only of the time, i. e. of an hour, to go one league ; this number multiplied by 22 gives 66-13 hours and, that is 13 hours and 12 minutes. In the preceding statements, the known numbers and those required depend on each other in a manner that it will be well to examine. To do this we may resume the first question, in which it is required to find the price of 18 yards, of which 13 cost 130 dollars. It is plain that the price of this piece would be double, if the number of yards were double; that the price would be triple, if the number of yards were triple, and so on; also, that for half or two thirds of the piece, we should have to pay butor of the whole price. Hence, if there be two pieces of the same cloth, the price of the second ought to contain that of the first as many times as the length of the second contains the length of the first; and this circumstance is stated in saying, that the prices are in proportion to the lengths. The relation, or ratio of the lengths, then, is that number, whether whole or fractional, which denotes how many times one of the lengths contains the other. If the first piece had 4 yards, and the second 8, the ratio of the former to the latter would be 2. In the given example, the first piece had 13 yards and the second 18; the ratio of the former to the latter is then 18 or 1. As the prices have the same ratio to each other that the lengths have, 180 divided by 130 must give 1§, which is the case. The four numbers 13, 18, 130, 180, written in this order, are such that the second contains the first as many times as the fourth contains the third, and thus they form what is called a proportion. A relation is not changed by multiplying or dividing each of its terms by the same number. To denote that there is a proportion between the numbers 13, 18, 130 and 180, they are usually written thus, 13: 18:: 130: 180, which is read 13 is to 18 as 130 is to 180; that is, 13 is the same part of 18 that 130 is of 180, or the ratio of 13 to 18 is the same as that of 130 to 180. The first term of a relation is called an antecedent, and the second a consequent. In a proportion there are two QUESTIONS, 2. What is the first term of a relation called, and what the second? antecedents and two consequents, viz :—the antecedent of the first ratio, and that of the second, the consequent of the first ratio, and that of the second. In the proportion 13 : 18 : : 130 : 180, 13 and 130 are the antecedents, and 18 and 180 the consequents. To ascertain that there is a proportion between the four numbers 13, 18, 130 180, we must see if the fraction and 188 are equal-and to do this, we reduce the latter to its lowest term; and if they be equal, as is supposed by the nature of proportion, it follows that by reducing them to the same denominators, the numerators will become equal, and consequently that 18 multiplied by 130 will give the same product as 180 by 13. This is actually the case, and proves that if four numbers are in proportion, the product of the first and last, or of the two extremes, is equal to the product of the second and third, or the two means. We see, that if four numbers were not in proportion, they would not possess the above property; for the fraction which expresses the first ratio not being equivalent to that which expresses the second, the numerator of the one will not be equal to that of the other, when they are reduced to a common denominator. The order of the terms of a proportion may be changed, provided they be so placed that the product of the extremes be equal to that of the means. In the proportion, 13: 18 130 180, the following arrangement may be made: 13: 18 :: 130: 180 13: 130 :: 18 180 180: 130:: : In each one of these, the product of the extremes is formed 18: 13 of the same factors, and the 180 18: 130 : 13 product of the means of the 18: 13: 180: 130 same factors. The second ar18: 180: : 13: 130 rangement is one of those 130 13: 180: 18 which most frequently occurs. 130 180:: 13: 18 Since the product of the means is equal to that of the extremes, one product may be taken for the other; and, as in dividing the product of the extremes by one extreme, we must necessarily find the other as the quotient, so in dividing by one extreme the product of the means, we shall find the other extreme. For the same reason, if we divide the product of the extremes by one of the means, we shall find the other mean. QUESTIONS. 3. Give an example. 4. How do you ascertain whether there is a proportion between four numbers? 5. To what is the product of the extremes equal? We can then find any one term of a proportion, when we know the other three; for the term sought must be one of the extremes or one of the means. The operation by which, when any three terms of a proportion are given, we find a fourth, is called the "Rule of Three." There must always be three terms given, two of which are of the same kind, and the other of the kind of the answer required." Lacroix. SIMPLE PROPORTION : OR, THE RULE OF THREE DIRECT. When the ratios are in the order in which the question is proposed, the proportion is direct. The ratio of the first to the second term, is the same as the ratio of the third to the answer, or term sought. In Direct Proportion, more requires more, and less requires less. If 3 men can build 12 rods of wall in a given time, how many rods can 6 men build in the same time? It is evident that 6 men would perform a greater amount of labor, in a given time, than 3 men; and, generally, the more men the more work in the same time; and the more time, the more work by the same number of men-and the reverse. If 3 men build 12 rods, then 6 men would build § of 12-24 rods: Or, if 3 men build 12 rods, then 1 man would build of 12-4 rods, and 6 men 4×6=24 rods. The ratio of 3 to 12 is the same as the ratio of 6 to 24: for 12=4, and 24-4; also, the ratio 3 to 6 is the same as the ratio of 12 to 24, for 2 and 24=2. By the analysis of the foregoing question, we find that the name of the answer and that number which follows "how many," are to be multiplied together, and the product divided by that number which follows "if.” The same is true, in all questions in Direct Proportion. Hence RULE. the Place the term of demand on that number which follows HOW MANY, HOW FAR, WHAT WILL, WHAT COST, &c. on the QUESTIONS. 6. Having the extremes and one of the means given, how may the other mean be found? 7. Having the means and one of the extremes given, how may the other extreme be found? 8. What is the Rule of Three? 9. How are the ratios? 10. What is meant by the order in which the question is proposed? 11. How is a question known to belong to the Rule of Three Direct? illustrate. 12. Rule for stating the question? right of the perpendicular line, and that which is of the same name, or which follows "IF," or "AT," on the left, and the remaining number, or that which is of the same kind with the answer required, on the right. Cancel, multiply and divide, as before directed. (See p. 44.) NOTE. When either of the terms is a compound quantity, it may be reduced to the lowest denomination mentioned, or the lowest de nomination may be reduced to a fraction of the highest. EXAMPLES. 2. If 5 yards of cloth cost $10, what will 15 yds. cost? How many $15 yards? Operation. If 5 yards cost $10, then one yard would cost $2, and 15 yards would cost 15 times 2, or $30. It will be seen by this example, that the ratio of the first term to the second is the cost of 1 yard; and as the ratio of the third term to the fourth, is the same as the ratio of the first to the second-10=2, and 33=2, it follows, that if we multiply the third term by the ratio of the first to the second, we shall obtain the fourth term. Again: the first term is to the third as the second is to the fourth; for 5=3 and 38=3, therefore, whether we multiply 10 by 3, the ratio of 5 to 15, or 15 by 2, the ratio of 5 to 10, the result is the same. 3. If 12 yards of cloth cost $48, what will 4 yds. cost? Answer, $16. 4. If 4 bushels of wheat cost $8, how much will 16 bushels cost? Answer, $32. 5. If a man earn $24 in 12 days, how much does he earn in 6 days? Answer, $12. 6. If 8 yards of cloth cost $12, how much will 10 yds. cost? Answer, $15. 7. If 10 yards of cloth cost $15, how much will 8 yds. cost? 8. If 6 acres of land are bought for $180, for how much may 15 acres be bought? Answer, $12. Answer, $450. 9. If 15 acres of land cost $450, what will 6 acres cost? Answer, $180. QUESTIONS. 13. When either of the terms is a compound quantity, what is the rule? 14. Analyze question 2d? 15. Rule for mixed numbers? 16. On which side of the line is the numerator of a fraction to be placed? K cost? 10. If 18 yards of cloth cost $36, what will 20 yards Answer, $40. 11. If 7 men be paid $8 for a certain amount of labor, what ought 25 men to receive at the same rate? Answer, $30. NOTE. Mixed numbers must be reduced to improper fractions, and the numerators placed on that side of the line where the whole numbers would be placed. It may be observed, that the numerator of a fraction always occupies the same side of the line which a whole number would occupy, standing in the place of the fraction. 12. If 2 horses plough 54 acres in a day, how many acres would 18 horses plough in the same time? Answer, 46 acres. 13. If 3 dollars will buy 62 yards of cloth, how many yards will $40 buy? 14. If 12 dolls. buy 4 yds. of cloth, how many yds. will $174 buy? Answer, 81 yards. Answer, 57 yards. 15. If 14 of a bushel of wheat cost $25, how much will 60 bushels cost? 16. If of a yard cost of a dollar, how much will of a yard cost? 17. If 3 horses consume 4 how many tons will 22 horses Answer, $115. Answer, $21. tons of hay in 4 months, consume in the same time? Answer, 33 tons. 18. If 1 yd. of ribbon cost 8 pence, how many dollars will 72 yds. cost? How many $72 yd. S. $8 Ans. Answer, $8. NOTE. When the answer is required in a different denomination from that given in the supposition, follow the tables from the denomination given to the denomination required. In the last example, the price of 1 yd. is 8d. The answer is required in dollars; there fore, continue the statement by saying 12 pence make 1 shilling and 6 shillings make 1 dollar, the denomination required. Then by cancelling 6 times 12 on the left cancels 72 on the right, 8 being the only number left on the right of the line, and there being no number on the left greater than 1, 8 is the answer in dollars. QUESTIONS. 17. Repeat the note under question 18. 18. What is the ratio of each term in the supposition to each corresponding term in the demand, in question 18? |