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132. If the first term, or either the second or the third, (not both,) can be divided by any number without remainder, the quotients may be used instead of the terms from whence they arise.
33. If 21 yards of lace cost 35 shillings, what cost 29 yards : Stating ; 21yd. : 35s. : : 29yd. Divide the 1st and 2d by 7, and there will arise 3 ; 5 : ; 29, and working with these numbers instead of the numbers in the question, we obtain the answer 21. 8s. 4d. 34. If 6 horses eat 21 bushels of oats in a week, how many bushels will serve 30 horses the same time 2 Stating; 6h. : 21bu. : : 30h. Here dividing the 1st and 3d by 6, we shall have l ; 21 : : 5, whence we have the answer 105 bushels.
ProMiscuous Examples for practice. 37. If a pound of tobacco cost 3s. 9d, what is that per cwt. Ans. 211. 38. Paid ll. 19s. 10d., for 11 men's wages; what sum will pay 15 men for the same time? Ans. 21. 14s. 4d.). 39. Gave 16t. 1s. 9d. for reaping 33 acres of wheat; what will the reaping of a field of 5 acres cost me at that price? Ans. 2l. 8s. 9d. 40. Bought 1 cwt. of sugar for 31. 14s. 8d. what is that per lb. ? Ans. 8d. 41. If 32EE, 34r. ln, of cloth cost 81. 3s. 3d, what is that a yard 2 Ans. 4s. 42. If the expence of housekeeping for 5 days bell. 17s. 9d. what is that per year? Ans. 137 l. 15s. 9d. 43. What sum will purchase an ox, weighing Scwt. 1gr. 231b. at 3s. 8d. per stone 2 Ans. 211. 14s. 0d.;. 44. What is the value of 19 quarters of wheat, at 24l. 6s. 8d. per load Ans. 92l. 9s. 4d. 45. What will 7+ hundred of faggots cost, at 5s. 5d. a score ? Ans. 10l. 3s. 1d.o. 46. How many hours will a person require to count 245000l. supposing he can count 250l. every 3 minutes ? Ans. 49 hours. 47. At 3s.6d. per week, how long can I keep my horse at grass for 5l. 2 Ans. 28 weeks 4 days. 48. If I rent an estate 735l. 10s. 6d. per annum, and pay 245l. 3s. 6d. poor's-rate, how much is that in the pound 2 Ans. 6s. 8d. 49. The week's wages of 9 Irish potatoe-diggers amount to 4 guineas; how many does a person employ who pays them weekly 106l. 17s. 4d. at the same rate Ans. 229. 50. A foreigner on his arrival in England exchanges 3808 florins for 24ll. 19s. 4d.; how much does he receive for each 2 Ans. 1s. 3d.4. 51. If ink be sold at 5s. 6d. per gallon, what must be given for a cask holding 12gal. 34t. 1pt. 2 Ans. 31. 10s. 9d.4. 52. If 13 horses eat up 10a, 2r. 10p. of grass, how many acres will be sufficient to supply the horses of a regiment 1200 strong for the same time 2 Ans. 975 acres. 53. What is the value of 5 fother of lead, at 16s. 4d. per cwt. 2 Ans. 791. 12s. 6d.
teaches from three numbers given to find a fourth, such that whenever the third is less than the first, the fourth will be proportionally greater than the second ; and when the third is greater than the first, the fourth will be always less than the second.
135. Rule I. State the question, and reduce the term as in the foregoing rule.
II. Multiply the first and second terms together, and divide the product by the third; the quotient will be the answer in the same denomination the second term was left in".
* It has been observed, that “ Direct and Inverse Proportion are parts of the same general rule, and in a scientific arrangement it would be best to consider them in that manner.” Thus, from the nature of any given question, it is easy to determine whether the fourth term ought to be greater or less than the second; if greater, the less extreme must be made the divisor, and the
Method of proof. Reverse the question as in the Rule of Three Direct.
1. If 8 men can do a piece of work in 20 days, in what time will the same be accomplished by 16 men :
OPERATIon. one. d. one. 8 : 20 ; ; 16 Earplanation. 8 Here, having stated the question, I mul
- tiply the first and second terms together 16) logo days Ans. viz. 20 × 8, and divide the product 160 by
16 the third; the quotient 10 is the numO ber of days required.
2. What quantity of calico, 5 quarters wide, will be sufficient to lime a garment, containing 3yd. 34r. of cloth, a yard and an half wide 2 qr. 9d, gr. qr.
3. If 8 cwt. be carried 51 miles for a certain sum, how far ought 34 cwt. to be carried for the same money? Ans. 12 miles.
4. How many yards of oil cloth, 3 quarters wide, will be suf. ficient to cover the floor of a passage 2 yards and a quarter wide and 20 yards long Ans. 60 yards.
5. Lent my friend 120l. for 5 months; what sum ought he to lend me for 9 months to requite my kindness? Ans. 66l. 13s. 4d.
other the multiplier; but if less, then the greater extreme must be made the divisor, and the less the multiplier. The truth of the rule may be shewn from the first example; for if 8 men can do a piece of work in 20 days, 16 men will do twice as much in the same time, or as much in half the time, that is in 10 days, which is the answer: and the same may be shewn of any other example under the rule. ar
6. If when the quarterm loaf cost 6d. the threepenny loaf weigh 2lb. 20z. 12dr. what ought it to weigh when the quartern costs 8d. 2 Ans. llb. 1002. lar. 7. lf 2 cwt. be carried 204 miles for a certain sum, how much ought to be carried 24 miles for the same money? Ans. 17ewt. 8. How much must be cut off from a board 8 inches wide, to make a top to a stool 12 inches long and 12 wide 2 Ans. 18 inc. 9. How many crowns are equal in value to 200 half-guineas 2 Ans. 420. 10. A field of wheat can be reaped by 20 men in 4 days; now supposing only 8 men can be hired, how long will they require to reap the same 2 Ans. 10 days. 11. A person drinks 100 bottles of wine, at 2s. 6d. per bottle, in a year; now supposing the price of the wine increases to 4s. 3d. the bottle, how much may he drink without increasing the expence 2 Ans. 58 bottles, 42 rem. 12. A family takes 12 loaves, at 1s. 2d. per loaf, in a week, but bread rising, they make shift with 9 loaves, which cost exactly as much; what is the increased value of the loaf 2 Ans. 1s. 6d. #, 6 rem.
136. COMPOUND PROPORTION.
When five terms are given to find a sixth, this rule is called The Double Rule of Three, or The Rule of Five; also, whatever number of terms is given, it is usually called by the general name of Compound Proportion.
137. Rule I. Let that term be put in the second place which is of the same kind with the answer required. II. Place the terms of supposition one above another in the first place, and the terms of demand one above another in the third, so that the first and third terms in each row may be of the same kind; let them be reduced to the same denomination, and the second to the lowest mentioned in it. III. Examine each stating separately (using the seeond term in common for each) by saying, If the first term give the second, does the third term require more or less than the second * if more, mark the less extreme for a divisor; but if less, mark the greater extreme. III. Multiply the unmarked numbers together for a dividend,