(If g make 1 18? What 13? What 1? What l§? What l? What 2 yards? What ? yard, then 1 yard and are .) What will of a yard cost? What ...? What? What? What 24? What 3 yards? 15. The interest of 100 dollars for 1 year is 6 dollars, at 6 per cent.; what is it for 2 years? For 3? For 5? For 71 For 9? For 12? For 20? 16. If 6 men can do a piece of work in 12 days, how long will it take 1 man to do the same? (1 man will be six times as long as 6.) How long will it take 2 men? (2 men will do it quicker than 1 man.) 3 men? 12 men? 17. If 4 men build a wall in 20 days, how many men would it require to build the same in 40 days? ( as many men.) In 80 days? After the same manner perform the following ✅ Exercises for the Slate. 1. If 20 yards of cloth cost $40, what will 8 yards cost? 1 yard is of $40; that is, 40+20=$2 a yard; then 8 yards are 8 × 2➡ 816, Ans. 2. What can you buy 15 tons of hay for, if 3 tons cost $36 (Find what 1 ton will cost first.) A. $180. 3. If 2 bushels of oats cost 40 cents, what will 24 bushels cost? A. $4,80. 4. What will 25 lbs. of sugar cost, at 17 cents a pound? 17 × 25=$4,25, Ans. 5. If $4,25 buy 25 lbs. of sugar, how much is it a pound? A. 17 cents. 6 If 3 pair of shoes cost $4,50, what will 12 pair cost?-18 What will 8?-12. What will 15-2250. What will 16?—24. A. $76,50. 7. If 2 pair of stockings cost 50 cents, what will 3 pair cost 75. What will 15?-375. What will 25?-625. What will 60?-20. What will 96 -24. What will 267 ?-6675. A. $121,50. 8. What will 600 bushels of rye cost, at 84 cents a bushel?504. What will 10?-840. What will 40?-3360. What will 800-672. What will 1000?-840. What will 2?-168. A. $2059,68. 9. If CO cents buy 4 lbs. of tobacco, how much will 30 cents buy ?-2. How much will 90 cents?-6. How much will 120 sents?-8. How much will $2,10?-14. How much will $2,40', 16. 4. 46 lb 10. If 1 pair of gloves cost 75 cents, what will 1 dozen pain Cost?-9. What will 1 doz. ?-1350. What will 2 duz, 118. What will 2 doz. ?-2250. What will 3 doz. ?-27. A. $90. 11. If 8 doz. pair of shoes cost 27 dollars, what will 1 pair cost? 75. What will 2 doz. ?-2250. What will 2 doz. 718. What will 1 doz. ?-1350. What will 1 doz. ?-9. A. $63,75. 12. If 5 tons of hay will keep 25 sheep over the winter, how many sheep can be kept on 7 tons, at the same rate ?-35. On 8 tons?-40. On 15 tons ?-75. On 60 tons ?-300. On 80-400. A. 850. 13. Boarding at $2,25 a week, how long will $9 last me?-4. How long will $13,50?-6. How long will $18?-8. How long will $20,25?-9. How long will $49,50?-22. A. 49 weeks. 14. If a man receive $50 for 2 months' wages, what is that a year?-300. What will 8 months' come to?-200. 16 months' come to?-400. 1 year's come to?-450. 2 years' come to ?600. 10 years' come to ?-3000. A. $4950. 15. What will 6 pieces of cloth, each piece containing 20 yards, come to, at $1,50 a yard ?-180. What will 1 piece come to?-30. What will 3 pieces ?-90. What will 5 pieces?-150. What will 10 pieces?-300. A. $750. 16. Bought 5 hhds. of rum, each containing 60 gallons, for a gallon; what do they come to ?-600. What will 4 hhds. come to?-480. What will 20 hhds. ?-2400. A. $3480. 17. William's income is $1500 a year, and his daily expenses are $2,50; how much will he have saved at the year's end? A. $587,50. 18. If William's income had been $2000, how much would he have saved?-108750. If $2500?-158750. 1 $3600?-268750. If $4000?-308750. A. $8450. 19. If a hhd. of molasses cost $20,16, how much is it a gal -lon (Divide by the number of gallons in a hhd.)-32. How much is it a quart? (Divide by the number of quarts in a hhd.)8. How much is it a pint?-4. How much is it a gill?-1. A. 45 cents ་ OPERATION. 24 63 30. If 8 pair of shoes cost 63 cents, what will 24 pair cost? 24 252 126 8)1512 Ans. $1,89 before. 21. If 3 yards of cloth cost 24 cents, what will 6 yards cost? OPERATION. Yds. Yds. Cts. 24 × 6=144÷3=48, the Ant 3 6 24 6 3) 144 Or, as we know that 6 yards cost times as much as 3 yds., that is, §=♣ by simply multiplying 24 by 2, it makes 48, the answer, the same as before. This is a much shorter process; and, could we discover the principle, it would oftentimen render operations very simple and short. naturally be led to the consideration of ratio relation; that is, the relation which necessarily exists between two of mere numbers. Ans. 8,48 wearching for this, we shall Q. What is the finding what part one number is of another called! A. Finding the ratio, or relation of one number to another. Q. What is ratio, then? A. The number of times one number or quantity is contained in another. Q. What part of 10 is 9-7 or, what is the ratio of 10 to 91 4. H. Q. What part of 3 oz. is 12 oz. 1 or,what is the ratio of 3 es. to 12 os.? A. 24, ratio. Q. What part of 4 yards is 9 yds. 1 or, what is the ratio of 4 to 91 A. = 24. Q. Hence, to find the ratio of one number to another, how do you proceed? A. Make the number which is mentioned last (whether it be the larger or smaller), the numerator of a fraction, and the other number the denominator; that is, always divide the second by the first 1. What part of $1 is 50 cents? or, what is the ratio of 1 to 50 cents? A. $1100 cents; then, the ratio, Ans. 2. What part of 5 s. is 2s. 6d.? or, what is the ratio of to 2 s. 6 d.? 2s. 6d. = 30 d., and 58.60 d.; therefore, 8, the atio, Ans. A., the ratio . Of 4 to 20? A.5. 3. What is the ratio of £1 to 15 s.? 4. What is the ratio of 2 to 3? A. Of 20 to 4? A. . Of 8 to 63? A. 77. Of 200 to 900? A.44 Of 800 to 900? A. 11. Of 2 quarts to 1 gallon? A. 2. Let us now apply the principle of ratio, which we were in pursuit of, to practical questions. PROPORTION. 22. If 2 melons cost 8 cts., what will 10 cost It is evident, that 10 melons will cost 5 times as much as 2; that is, the mtio of 2 to 10 is 45; then, 5 x 8 = 40, Ans. But by stating the ques tien as before, we have the following proportions :- Q. When, then, numbers bear such relations to each other, are the numbers said to form? A. A proportion. Q. How may proportion be defined, then? A. It is an equality of ratios. Q. How many numbers must there be to form a ratio? A. Two. Q. How many to form a proportion? A. At least, three. To show that there is a proportion between three or m aumbers, we write them thus : Melons. Melons. Cents. Cents. 2 : 10 :: 8: 40, which is read, 2 is to 10 as 8 is to 40; or, 2 is the same part of 10 that 8 is of 40; or, the ratio of 2 to 10 is the same as that of 8 to 40. Q. What is the meaning of antecedent? A. Going before. Q. What is the meaning of consequent ? A. Following. Q. What is the meaning of couplet ? A. Two, or a pair. Q. What may both terms of a ratio be called! A. A couplet. Q. What may each term of a couplet be called, as 3 to 4. A. The 3, being first, may be called the ante cedent; and the 4, being after the 3, the consequent. Q. In the following proportion, viz. 2: 10 :: 8: 40, which are the antecedents, and which are the consequents? A. 2 and 8 are the antecedents, and 10 and 40 the consequents. What are the ratios in 2: 10 :: 8: 40? 4. In the last proportion, 2 and 40, being the first and last terms, are called extremes; and 10 an:! 8, being in the middle, are called the means. Also, in the same proportion, we know that the extremes 2 and 40, multiplied together, are equal to the product of the means, 10 and 8, multipled together, thus; X 40=80, and 10 X 880. Let us try to explain the reason of this. In the foregoing proposition, the first ratio,, (=5,) being equal to the second ratio,, (5,) that is, the fractional ratios being equal, it follows, that reducing these fraes tions to a common denominator will make their numerators alike; thus, and, become fg and f; in doing which, we multiply the |