ART. 77. Reduction is the process of changing the denomination of a number, without altering its value. Ex. Since 3 feet make 1 yard, yards may be changed to feet by multiplying by 3; and, feet to yards, by dividing by 3: 5 yards=5X3=15 feet: and 15 feet 15÷3-5 yards. Hence, as shown in United States money, Reduction Descending consists in changing a number from a higher to a lower denomination: Reduction Ascending, in changing a number from a lower to a higher denomination. REM.-The Tables teach the names of the different units, and the number of units of one order or denomination which make a unit of the next higher order: they are analogous to the Table of Orders in Simple Numbers. ART. 78. DRY MEASURE Is used in measuring grain, vegetables, fruit, coal, &c. The standard unit of dry measure is the Bushel; a circular measure of 18 inches diameter, 8 inches deep, and contains 2150 cubic inches. Those who resort to TABLES OF UNIT VALUES, to avoid each successive step of an operation, lose a valuable exercise. NOTES.-1. There are two other denominations of Dry Measure, the quarter and chaldron. The quarter contains 8 bushels, of 70 pounds each, used in England in selling wheat. The chaldron, in England, and in some of the U. S., contains 36 bu.; in other States 32 bu., and is used for measuring coal. 2. When Grain and Seeds are bought and sold by weight, 60 pounds of Wheat, 60 of Clover seed, 56 of Rye, Corn, or Flax seed, 32 of Oats, 42 of Timothy seed, and 48 of Barley, make 1 bushel. For information in detail in regard to foreign and domestic weights and measures, see "Ray's Higher Arithmetic." REVIEW.-77. What is Reduction? Give an example. In what does reduction descending consist? In what does reduction ascending consist? REM. What do the Tables teach? 83 TO TEACHERS.-Numerous questions should be asked on each Table, similar to the following: 1. How many pints in 1 quart? in 2? in 3? in 4? in 5? in 6? in 7? in 9? in 10? 2. How many quarts in 1 peck? in 2? in 3? in 4? in 5? in 6? in 7? in 8? in 9? in 10? 3. How many pecks in 1 bushel? in 2? in 3? in 4? in 5? in 6? in 7? in 8? in 9? in 10? 4. How many quarts in 2 pints? in 5? in 6? in 8? in 9? in 11? in 12? in 13? in 14? 5. How many pecks in 8 quarts? in 16? in 24? in 35? in 40? in 49? in 56? in 65? 6. How many bushels in 4 pecks? in 12? in 20? in 11? in 15? in 27? in 32? in 39? ART. 79. THE PRECEDING EXAMPLES SHOW, THAT To reduce quarts to pints, multiply the number of quarts by the number of pints in a quart. To reduce pecks to quarts, multiply the number of pecks by the number of quarts in a peck. To reduce bushels to pecks, multiply the number of bushels by the number of pecks in a bushel. Hence, REDUCTION DESCENDING is performed by Multiplication: the multiplier being that number of the lower order or denomination, which makes a UNIT of the next higher. ART. 80. To reduce pints to quarts, divide the pints by the number of pints in a quart. To reduce quarts to pecks, divide the quarts by the number of quarts in a peck. To reduce pecks to bushels, divide the pecks by the number of pecks in a bushel. Hence, REDUCTION ASCENDING is performed by Division: the divisor being that number of the lower order or denomination, which makes a UNIT of the next higher. REVIEW. 78. For what is Dry Measure used? Repeat the Table. What is the standard unit of Dry Measure? 7. Reduce 3 bushels to pints. SOLUTION. To reduce bushels to pecks, multiply by 4, because there are 4 pecks in a bushel. To reduce pecks to quarts, multiply by 8, because there are 8 quarts in a peck, or 8 times as many quarts as pecks. To reduce quarts to pints, multiply by 2, because there are 2 pints in a quart. 8. Reduce 192 pints to bushels. SOLUTION. To reduce pints to quarts, divide by 2, because there are 2 pints in a quart. To reduce quarts to pecks, divide by 8, because there are 8 quarts in a peck. To reduce pecks to bushels, divide by 4, because there are 4 pecks in a bushel. OPERATION. 3 Bushels. 4 12=pk. 8 96=qt. 2 192=pt. OPERATION. 2)192 Pints. 8)96=qt. 4)12=pk. The two preceding examples show that Reduction Descending and Ascending prove each other. 9. Reduce 7 bushels, 3 pecks, 4 quarts, 1 pint, to pints REVIEW.-78. NOTE 1. What other denominations of Dry Measure! What does the quarter contain? For what is it used? What does the chaldron contain? For what is it used? NOTE 2. How many pounds in a bushel of wheat? How many in other grains? 79. How are quarts reduced to pints? Pecks to quarts? Bushels to pecks? How is reduction descending performed? What the multiplier? 80. How are pints reduced to quarts? Quarts to pecks? Pecks to bushels ? How is reduction ascending performed? What is the divisor? 10. Reduce 505 pints to bushels. SOLUTION. To reduce pints to quarts, divide by 2, and there is 1 left; as the dividend is pints, his remainder is 1 pint. To reduce quarts to pecks, divide by 8, and 4 quarts are left. To reduce pecks to bushels, di vide by 4, and 3 pecks are left. OPERATION. Pt. in a qt. 2)505 Qt. in a pk. 8)252 qt. 1 pt. 7 bu. 3 pk. Ans. 7 bu. 3 pk. 4 qt. 1 pt. The remainder is always of the same denomination as the dividend, Art. 38. Hence, ART. 81. GENERAL RULES. TO REDUCE FROM A HIGHER TO A LOWER ORDER, Rule.-Multiply the highest denomination given, by that number of the next lower, which makes a unit of the higher; add to the product the number, if any, of the lower denomination. Proceed in like manner with the result thus obtained, till the whole is reduced to the required denomination. TO REDUCE FROM A LOWER TO A HIGHER ORDER, Rule.-Divide the given quantity by that number of its own denomination which makes a unit of the next higher. Proceed in like manner with the quotient thus obtained, till the whole is reduced to the required denomination. The last quotient, with the several remainders, if any, annexed, will be the answer. PROOF.-Reverse the operation: that is, reduce the answer back to the denomination from which it was derived. If this result is the same as the quantity given, the work is correct. REVIEW.-81. What is the general rule for reducing from a higher to a r order? From a lower to a higher? What the method of proof? ART. 82. TROY OR MINT WEIGHT Is used in weighing gold, silver, jewels, liquors, &c. NOTE.-The standard unit of weight in the United States, is the Troy pound, containing 5760 grains. For interesting historical and other information with respect to coins, see "Ray's Higher Arithmetic." TEACHERS should ask questions on each Table, as on Dry Measure. 1. Reduce 13 lb. 11 oz. 16 pwt. 14 gr. to grains. SUGGESTION.-When the denominations to be added are small, add while multiplying; when large, beginners should add after multiplying. OPERATION. lb. Oz. pwt. gr. 167 oz. (Brought up.) 3340 pwt. 16 pwt. to be added. 3356 pwt. 24 REVIEW.-82. For what is Troy Weight used? Repeat the Table NOTE. What is the standard unit of weight in the United States ? |