REDUCTION. 175. A Compound Number is a concrete number whose value is expressed in two or more different denominations. (110.) 176. Reduction is the process of changing a number from one denomination to another without altering its value. Reduction is of two kinds, Descending and Ascending. 177. Reduction Descending is changing a number of one denomination to another denomination of less unit value; thus, $1 = 10 dimes = 100 cents = 1000 mills. 178. Reduction Ascending is changing a number of one denomination to another denomination of greater unit value; thus, 1000 mills 100 cents = 10 dimes = $1. 179. A Scale is a series of numbers, descending or ascending, used in operations upon compound numbers. NOTE. In simple numbers and decimals, the scale is uniformly 10; in compound numbers the scales are varying. NOTE. This table is given here for the purpose of presenting all the compound numbers together; but Decimal Currency has been previously treated of so fully, that additional exercises are not considered necessary. Reduction What is a compound number? Reduction is what? descending? Reduction ascending? A scale? Scale of simple numbers and decimals? Scale of compound numbers? Repeat the table of U. S. money. 4 farthings (far. or qr.) make 1 penny, 181. English Money is the currency of Great Britain. d. TABLE. 12 pence 66 1 shilling, S. 20 shillings 66 1 pound or sovereign,..£, or sov. SCALE ascending, 4, 12, 20; descending, 20, 12, 4. NOTES. 1. The equivalents of each unit in all lower denominations may easily be learned, when but few denominations and small numbers are used. 2. Farthings are generally expressed as fractions of a penny; thus, 1 far., sometimes called 1 quarter, (qr.) =4d.; 3 far. =&d. CASE I. 182. To perform reduction descending. 1. Reduce 21 £ 18 s. 10 d. 2 far. to farthings. OPERATION. 21 £ 18 s. 10 d. 2 far. 20 438 s. 12 5266 d. 4 ANALYSIS. We multiply 21 £ by 20, because there are 20 times as many shillings as pounds, and add 18 shillings to the product, making 438 shillings in 21 £ 18 s. We next multiply 438 shillings by 12, because there are 12 times as many pence as shillings, and add 10 pence to the product, making 5266 pence in 21 £ 18s. 10d. Lastly, we multiply the 5266 pence by 4, because there are 4 times as many farthings as pence, and add 2 farthings to the product; and we obtain 21066 farthings in the given number. Hence the following Ans. 21066 far. RULE. I. MULTIPLY the highest denomination of the given number by that number of the scale which will reduce it to the next lower denomination, and add to the product the given number, if any, of that lower denomination. Repeat the table of English money. The scale. Case I is what? Give explanation. Rule. II. Proceed in the same manner with the results obtained in each lower denomination, until the reduction is brought to the denomination required. CASE II. 183. To perform reduction ascending. 1. Reduce 21066 farthings to pounds. OPERATION. 4) 21066 far. 12) 5266 d. 2 far. 210) 4318 s.+10 d. 21 £.188. Ans. 21 £ 18 s. 10 d. 2 far. ANALYSIS. We first divide the 21066 far. by 4, because there are as many pence as farthings, and we find that 21066 far. 5266 d. + a remainder of 2 far. We next divide 5266 d. by 12, because there are as many shillings 438 s. +10 d. Lastly, we are as many pounds as as pence, and we find that 5266 d. divide the 438 s. by 20, because there shillings, and we find that 438 s.21 £+18 s. The last quotient with the several remainders annexed in the order of the succeeding denominations gives the answer 21 £ 18 s. 10 d. 2 far. Hence the following RULE. I. DIVIDE the given number by that number of the scale which will reduce it to the next higher denomination. II. Divide the quotient by the next higher number in the scale; and so proceed to the highest denomination required. The last quotient, with the several remainders annexed in a reversed order, will be the answer. NOTE. Reduction descending and reduction ascending mutually prove each other. EXAMPLES FOR PRACTICE. 1. In 14194 farthings how many pounds? 2. In 14 £ 15 s. 8 d. 2 far. how many farthings? Case II is what? Give explanation. Rule. Proof, how performed? WEIGHTS. 184. Weight is a measure of the quantity of matter a body contains, determined according to some fixed standard. Three scales of weight are used in the United States and Great Britain, namely, Troy, Apothecaries', and Avoirdupois. I. TROY WEIGHT. 185. Troy Weight is used in weighing gold, silver, and jewels; in philosophical experiments, &c. TABLE. 24 grains (gr.) make 1 pennyweight,. .pwt. or dwt. 20 pennyweights 12 ounces 66 1 ounce, Oz. lb. SCALE-ascending, 24, 20, 12; descending, 12, 20, 24. EXAMPLES FOR PRACTICE. 3. In 5 lb. 7 oz. 12 pwt. 9 gr., how many grains? 4. In 43457 grains how many pounds? Define weight. Troy weight. Repeat the table. Give the scale. 5. Reduce 41760 grains to pounds. Ans. 7 lb. 3 oz. 6. A miner had 14 lb. 10 oz. 18 pwt. of gold dust; how much was it worth at $.75 a pwt.? Ans. $2683.50. 7. How many spoons, each weighing 2 oz. 15 pwt., can be made from 5 lb. 6 oz. of silver? Ans. 24. 8. A goldsmith manufactured 1 lb. 1 pwt. 16 grs. of gold into rings, each weighing 4 pwt. 20 gr.; he sold the rings for $1.25 apiece; how much did he receive for them? Ans. $62.50. II. APOTHECARIES' WEIGHT. 186. Apothecaries' Weight is used by apothecaries and physicians in compounding medicines; but medicines are bought and sold by avoirdupois weight. 112 96 288 = 5760 SCALE ascending, 20, 3, 8, 12; descending, 12, 8, 3, 20. EXAMPLES FOR PRACTICE. Define apothecaries' weight. Repeat the table. Give the scale. |