A carpenter buys 9 pounds of nails for 45 cents. He afterwards buys 87 pounds at the same rate. How much did he pay for them?
An answer to this question is readily found by multiplying the 87 pounds by 45 cents, the price of the 9 pounds, and dividing the product, 3915, by 9; the quotient, 435 cents, is the answer to the question.
365. - The "Rule of Three." - The process by which this problem is solved is known as the Rule of Three, or Proportion.
In cases of this kind there are three quantities given, to find a fourth. Previous to working the question it is usual to make a statement, placing the three given quantities in such order that the quantity which is of like kind with the answer shall occupy the second place; the quantity upon which this depends for its value is put in the first place, and the remaining quantity, which is of like kind with that in the. first place, is assigned to the third place.
When thus arranged, the second and third quantities are multiplied together and the product is divided by the first quantity; the quotient, the answer to the question, is a fourth quantity. These four quantities are related to each other in this manner, namely: the first is in proportion to the second as the third is to the fourth; or, taking the quantities of the given example, and putting them in a formal statement with the customary marks between them, we have -
9: 45 :: 87: 435,
which is read: 9 is to 45 as 87 is to 435; or, 9 is in proportion to 45 as 87 is to 435; or, 9 bears the same relation to 45 as 87 does to 435..
366. - Couples: Antecedent, Consequent. - These four quantities are termed Proportionals, and may be divided into two couples; the first and second quantities forming one couple, and the third and fourth the other couple. Of each couple the first quantity is termed the antecedent, and the last the consequent. Thus 9 is an antecedent and 45 its consequent; so, also, 87 is an antecedent and 435 its consequent.
367. - Equal Couples: an Equation. - These four quantities may be put in form thus:
45/9 = 435/87
Each couple is here stated as a fraction: each has its antecedent beneath its consequent, and the two couples are separated by a sign, two short parallel lines, signifying equality. This is an equation, and is read thus: 45 divided by 9 is equal to 435 divided by 87; or, as ordinary fractions: 45 ninths are equal to 435 eighty-sevenths.