3. Reduce £1000 of the different currencies to Federal Ans. £1000. = 4000. currency New England currency= 3333.333. Pennsylvania currency 2666-666. = = 2500. 106. The following are the rates at which some of the foreign coins are estimated at the custom-houses of Florin or Guilder of the United Netherlands $0.40. Thaler of Prussia and N. States of Germany $0.69. Brunswick, Newfoundland, and Canada. $4.00. 107. The quotient arising from dividing one quantity by another of the same kind or denomination, is called a ratio. Thus, the ratio of 12 to 2=26. 12 to 3=2=4. 12 to 4= 3. 12 to 6-2-2. 12 to 12==1. Hence, we see that the ratio of two quantities shows how many times greater the one is than the other. It is therefore evident, that there cannot exist a ratio between two quantities of different denominations. There is no ratio between 12 feet and 3 pounds, for we cannot say how many times 12 feet is greater than 3 pounds. But there is a ratio between 12 feet and 3 feet, which is There is the same pounds. ratio between 12 pounds and 3 The ratio is itself an abstract number; it is not a denominate number. The ratio of 12 feet to 3 feet is 4 units simply; it is neither 4 feet nor 4 pounds, but simply 4 times I; showing that 12 feet is 4 times as great as 3 feet. In this way we find The ratio of 10 yards to 5 yards When the ratio of two quantities is the same as the ratio of two other quantities, the four quantities are in proportion. Thus, the ratio of 8 yards to 4 yards, is the same as the ratio of 12 dollars to 6 dollars; therefore, there is a proportion between 8 yards, 4 yards, 12 dollars, and 6 dollars. The usual method of denoting that four terms are in proportion, is by means of points, or dots. Thus, the above proportion is written 8 yards 4 yards: 12 dollars: 6 dollars; in which two dots are placed between the first and second terms, and between the third and fourth; and four dots between the second and third. The proportion is read 8 yards is to 4 yards as 12 dollars is to 6 dollars. Of these four terms, the first and fourth are called extremes; the second and third are called means. Since in a proportion the quotient of the first term divided by the second, is equal to the quotient of the third term divided by the fourth, we have, using the above proportion, 4 yards 6 dollars If we reduce these fractions to a common denominator, (ART. 40,) they will become mon denominator 4×6, which is in effect multiplying each fraction by 4×6, we have 8×6 or 48=12×4 or 48; that is, the product of the extremes is equal to the product of the means. Again, 8×6=48 4, and 8×6=48 12. Hence, if the product of the extremes be divided by either mean, the quotient will be the other mean. Hence, if the product of the means be divided by either extreme, the quotient will be the other extreme. From the above properties, we see that if any three of the four terms which constitute a proportion are given, the remaining term can be found. 108. The method of finding the fourth term of a proportion, when three terms are given, constitutes the RULE OF THREE. What is the quotient arising from dividing one number by another of the same kind called? What is the ratio of 12 to 2? Of 12 to 3? Of 12 to 4? What does the ratio of two quantities show? Can a ratio exist between two quantities of different denominations? Is there a ratio between 12 feet and 3 pounds? Can the ratio be a denominate number? How are four quantities related when the ratio of the first to the second is the same as the ratio of the third to the fourth? Which are called extremes? Which are called means? To what is the product of the extremes equal? If the product of the extremes be divided by one of the means, what will the quotient be? How many terms of a proportion must be known in order to find the others? 1. Let us endeavor to find the value of 24 yards of cloth, on the supposition that 8 yards are worth $12. It is obvious that the value sought must be as many times greater than $12 as 24 yards is greater than 8 yards. Hence, there is the same ratio between $12 and the value sought, as there is between 8 yards and 24 yards. sequently, we have this proportion : 8 yards 24 yards $12 value sought. Con Taking the product of the means, we have 24 × 12= 288. This, divided by the first term, gives 28-36 for the fourth term sought, which must be of the same kind as the third term; therefore, $36 is the value of 24 yards. NOTE. When we take the product of the means we do not multiply the 24 yards by 12 dollars, but simply multiply 24, the number denoting the yards, by 12, the number denoting the dollars. The product, 288, is neither yards nor dollars, but 288 units. When we divide this product by the first term of the proportion, we do not divide by 8 yards, but simply divide by 8, the number denoting the yards. The quotient, 36, gives the fourth term of the proportion; and since the fourth term is of the same denominate value as the third term, our fourth term, or answer, must be 36 dollars. 2. What will 312 pounds of coffee cost, if 25 pounds cost $3.25? In this example, the ratio of 25 pounds to 312 pounds, is the same as the ratio of $3.25 to the number of dollars sought. Hence, |