18? What is? What 1% ? What 18? What 18? What 2 yards ? What ?? (If make i yard, then 1 yard and ž are ..) What will of a yard cost? What-11? What **? What 10? What ? 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 7? For 9? For 12? For 20 ? 16. If 6 men can do a piece of work in 12 days, how long will it take I 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 ? (1 as many men.) In 80 days? Exercises for the Slate. 1. If 20 yards of cloth cost $40, what will 8 yards cost ? 1 yard is z of $40; that is, 40 = 20=$2 a yard; then 8 yards are 8X2= $16, 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. $1,80. 4. What will 25 lbs. of sugar cost, at 17 cents a pound? 17 x 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 152-2250. 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 80 ?-20. What will 96 ?-24. What will 2677-6675. 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 60 cents buy 4 lbs. of tobacco, how much will 30 cents buy ?-2. How much will 90 cents ?-6. How much will 120 cents ?-8. How much will $2,10 ?-14. How much will $2,40?16. A. 46 lbs. 10. If 1 pair of gloves cost 75 cents, what will 1 dozen pair cost?-9. What will it doz. ?-1350. What will 2 doz.? 18. What will 21 doz. ?-2250. What will 3 doz. ?-27. A. $90 11. If 3 doz. pair of shoes cost 27 dollars, what will 1 pair cost?-75. What will 24 doz. ?-2250. What will 2 doz.?18. 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 $20,252-9." How long will $49,50 7-22. A. 49 weeks. 14. If a man receive $50 for 2 'inonths' wages, what is that a year ?-300. What will "8 months' come to?-200. 16 months' come to ?-400. 14 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 picce come to ?-30. What will 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 $2 a gallon; what do they come to ?-600. What will 4 hhds. come to?-480. What will 20 hhds.?-2400. A. $2480. 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. If $3600 ?-268750. If $4000 ?-308750. A. $8450. 19. If a hhd. of molasses cost $20,16, how much is it a gallon? (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. The foregoing questions have been solved by a method termed analysis. This method is thonght to accord with the natural operations of the human mind. Men in business scarcely recognise any other. The forınality of statements is rarely if ever made by them; and, when it is made, they do it more for the sake of testing the correctness of the other method, than for any practical purpose. They may have adopted a statement in the commencement of their business, from the circumstance of having been taught it at school; but the longer they continue in business, the less occasion they have for it. There is another method, which consists in ascertaining the ratio or relation which one number has to another. This is used more or less by all, but more extensively by scientific men, and those well versed in mathematical principles. 24, 20. If 8 pair of shoes cost 63"cents, what will 24 pair cost? 3 of 6377 cents, the price of 1 pair, which we multiply by 24 to get the price of 24 pair; thus, 24 x 7=$1,89. But since 7 is a fraction, it would be more convenient to multiply OPERATION. by 24 first, and divide by 8 afterPair. Pair. Cents. wards, as this cannot make any 8 63 difference; and that we may make no mistake in the operation, we 24 will make a statement by writing the 63 cents on the right, as a third 25 2 term (see operation) ; on the left 126 of which we write the multiplier, 24, as a 2d term, and, as a first term, 8)1512 the divisor, 8: then, 63 X 24= 15128 $1,89, the Answer, as Ans. $1,89 before. 21. If 3 yards of cloth cost 24 cents, what will 6 yards cost ? OPERATION. Yds. Yds. Cts. 24 X 6=144 · 3=48, the Ans. 3 6 24 Or, as we know that 6 yards cost 2 6 times as much as 3 yds., that is, = , 3) 144 by simply multiplying 24 by 2, it makes 48, the answer, the same as before. This is a much shorter process; and, could we Ans. $,48 discover the principle, it would oftentimes render operations very simple and short. In searching for this, we shall naturally be led to the consideration of ratio, or relation ; that is, the relation which necessarily exists between two or more numbers. A. 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 ? or, what is the ratio of 10 to 9 ? 9 Q. What is the ratio of 17 to 18 ? A. 19. A. 14 Q. What is the ratio of 18 to 17 ? 4, ratio. 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. A. $1 1. What part of $1 is 50 cents? or, what is the ratio of $1 to 50 cents ? 100 cents; then =}, the ratio, Ans. 2. What part of 5 s. is 2 s, 6 d. ? or, what is the ratio of 58. to 2 s. 6 d.? 2 s. 6 d. = 30 d., and 5s.=60 d.; therefore, i=1, the ratio, Ans. 3. What is the ratio of £1 to 15 s. ? A. Jó=, the ratio 4. What is the ratio of 2 to 3? A. 3. Of 4 to 20? A.5. Of 20 to 4? A. . Of 8 to 63? A. 77. Of 200 to 900 ? A.4. Of 800 to 900 ? A. 13. 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. If 222 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 ratio of 2 to 10 is 10 =5; then, 5 X 8= 40, Ans. But by stating the quesLion as before, we have the following proportions : OPERATION. In this example, we make a new discovery, viz. that the ratio of 8 2 10 8 to 40 (which is 4 =5), is the 10 same as 2 to 10, which is also 5, that is, 2 is the same part of 10 2) 80 that 8 is of 40. mer ber: ext Q. When, then, numbers bear such relations to each other, what are the numbers said to form ? A. A proportion. wa F fouz last To show that there is a proportion between three or more numbers, we write them thus: Melors. Melons. Cents. Cents. 10 :: 8 : 40, which is read, 2 is to iù 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. The 3, being first, may be called the antecedent; 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. Q. What are the ratios in 2 : 10 :: 8 : 40 ? Q. In the last proportion, 2 and 40, being the first and last terms, are called extremes; and 10 and 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, multiplied together, thus; 2 x 40=80, and 10 X 8 Let us try to explain the reason of this. In the foregoing proposition, the first ratio, 2,1=5,) being equal to the second ratio, (=5,) that is, the fractional ratios being equal, it follows, that reducing these fractions to a common denominator will make their numerators alike; thus, and 40 become 16 and i. ; in doing which, we multiply the nu =80. 10 |