and dividing these equal quantities by 9, the quotients must be For another example, find the 2d term of the proportion equal, and we have, x = 72 9 = 8, the required term. = X = 48, the 2d term. In each of these examples, the unknown term might, by the changes allowed in proportions, have been put in the fourth place: 69 x 12, is the same as 9: 6 = 12 : x, and 12: x 9:36, might be written 9: 36 12: x. It is usual, in stating a proportion, to put the unknown term always in the fourth place. = Application of the Principles of Proportion. (78.) In all questions which admit of solution by proportion, we have given, the ratio between two quantities, and one term of an equal ratio between two other quantities. Take for an example, the following problem : What is the price of 4 yards of cloth, when 12 yards are sold for 18 dollars? Here the ratio is given between the length of the two pieces of cloth, and one term of the ratio of the corresponding prices; and it is evident that the ratio of the prices should be equal to the ratio of the lengths; or, in other words, that if one piece is twice, three times, or four times as long as the other, the price should be twice, three times, or four times as great. In stating a proportion from this problem, we have only to put these ratios equal, and arrange them so that the unknown term may be the fourth. We have, therefore, as 12 yards is to 4 yards, so is the price of 12 yards, namely, 18 dollars, to the price of 4 yards; or putting for the unknown term, and writing it in the form of a proportion. 2. If 21 men build 18 feet of wall in a given time, how many feet of wall will 31 men build in the same time? Here, the ratio between the number of men must, evidently, be equal to the ratio between the quantities of work accomplished, and the problem gives the proportion, NOTE. In stating a proportion, it is necessary to observe, that as the known term of the second ratio is always put for the antecedent, the corresponding term of the first ratio must also be the antecedent. In the first example, 18, the known term of the second ratio corresponds to 12, the antecedent of the first; 18 dollars being the price of 12 yards: and in the second example, 18 and 21 correspond to each other; 18 feet of wall being the work that 21 men will perform in the given time. 3. If I buy 368,21 acres of land for 12020,25 dollars, and sell 112,5 acres of the same land at the same price per acre, how much am I to receive for it? 4. What is the wages of a man for 311 days, at 16 dollars per month of 30 days? Ans. $165,866. 5. How many yards of matting that is 3 feet broad will cover a floor that is 27 feet long and 20 feet broad? Ans. 60. 6. The joint stock of two merchants is 13,150 dollars; they gain in trade 2600, what part of this belongs to A, whose capital is 8300 ? 13150: 8300 = 2600 : x. This proportion can be simplified by dividing the two first terms by 50(68), which gives, 263: 166-2600: x; x=1641,06, the answer. 8* Inverse Proportion. (79.) Quantities are said to be in inverse proportion, when they are of such a nature that the ratio of any two of them diminishes, when the ratio of the other two increases. For example; the greater the number of men engaged in a piece of work, the sooner it will be accomplished, and the faster a man travels, the less time will it take him to go a given number of miles. For an example of this kind of proportion, 1. Let it be required to determine in what time 16 men will do a piece of work which 11 men can do in 24 days. Evidently, the more the men employed, the shorter the time, and hence instead of stating the proportion, 11:16 24: x; the first ratio must be inverted, for since, from the nature of the problem, x must be less than 24, the consequent of the first ratio must be less than the antecedent, and the proportion must be written 16: 11 24 r. Reducing this proportion 24. we find for the fourth term x = 161. Take for another example; 2. Two men starting at the same time, walk a certain distance, A walks at the rate of 3 miles an hour, and arrives in 13 hours; in what time will B arrive who walks 4 miles an hour? 4:313: x. x= 92 hours. 3. An engineer having raised 100 yards of a certain work in 24 days, with 5 men; how many must he employ to finish a like quantity of work in 15 days? Ans. 8. 4. A garrison of 536 men have provision for 12 months; how long will these provisions last if the garrison be increased to 1124 men? Ans. 174 days. PROMISCUOUS EXAMPLES. 281 5. What is the interest of 275 dollars for a year at 5 per cent? 100: 275 51: x. Reducing the fractions to decimals, 100: 275,755,25: x. Dividing the first ratio by 25, 4: 11,035,25: x. x = 14,476875. 6. What length must be cut off a board 72 inches wide to contain a square foot? Ans. 181. 7. A wall that is to be built to the height of 36 feet, was raised 9 feet by 16 men in 6 days; how many men must be employed to finish it in 4 days at the same rate of working? Ans. 72. 8. The circumference of the earth is about 24877 miles; at what rate per hour is a person at the equator carried round, one whole revolution being made in 23 hours 56 minutes? Ans. 1039 miles. COMPOUND PROPORTION. (80.) A required quantity may be so related to other quantities, that in order to determine it, we are under the necessity of making use of two or more proportions. The following is a question of this kind :— If the wages of 4 men, for 7 days, be 42 dollars, what will be the wages of 14 men for 9 days? This question involves two proportions. Allowing the same number of men, the wages must be proportional to the number of days. This gives the proportion; for 10 days. In the next place, the wages must be proportional to the number of men employed; and from this condition we have the proportion, (81.) This method may be abbreviated by the following RULE. Place that quantity which is of the same kind as the required quantity, in the 3d term of a proportion, and x to denote the unknown term, in the 4th. Arrange above one another the several given ratios, according as they may be direct or inverse when compared with the ratio of the 3d term to the required term. Multiply these ratios. together, term by term, continually. The result will be a simple proportion, which reduced, gives the answer. For an example, be the second ratio. the time, which is take the preceding problem; 42: x will The given ratios are, 1st, the ratio of days. days. 7: 10; 2d, the ratio of the number of men, which is 4: 14. Arranging these according to the rule, we have Dividing the first ratio of this last proportion by 28, we have (68), 1:542x, whence, 210, the answer. Take for another example, the following, in which several ratios are given, some direct, and some inverse. If 180 men, in 6 days, when the days are 10 hours long, can dig a trench 200 yards long, 3 wide, and 2 deep; in how many days of 8 hours long, will 100 men dig a trench 360 yards long, 4 wide, and 3 deep? length of the trench direct, 200: 360 Ratio of the width of the trench direct, 3:4 depth of the trench direct, length of the days inverse, Multiplying according to the rule, 2:3 8:10 6:x. 96000: 7776006: x. By dividing both terms of the first ratio of this last propor tion by 1200, we have, 80: 648 6: x. |