KEY TO CARD No. 31. (for Ans. OPERATION. Let us query what propor6. S. b. tion this lesson is in. Is the 144: 50 :: 12 third term greater than the 50 first? we see it is: does the nature of the question show, 600s. that 120 gallons, the third Pence in a shilling, 12 term, will cost more than 32 gallons ? If so, the proportion First term, 144)7200(50d. || is direct. 720 3080, 2d. term. ...0 120,3d.term. In this last question, the Ist. term, 32)369600(11550d. third term is less than the 32 first, and the nature of the question requires the fourth . 49 term or answer to be less 32 than the second : Therefore less requires less. 176 LESSON 8. 160 If a barrel of wine contain . 160 ing 32 gallons cost £12 16 8 160 what will a pipe of 120 gallons cost? Answer, £48 2 6. ...0 Reduce the given sum into If the second term be in pence. £ 12 16 8 pence, the answer or fourth term will be in pence. Re20 duce them to pounds. 256 12)11550 pence. 12 210) 96|2s. 6d. over. £ 48 2s. over. g. LESSON 9. If 32 gallons of wine cost £ 12 16 8 how much is that a gallon? Answer, £0 8s. Od. Igr. As 32 gal. : £12 16 8 :; 1 gal. to a fourth number. S. That is to say, as 32 gallons are to £ 12 16 8, so is 1 gallon to a fourth number Note. As the third term is one, so multiplication is needless, and division only is necessary; therefore divide the second term £ 12 16 8 by 32 the first term. Divide as in compound division, by 4 and by 8, or by 8 and by 4. £. d. 9 08 0 1 fourth term, or answer. Proof of Lesson 8 and 9, by Compound Multiplication. Multiply £o 80 I by £o 8 0 1 120 gallons. Multiply by 10 10 and by 12. 4 0 2 2 12 £48 2 6 0 LESSON 10. If a servant procure 37} cents a week, how much will he gain in 52 weeks? Answer, 19 dol. 50 cents. w. cents. 1 37.5 :: 52 } cent=.5 of a cent. 52 1950 cents = 19 dol. 50 cents. W. : 750 1875 1950.0 cents. LESSON 11. When a man earns 125 dollars a year, that is, in 365 days, how much may he spend each day and lay up 53 dollars 82, cents at the year's end ? Ans. 195 cents each day. DIRECTIONS. In the first place annex two ciphers to 125 to reduce the dollars to cents, thus, 12500 cents. Subtract 53 dol. 82} cents or 5382.5 Cents remain, 7117.5 This sum 5382.5 cents is, as if it were taken out from the whole sum, and deposited in the chest. Then there remain 7117.5 cents for spending money to be divided into 365 parts. $. cents. $. 365 : 7117.5 :: 1 EXPLANATION. 365)7117.5(19.5 365 As 365 days are to 7117.5 cents, so is 1 day to 19.5 cents. 3467 or, 3285 If 365 days give 7117.5 cents, what will 1 day give? INVERSE PROPORTION. We must now study inverse proportion. When the third term is greater than the first, and the nature of the question requires the fourth term to be less than the second, then the proportion is inverse. This is called more requires less. EXAMPLE. If 4 men can harvest a field of wheat in 2 days, in what time can 8 men do the same work ? Answer, 1 day RULE. Multiply the first and second terms together and divide the product by the third, the quotient will be the four term or answer. 8)8(1 day, answer. In this example the fourth term is as much less than the second, as the third is greater than the first: For, 8 is twice the quantity of 4, but 1 is only half the quantity of 2. Therefore this sort of proportion is properly named inverse ; or, indirect, or, if you please, contrary. AGAIN. When the third term is less than the first and requires the fourth to be greater than the second, the proportion is also inverse. This is called less requires more. EXAMPLE If 12 persons in a family have provisions for 36 days, how many persons will the same provisions serve 18 days? Answer, 24 persons. d. p. d. 36 : 12 : : 18 to a fourth number, 24. That is to say, if 36 days require 12 persons to devour the food, what number of persons will 18 days require? Here the third term is less than the first and requires the fourth to be contrary, that is, as much greater than the second. For, the third term is only half the quantity of the first, but the fourth term is double the quantity of the second. OPERATION. LESSON 12. II 864 feet in length of plank 12 inches wide will floor a room, how much in length will be required when the planks are 18 inches wide ? Answer, 576 feet in length. in. w. mo. mo. OPERATION. in. w. f. 1. Here we say, if 12 inches 12 : 864 : : 18 12 wide require 864 feet in length, what will 18 inches 18)10368(576 wide require ? 90 The third term is greater 136 than the first, and requires 126 the fourth to be less than the . 108 second. 108 LESSON 13. If 500 dollars draw 35 dollars of interest in 12 months, what sum will give the same interest in 3 months ? Answer, 2000 dollars. OPERATION. If 12 months require 500 12 dols. 3 months will require four times that sum, 3)6000(2000 LESSON 14. If 24 horses will eat 36 bushels of oats in 1} day, what number of horses will eat the same quantity in 3 days? Answer, 12 horses. d. h. d. 1.5 : 24 : : 3 14 day=1.5 day. 1.5 If 1.5 day require 24 horses 120 to devour 36 bushels, then 3 24 days will require a less num ber. 3)36.0(12 LESSON 15. If 56 pounds of butter cost £0 18 8 what is that a pound? Answer, 4 pence. lb. d. lb. 18$. 8d. = 224 pence. 56 : 224 :: 1 The third term is less than the 56)224(4 first and requires the fourth to be less than the second. This is less requires less or direct proportion. |