numbers, they should first be changed to improper fractions, or the fractional part to a decimal. 4. When the vertical line is used, the divisor and (?) are written on the left, and the factors of the dividend on the right. SECOND METHOD. 435. The following method of solving examples in simple proportion without making the statement in form, may be used by those who prefer it. Every question in simple proportion gives three terms to find a fourth. Of the three given terms, two will always be like numbers, forming the complete ratio, and the third will be of the same name or kind as the required term, and may be regarded as the antecedent of the incomplete ratio; hence the required term may be found by multiplying this third term, or antecedent, by the ratio of the other two, (428, II). From the conditions of the question we can readily determine whether the answer, or required term, will be greater or less than the third term; if greater, then the ratio will be greater than 1, and the two like numbers must be arranged in the form of an improper fra tion, as a multiplier; if less, then the ratio will be less than 1, ar the two like numbers must be arranged in the form of a proper fraction, as a multiplier. 1. If 4 tons of hay cost $36, what will 5 tons cost? OPERATION. $36 × $45, Ans. ANALYSIS. In this example, 4 tons and 5 tons are the like terms, and $36 is the third term, and of the same kind as the answer sought. Now if 4 tons cost $36, will 5 tons cost more, or less, than $36? Evidently more: and the required term will be greater than the third term, $36, and the ratio greater than 1. We therefore arrange the like terms in the form of an improper fraction,, for a multiplier, and obtain $45, the answer. 2. If 7 men build 21 rods of wall in a day, how many rods will 4 men build in the same time? In this example, men and 4 men are the like terms, and 21 rods is the third term, and Since 4 men will perform less of the same kind as the answer sought. work than 7 men in the same time, the required term will be less thar 21, and the ratio less than 1. We therefore arrange the like terms in the form of a proper fraction, 4, and obtain by multiplication, 12 rods, the answer. 436. Hence the following RULE. I. With the two given numbers, which are of the same name or kind, form a ratio greater or less than 1, according as the answer is to be greater or less than the third given number. II. Multiply the third number by this ratio; the product will be the required number or answer. NOTES.-1. Mixed numbers should first be reduced to improper fractions, and the ratio of the fractions found according to 418. 2. Reductions and cancellation may be applied as in the first method. The following examples may be solved by either of the foregoing methods. EXAMPLES FOR PRACTICE. 1. If 12 gallons of wine cost $30, what will 63 gallons cost? 15 2. If 9 bushels of wheat make 2 barrels of flour, how many barrels of flour will 100 bushels make? Ans. 223. 3. If 18 bushels of wheat be bought for $22.25, and sold for $26.75, how much will be gained on 240 bushels, at the same rate of profit? Ans. $60. 4. If 6 bushels of oats cost $3, what will 91 bushels cost? 5. What will 87.5 yards of cloth cost, if 1 yards cost $.42?2 6. If by selling $1500 worth of dry goods I gain $275.40, what amount must I sell to gain $1000? 544662+ 7. If 20 men can perform a piece of work in 15 days, how many men must be added to the number, that the work may be accomplished in of the time? Ans. 5. 8. If 100 yd. of broadcloth cost $473.079,-how much will 3.25 yd, cost? 75.375 9. If 1 lb. 4 oz. 10 pwt. of gold may be bought for $260.70, ow much may be bought for $39.50? Ans. 2 oz. 10 pwt. 10. In what time can a man pump 54 barrels of water, if he pump 24 barrels in 1 h. 14 min.? Ans. 2 h. 46 min. 30 sec. 11. If of a bushel of peaches cost $13, what part of a bushel can be bought for $76? Ans. bu. 12. If the annual rent of 46 A. 3 R. 14 P. of land be $374.70, how how much will be the rent of 35 A. 2 R. 10 P.? 2857 13. If a man gain $1870.65 by his business in 1 yr. 3 mo. much would he gain in 2 yr. 8 mo., at the same rate? 3, 14. Two numbers are to each other as 5 to 72, and the less is 164.5, what is the greater? Ans. 246.75. 15. If 16 head of cattle require 12 A. 3 R. 36 P. of pasture during the season, how many acres will 132 head of cattle require? Ans. 107 A. 7 P. 16. If a speculator in grain gain $26.82 by investing $325, how much would he gain by investing $2275.50?/84,28417. What will be the cost of paving an open court 60.5 ft. long and 44 ft. wide, if 14.25 sq. yd. cost $341? 709.86+ 18. At 64 cents per dozen, what will be the cost of 101 gross of steel pens? 8.064 19. If when wheat is 7s. 6d. per bushel, the bakers' loaf will weigh 9 oz., what ought it to weigh when wheat is 6s. per bushel? Ans. 11 oz. COMPOUND PROPORTION. 437. Compound Proportion is an expression of equality between a compound and a simple ratio, or between two compound ratios. It embraces the class of questions in which the causes, or the effects, or both, are compound. The required term must be either· a simple cause or effect, or a single element of a compound cause or effect. FIRST METHOD. 1. If 8 men mow 40 acres of grass in 3 days, how many acres will 9 men mow in 4 days? is to 9 men 4 days, as 40 acres is to (?), or how many acres. Dividing the continued product of all the elements of the means by the elements of the given extreme, we obtain ( ? ) = 60 acres. 2. If 6 compositors in 14 hours can set 36 pages of 56 lines each, how many compositors, in 12 hours, can set 48 pages of 54 lines each? OPERATION. ( ? ) | $ 66 54 ANALYSIS. :: :{ 36 56 : { } 48 54 In this example, an element of the second cause is required; and the statement is, 6 compositors 14 hours is to (?) compositors 12 hours as 36 pages of 56 lines each is to 48 pages of 54 lines each. Now, since the required term is an element of one of the means, we divide the continued product of all the elements of the extremes by the continued product of all the given elements of the means. Placing the dividend on the right of the vertical line and the divisors on the left, and canceling equal factors we obtain (?)=9. ( ) 9, Ans. 438. From these illustrations we deduce the following RULE. I. Of the given terms, select those which constitute the causes, and those which constitute the effects, and arrange them in couplets, putting (?) in place of the required term. II. Then, if the blank term (?) occur in either of the extremes, divide the product of the means by the product of the extremes; but if the blank term occur in either mean, divide the product of The extremes by the product of, the means. NOTES.1. The causes must be exactly alike in the number and kind of their the same is true of the effects. 2. The same preparation of the terms by reduction is to be observed as in imple proportion. SECOND METHOD. 439. The second method given in Simple Proportion, is also applicable in Compound Proportion. In every example in compound proportion all the terms appear in couplets, except one, called the odd term, which is always of the same kind as the answer sought. Hence the required term in a compound proportion may be found, by multiplying the odd term by the compound ratio composed of all the simple ratios formed by these couplets, each couplet being arranged in the form of a fraction. The fraction formed by any couplet will be improper when the required term, considered as depending on this couplet alone, should be greater than the odd term; and proper, when the required term should be less than the odd term. 1. If it cost $4320 to supply a garrison of 32 men with provisions for 18 days, when the rations are 15 ounces per day, what will it cost to supply a garrison of 24 men 34 days, when the rations are 12 ounces per day? BY CANCELLATION. (?)|4320 32 24 18 34 = $4896 ANALYSIS. In this example there are three pairs of terms, or couplets, viz., 32 men and 24 men, 18 days and 34 days, 15 ounces and 12 ounces; and there is an odd term, $4320, which is of the same kind as $4896, Ans. the required term. We arrange each couplet as a multiplier of this term, thus; First, if it cost $4320 to supply 32 men, will it cost more, or less, to supply 24 men? Less; we therefore arrange the couplet in the form of a proper fraction as a multiplier, and we have $4320 × 3. Next, if it cost $4320 to supply a garrison 18 days, will it cost more, or less, to supply it 34 days? More; hence the multiplier is the improper fraction, and we have $4320 × 3 × 1. Next, if it cost $4320 to supply a garrison with rations of 15 ounces, will it cost more, or less, when the rations are 12 ounces? Less; consequently, the multiplier is the proper fraction 4, and we have $4230 × 3 × 14 × 13=$1896, the required term. Hence the following 32 |