we found, when investigating fractions, that, when there were both factors and divisors in a computation, the result was the same, if we divided both a divisor and a factor by the same number, it appears that, if we divide 4 and 12 by 4, we shall have the same proportion, 1:7=3:3, in which X, the answer, is seen, by inspection, to be 21. 2. If a horse travel 30 miles in 6 hours, how many miles will he travel in 11 hours, at that rate? [Ask again, More, or less ?] 6:11 30: x Dividing by 6, 1:11= 5:x=55, by inspection. 3. At $54 for 9 barrels of four, how many barrels may be purchased for $186 ? [More, or less ?] 54 : 186 Dividing by 9, 6 : 186 1 : 31 =1:x= 31, by inspection. Many of these questions may be still further shortened, by abbreviating, mentally, while first stating them. Thus : 4. If 3 men perform a certain piece of work in 10 days, how long will it take 6 men to do the same ? Dividing by 3, 2:1=10:x=5, by inspection. Fellowship.-1. Two men own a ticket ; the first owns 1, the second , of it. The ticket draws a prize of $40. What is each man's share ? 1 fourth Second man, 3 fourths. 9 : * 1:2 6, First man, 4 40 : 3 First man's proportion, 4:1=40: x=10, by inspection. Second 4:3 Dividing by 4, 1:35 10:x=30 40, proof. 2. Two persons have a joint stock in trade. A puts in $250, and B $350 ; they gain $400. What is each man's share ? 600 Dividing by 200, mentally, 3 : 250 = 2: x= evidently ļ of 500 = $1663 $2331 Proof, $400 3. A bankrupt is indebted to A, $780, to B, $460, and to C, $760; his estate is worth only $600. How must it be divided ? A's debt, $780 460 760 66 138 67 3 : x 3 : 7 : 3 $2000 Dividing by 200, mentally, 10 : 780 = 3:x=234, by inspection. 228 $600, proof. Fractions.—1. If tú lb. of sugar cost 75 of a shilling, what will all of a lb. cost ? 11 33 30.43 15 By reversing our divisor, 36, the whole proportion is changed into multiplication of fractions. 30 33 7 Reversing, Х x 7 42 Dividing by 11 and by 15, 7 X 23 Х of a shilling 43 2. If It of a ship costs $252, what is 3 of her worth? 16 3 252 Reversing, 7 X 32 Х 1 3 36 Dividing by 16 and 7, $54, by inspection. 2 3 1 1 This question might as readily have been performed in one line, by reversing and dividing at the same time. Every question is not susceptible of such abbreviations; but a vast majority may be thus considerably shortened, and a large number entirely so, as above, so as to require no multiplication. The pupil should be encouraged even still further to shorten such questions, by resolving all the abbreviating processes into one, mentally, while stating the question. Such a habit is easily acquired. Children of both sexes, under nine years of age, have solved questions like the above, without writing them down, at all, merely by inspecting the book. Where questions cannot be sufficiently abbreviated to be solved by inspection, recourse must be had to the rule, Product of means = product of extremes. Compound Proportion.-Proportion is said to be compound, when the imperfect ratio is not equal to another given ratio, but is compounded of several relations, or ratios. Take, for instance, the following question : 1. If a man travel 273 miles in 13 days, travelling only 7 hours in a day, how many miles will he travel in 12 days, travelling 10 hours in a day? Here it will be perceived, that the question, -How many miles ?-depends neither entirely on the number of days, nor on the number of hours travelled in each day, but is influenced by both. It might be resolved into two questions of simple proportion ; but it is more easily and simply treated as one, of compound proportion, solved, however, on the same principles. Dividing 273 by 13 and by 7, or by their product, 91, 1:12 3:x=360, by inspection. 1:10 S 2. If 6 men build a wall, 20 feet long, 6 feet high, and 4 feet thick, in 16 days, in what time will 24 men build one, 200 feet long, 8 feet high, and 6 feet thick ? } 24 : 6 Contracting, 8-94:6 2 ) 4 20 : 200 20 : 200 10116-:x= -80, 16:x. 6:8 6:& by inspect. 4:6 4:& This was done by dividing 24 and 8 by 8 ; 20 and 200 by 20 ; 3 and 6 (first ratio) by 3; 6 and 6 by 6 ; 4 and 16 by 4. It is hardly necessary to observe, that, in these abbreviations, all the 1s have been omitted, as the multiplying or dividing by that number can produce no change. 3. If 56 lbs. of bread be sufficient for 7 men 14 days, how much bread will serve 21 men 3 days ? 7:21 56 : X. 14 : Contracting by 7 and 14, 1:31 4:x=36, by inspect. 1:35 There being a greater variety of numbers in compound proportion, it admits of contractions more frequently than simple proportion, though there may be some questions, which are not susceptible of any. When multiplication has to be performed, it should be recollected, that the left-hand extreme and the first mean, consist of several numbers, the product of which being severally taken, we proceed as in simple proportion. The teacher should be careful to impress on his pupils the necessity of asking the question, More, or less ? previously to the writing down of every ratio. Thus, in the last question, let the pupil say, How much bread ? 56 : x. If 56 be sufficient for 7, how much for 21 ? More, or less ? More. Then the consequent must be the greater ; 7: 21. If 56 be sufficient for 14 days, how much for 3? More, or less ? Less. Then the antecedent must be the greater ; 14 : 3. 4. If $100 gain $6, in one year, what will $400 gain in 9 months ? 100: 400 Dividing by 100 1:2 1:x=18. 12:9 and 6 and 2, 11:9 Interest.-Let the subject of interest be explained from any of the popular books on arithmetic ; adding, The words, per cent., per ann., are either expressed or } 6:2. } understood, in every question respecting interest, immediately after the rate. Per cent. means for every hundred. When the rate is not expressed, six is always understood. For instance, in the following question, What is the interest of $11.04 for 1 year, at 3 per cent. ? the words, per annum, are understood. And in the question, What is the interest of $150 for 16 days ? the words, at 6 per cent. per ann., are understood, and must be supplied, in stating the question. From the want of a clear understanding of the terms employed, many pupils find the subject of interest exceedingly difficult. Let the teacher repeatedly question bis class, till he is sure they are thoroughly understood. Case I.—Principal, time, and rate, given, to find the interest or amount. 1. What is the interest of $11.04, for 1 year, at 3 per cent. ? 100 : 11.04 51:.1104 Dividing by 100, 3:x=.3312, by inspec. 1: is 2. What is the interest of $150, for 16 days ? 100 : 150 6: x. 360 : 16 and 6. Divide by 6, that is, the upper by 3, the lower by 2. 1:.05 X : .4, by inspection. 8 1:10 Divide by 100. 6:x==$7200, by inspection. 1:120 4. What is the interest of $520.04, for 30 years and 6 months ? 100 : 520.04 12:366 Divide by 100 and by 12, that is, 366 by 2, and 6 by 6. } } Divide by 1.000 ( 6: 16 }1:4. } } 1: {i } } 6:1. |