60 COMPOUND DIVISION. COMPOUND DIVISION. COMPOUND DIVISION teaches to divide nuinbers of different denominations. RULE.—When the divisor does not exceed 12, divide as in short division: when the divisor exceeds 12, divide after the mode of long division. Divide the highest denomination, under which write the quoient, and if there be a remainder, reduce it to the next less denomination, adding thereto the number given, if any, of that denomination, and divide as before; and so proceed, and the quotient Ehus found will be the answer. PROOF-By Compound Multiplication. OF WEIGHTS AND MEASURES. EXAMPLES OF SHORT DIVISION. tb 3 3 gr. yd. fl. in. 4.) 2567 3 2 (5.) 3)41 626 36 (6.) 9.18 2 3 Y. M. W. D. H. min. sec. -7.) 6)487 12 3 6 12 48 36 (8.) 10)11 20 51 20 A. R. P. c.d. c.fl. c.in. bu. p. gt. (9.) 12)126 3 24 (10.) 4)22 122 432 (11.) 12)407 6 4 sig. EXAMPLES OF LONG DIVISION. 1. If 26 yds. of cloth be sold for $32.50, what is the price of 1 yd.? 2. In 16 lots of land there are 69A. IR. 24P.-how much in each? Ans. 4A. IR. 14P. 3. Bought 48lb. of cheese for $528: what did I pay per pound? Note. If the divisor be complex (more than one denomination) reduce the Hivisor and dividend to the same denomination. 4. Paid $6.84 for 281 yards of muslin: how much did I pay per yard? Questions.—How must the different numbers in Compound Addition and Subtraction be placed? Where must you begin to add or subtract? Why do you begin to add or subtract at the right hand denomination ? How must the first column be added? How divided? What sum must you set down? Where do you set the remainder ? What do you carry to the next denomination? What do you do next? How must each denomination be subtracted ? When any one of the denominations in the subtrahend (or lower line) is less than the denomination in the minuend (or upper line) over it, what is the method to be taken? How is the difference, or remain.er, between two given dates found ? In a certain year all the countries except one, changed Old Style to New; in what year did that occur? What country is it where the Old Style is still in use? When one date is in Old Style and the other in the New, what is the method or rule given? What does Compound Mul. tiplication teach? Where do you begin or place the multiplier? Why do you begin to multiply at the right hand ? When you multiply each denomination, what do you divide by? What do you set down, and what must you carry? How do you prove Compound Multiplication ? What does Compound Division teach? &c. &c. VULGAR FRACTIONS. A VULGAR FRACTION is a part of a unit or whole number, expressed by two numbers; one placed directly above the other, with a line between them; thus, & signifies three-fourths of one, and $ signifies one quarter of one. The number above the line is called Numerator, because it shows the number of parts the fraction contains. The number below the line is called the denominator, because it shows the number of these parts it requires to make a unit, or whole thing. NOTATION AND NUMERATION. If an orange be cut into four equal parts, by what fraction is 1 part expressed !-2 parts ?-3 parts ?-4 parts? how many parts make unity, or whole orange? NOTE.—To find the fractional part of a whole number, multiply by the numerator, and divide by the denominator. Ex. What is the of 12?—thus, 12x3=36-439, ans. REDUCTION OF VULGAR FRACTIONS To reduce a vulgar fraction to its lowest terms. RULE.--Divide the larger number by the less, and the last divisor by the remainder, till nothing remain; the last divisor is the | largest common measure or divisor sought, by which divide both parts of the fraction, for the fraction required. EXAMPLES. Ans. 1 13 1. Reduce to its lowest terms. 48)56(1 48 48 Ans. 4. Reduce 411 to its lowest terms. Ans. REDUCTION OF A VULGAR FRACTION TO A DECIMAL. Rulë.—Place a decimal point at the right of the numerator; then annex as many ciphers as are necessary, divide by the denominator, and the quotient will be the answer required. NOTE.-The quotient will have as many decimal places as there are decimals in the dividend. EXAMPLES. 1. Reduce to a decimal. Thus, 4)1.00 2. Reduce } to a decimal. .25 Ans. 3. Reduce to a decimal. 4. Reduce to a deciinal. 5. Reduce to a decimal, and back to its original fraction. Questions.—What is a vulgar fraction? How expressed? What is the number above the line called? Why? The number below the line? Why? Fractions arise from what? The quotient arising is what? What is reduction of vulgar fractions? How reduced to its lowest terms? How is a vulgo fraction reduced to a decimal ? &c. qrs. na .4)3.5 qr. REDUCTION OF COMPOUND NUMBERS TO A DECIMAL. To reduce a given sum of lower denomination, to the decimal of a higher denomination. RULE.-Reduce the given sum to its lowest denomination mentiɔned, for a numerator, or dividend. Reduce the proposed integer to the same denomination the given sum is reduced to, for a divisor, or denominator; and the quotient arising will be the decimal required. EXAMPLES. or thus, 4)2.0 na. 4 .875 yd. 2. Reduce 50 cents to the decimal of $1. 3. Reduce 1 yard to the decimal of an ell English. 4. Reduce 2 pecks, 4 quarts, to the decimal of a bushel. 5. Reduce 2R. 20P. to the decimal of an acre. 6. Reduce 14h. 45m. 36sec. to the decimal of a day. Ans. .615 d. 7. Reduce 5 cents to the decimal of a dollar. Ans. $.05. 8. Reduce 2 quarters, 14 pounds to the decimal of a cwt. (This weight is becoming obsolete in this country.) Question. — How are lower denominations reduced to the decimal of higher denominations ? To find the value of any decimal, in the term of an integer of the inferior denomination. RULE.-Multiply the decimal by the number of parts in the next less denomination, and point off as many decimal figures in the product, as there are decimals in the given sum. Multiply the remainder (if any) by the parts in the next inferior denomination, and point off as before ; and so proceed through all the parts of the integer, and the several denominations standing on the left hand will be the answer. NOTE. A decimal of a dollar needs no operation to obtain its value. EXAMPLES. 1. Reduce .7875 lb. troy, to its proper value, or in terms of an integer of the inferior denomination. Ans. Soz. 9dwt. 2. Reduce .625 of an acre to its equivalent value. Ans. 2R. 20P. 3. What is the value of .125 of a gallon? Ans. 1 pt. 4. What is the value of .625 of a dollar, by inspection? 5. What is the value of .3 of a Julian year? Ans. 109d. 13h. 48m. 6. What is the value of .875 of a yard? Ans. 3qr. 2na. 7. In the spring of 1824, the compiler of this treatise bought 840 bushels of Connecticut potatoes, in Philadelphia, at $.1125 per bushel: what was the amount? Ans. $94.50. Questions.--How do you find the value of a decimal, in term of an integer, an inferior denomination? What is done with the remainder, if any? cts. P. or .50 or .50 or .50 or .25 or 5* is zo 18 or 2 is Or say, MULTIPLICATION AND PRACTICE. Multiplication and Practice comprise two methods of computing. Practice teaches the manner of calculating the value, or cost of articles, by taking aliquot parts; (aliquot parts of any number or quantity are such as will exactly measure it without a remainder,) as 3 is an aliquot part of 12, and 25 cents is an aliquot part of a dollar; because 4 times 25 cents equal one dollar. And 1, 2, or 3 pecks are each aliquot parts of a bushel : Or, 1, 2, or 3 quarters are aliquot parts of a yard, &c. Multiplication is here the same as Multiplication of Federal Money. TABLE OF ALIQUOT PARTS, &c. ft. C. .50 or 50 is 80 or 2 is 64' is .33; or 331 is 40 or 1 is 32 is or .25 .25 or 25 is 32 or .125 .20 or 20 is 20 is or .125 8 is 16. cwt. .087 or 8j is 56 = .06] or 61 28 .05 in. qr. yd. 14 = or .125 50 cts, is } a dollar, 9 or 1 25 “ is of 50 cts. 41 or 2 na. or .125 &c. 41 1 " The scholar's ingenuity will enable him to discover many abbreviations which we have not room to mention. He should select the shortest method of solving each question; making use of either rule. EXAMPLES. 1. What is the value of 1624 yds. of muslin, at 25 cts. per yd. ? By Practice. 4)1624 Ans. $4.06 2. What will 1754 bushels of clover seed cost, at $4.25 per bushel ? By Notation. Multiplication and Practice. Multiplication 100 bu. = $425. 4)175.5 175.5 50 bu. 212.5 41 4.25 25 bu. = 106.25 1 bu. = 2.125 702.0 8775 43.875 3510 175.5 Ans. $745.875 7020 Ans. $745.875 Ans. $745.87,5 m. 3. Required the value of cwt. of wheat, at $4.75 per cwt. ? Ans. $3.5625 By Notation Fractional. Practice. Multiplication of Decimals. or .50 To or .0625 |