ment, questions which would require two or more, by Simple Proportion. 2. If a man build 27 rods of wall in 3 days, when the days are 12 hours long, how many rods can he build in 9 days, when the days are 16 hours long? If a man, in 3 days, build 27 rods, in 1 day he would build of 27-9 rods. If in one day, 12 hours long, he build 9 rods, in 1 hour he would build of a rod, and in 16 hours 12X16=144=12 rods. If in 1 day, 16 hours long, he build 12 rods, in 9 days he would build 12×9=108 rods, the answer. 36 In this example, 3 days, 12 hours long, are equal to 12× 3=36 hours, and 9 days, 16 hours long, are equal to 16X9 =144 hours. We have, then, this proportion-36 h. : 144h. :: 27 rds.: 108 rods for 144-4 and 108-4. The ratio of the time in the supposition, to the time in the demand, is the same as the ratio of the term sought to the rods in the conditions of the question. That is, the ratio of 36 hours to 144 hours, expresses how many more rods can be built in 9 days, 16 hours long, than in 3 days, 12 hours long. It will be perceived, that the ratio of the time in the supposition to the time in the demand, is the product of two simple ratios. It is a ratio of the ratio of days to days, and hours to hours, (a ratio produced by the multiplication of simple ratios, is called a compound ratio.) Thus if a man in 3 days 12 hours long, build 27 rods of wall, the amount of wall built in 9 days, (the days being of equal length,) is expressed by the ratio of 3 to 9—3—3, that is, he could build 3 times the number of rods-27X3 =81 rods; but the days are 16 hours long: this circumstance, again, affects the result, and is expressed by the ratio of 12 to 16—12=13; that is, the amount of labor performed in 16 hours is greater by than the labor performed in 12 hours, 3 of 81 rds.=27, and 27+81=108 rods. If we multiply these simple ratios, 3×13=4, we have a compound ratio, the same as above, and 27X4=108 rods, the same answer. The ratio of the days to the hours may be expressed thus.-If, in 3 days, 12 hours long, 27 rods of days. hours. wall are built, how many rods can be built in of 1§?of 1944, and 27×4=108, the answer. L From the preceding illustrations we derive the following general RULE : General Rule by Proportion. General Rule by Cancelling. Make that number, which Draw a perpendicular line, is of the same kind with the and place the sign of the ananswer, the third term; and swer on the left, and that of the remaining numbers, number which is of the same take any two that are of the kind with the answer, on the same kind, and place one for right; and of the remaining a first term, and the other for numbers, take any two of the a second term, according to same kind, and place one on the directions in Simple Pro- the right and the other on portion; then, any other two the left, according to the diof the same kind, and so on rections in Simple Proportill all are used. Lastly, mul- tion; then, any other two of tiply the third term by the the same kind, and so on till product of the second terms, all are used. Then cancel, and divide the result by the multiply and divide, as before product of the first terms, and directed. the quotient will be the fourth term, or answer. EXAMPLES. EXAMPLES. 1. If a man build 27 rods 1. If a man build 27 rods of wall in 3 days, when the of wall in 3 days, when the days are 12 hours long, how days are 12 hours long, how many rods can he build in 9 many rods can he build in 9 days, when the days are 16 days, when the days are 16 hours long? hours long? Operation. 27 1008 288 rds. Operation. rods. days 316 hours 4 3 hours 1227 rods 108rods, Answer. Having placed 27 rods, the name of the answer, on the right; and, as a greater amount of labor can be ac 36)3888(108, Answer. |complished by 1 man in 9 days than in 3 days, and in 36 Having plac-16 hours than in 12 hours, we 238 ed 27 rods for place 9 and 16 on the right, 288 the third term, and the remaining numbers we place 9 days for the sec-jon the left. ond; for it is obvious, that a It has been shown by the greater amount of labor could foregoing illustration,that the be accomplished in 9 days by amount of labor done in one man, than in three days. 9 days, 16 hours long, comFor the same reason, 16 hours pared with 3 days, 12 hours is made to occupy the second long, is expressed by the proplace, and 12 hours the first. duct of the ratios of 3 to 9, 2. If 3 men can build 360 rods of wall in 24 days, how and 12 to 16. Hence, by cancelling: If we divide 9 on the right by 3 on the left, and 16,on the right, by 4; and 12 on the left, by 4, we have then, 3 on each side of the line, which cancel each other, and 4 remaining on the right is the product of the ratios, as before shown: 4 times 27 rods is 108 rods-the answer. QUES. 2d, by cancelling. many rods can 8 men build How many rods? in 27 days? Operation. 3 8: 360 327 3 3 24 360 Rods, 1080, Answer. 72)77760(1080, Answer. 72 576 576 3. How many men will it For Question 3d, by take to build a wall 75 rods Cancelling, see next page. long, 8 feet high, 3 feet thick, in 6 days, working 9 hours each day, if 20 men can build| a wall 100 rods long, 6 feet QUES. 3d, by cancelling. high, 4 feet thick, in 12 days, How many men? 75 5×5=40 working 12 hours each day? Statement. 5 1008 3 63 men Ans. 412 3 920 men. 40 men, Ans. 129600)5184000(40 Ans 518400 0 4. If of a yard of cloth, yd. wide, cost-what is 4. If of a yard of cloth, yd. wide, cost £, what is the value of yard, 12 yard the value of ğ yd., 12 yd. 5. If a man travel 240 5. If a man travel 240 mls. in 12 days, when the days are miles in 12 days, when the 12 hours long, how far can days are 12 hours long, how he travel in 27 days, when far can he travel in 27 days, the days are 16 hours long? when the days are 16 hours long? Operation. How far 27 days 9 Days 12 16 hours 4 4hours 12 240 miles 20 720 miles, Ans. 144)103680(720 mls. An. 1008 288 288 0 In Compound Proportion, the terms in the supposition and demand may be distinguished by cause and effect, or producing and produced terms. That which causes any thing, or produces an effect, as men, time, length, breadth, depth, &c., may be denominated a producing term. Thus, in the foregoing question, among the terms of supposition, one man, 12 days, 12 hours long, are the joint cause, or producing terms, and miles, the effect, or produced term. Among the terms of demand, 27 days, 16 hours long, are the joint cause, or the producing terms, and the rods required, are the effect, or the produced term. In all questions in Proportion, the answer required will be either cause or effect (a producing or produced term.) Hence the RULE. When the term required is a produced term, Draw a perpendicular line, and place the terms of demand. QUESTIONS. 1. What is meant by producing and produced terms? 2. Rule, when the term required is a produced term? |