.40 per yard.
3. A merchant sold a quantity of cloth for $120, and by doing so gained 50 percent. He then sold another quantity, for $120, and thereby lost 50 percent. Did he gain or lose by the bargain, and how much?
Variables, constants, and equations.
x amount of dollars the merchant paid for the first amount of cloth;
y amount he paid for the second amount of cloth;
120 = x*(1+.5) when he sold the first batch for $120, he gained 50%;
120 = y*(1-.5) when he sold the second batch for $120, he lost 50%,
t
total gain or loss;
t = 2*120 - x - y selling price
minus cost of buying.
Algorithm.
Compute: 120*(2 - 1/1.5 - 1/.5)
Execution.
The computation should be done mentally.
The answer is t = -80. This means that the merchant lost $80.