14. If 16 men finish a piece of work in 28 days, how long will it take 12 men to do the same work? First find how long it would take 1 man to do it; then 12 men will do it in of that time. 15. How many pieces of merchandise, at 20 given for 240 pieces, at 12 s. apiece? Ans. 377 days. s. apiece, must be Ans. 149 11 16 16. How many yards of bocking that is 14 yd. wide will be sufficient to line 20 yds. of camlet that is of a yard wide? First find the contents of the camlet in square measure; then it will be easy to find how many yards in length of bocking that is 14 yd. wide it will take to make the same quantity. Ans. 12 yards of bocking. 17. If 14 yd. in breadth require 20 yds. in length to make a cloak, what in length that is wide will be required to make the same? Ans. 341 yds. 18. If 7 horses consume 2 tons of hay in 6 weeks, how many tons will 12 horses consume in 8 weeks? If we knew how much 1 horse consumed in 1 week, it would be easy to find how much 12 horses would consume in 8 weeks. 21 = 11 tons. will consume consume If 7 horses consume 11 tons in 6 weeks, 1 horse of 11 of a ton in 6 weeks; and if a horse of a ton in 6 weeks, he will consume of = 11 168 11 169 =132 168 of a ton in 1 week. 12 horses will consume 12 times 32 132 = 21 in 1 week, and in 8 weeks they will consume 8 times 18 = 62 tons, Ans. = 19. A man with his family, which in all were 5 persons, did usually drink 74 gallons of cider in 1 week; how much will they drink in 22 weeks when 3 persons more are added to the family? Ans. 280 gallons. 20. If 9 students spend 107 £. in 18 days, how much will 20 students spend in 30 days? Ans. 39 £. 18 s. 420 d. DECIMAL FRACTIONS. ¶ 66. We have seen, that an individual thing or number may be divided into any number of equal parts, and that these parts will be called halves, thirds, fourths, fifths, sixths, &c., according to the number of parts into which the thing or number may be divided; and that each of these parts may be again divided into any other number of equal parts, and so on. Such are called common, or vulgar fractions. Their denominators are not uniform, but vary with every varying division of a unit. It is this circumstance which occasions the chief difficulty in the operations to be performed on them; for when numbers are divided into different kinds or parts, they cannot be so easily compared. This difficulty led to the invention of decimal fractions, in which an individual thing, or number, is supposed to be divided first into ten equal parts, which will be tenths; and each of these parts to be again divided into ten other equal parts, which will be hundredths; and each of these parts to be still further divided into ten other equal parts, which will be thousandths; and so on. Such are called decimal fractions, (from the Latin word decem, which signifies ten,) because they increase and decrease, in a tenfold proportion, in the same manner as whole numbers. 67. In this way of dividing a unit, it is evident, that the denominator to a decimal fraction will always be 10, 100, 1000, or 1 with a number of ciphers annexed; consequently, the denominator to a decimal fraction need not be expressed, for the numerator only, written with a point before it (') called the separatrix, is sufficient of itself to express the true value. Thus, The denominator to a decimal fraction, although not expressed, is always understood, and is 1 with as many ciphers annexed as there are places in the numerator. Thus, '3765 is a decimal_consisting of four places; consequently, 1 with four ciphers annexed (10000) is its proper denominator. Any decimal may be expressed in the form of a common fraction by writing under it its proper denominator. Thus, '3765 expressed in the form of a common fraction, is 3765 10000 When whole numbers and decimals are expressed together, in the same number, it is called a mixed number. Thus, 25'63 is a mixed number, 25', or all the figures on the left hand of the decimal point, being whole numbers, and '63, or all the figures on the right hand of the decimal point, being decimals. The names of the places to ten-millionths, and, generally, how to read or write decimal fractions, may be seen from the table following. From the table it will be seen, that the first figure on the right hand of the decimal point signifies so many tenth parts of a unit; the second figure, so many hundredth parts of a unit; the third figure, so many thousandth parts of a unit, &c. It takes 10 thousandths to make 1 hundredth, 10 hundredths to make 1 tenth, and 10 tenths to make 1 unit, in the same manner as it takes 10 units to make 1 ten, 10 tens to make 1 hundred, &c. Consequently, we may regard unity as a starting point, from whence whole numbers proceed, continually increasing in a tenfold proportion towards the left hand, and decimals continually decreasing, in the same proportion, towards the right hand. But as decimals decrease towards the right hand, it follows, of course, that they increase towards the left hand, in the same manner as whole numbers, ¶ 68. The value of every figure is determined by its place from units. Consequently, ciphers placed at the right hand of decimals do not alter their value, since every significant figure continues to possess the same place from unity. Thus, '5, '50, '500 are all of the same value, each being equal to, or But every cipher, placed at the left hand of decimal fractions, diminishes them tenfold, by removing the significant figures further from unity, and consequently making each part ten times as small. Thus, '5, '05, '005, are of different value, '5 being equal to, or ; '05 being equal to 8, or; and 005 being equal to 200. Decimal fractions, having different denominators, are readily reduced to a common denominator, by annexing ciphers until they are equal in number of places. Thus, '5, '06, 234 may be reduced to '500, '060, 234, each of which has 1000 for a common denominator. 69. Decimals are read in the same manner as whole numbers, giving the name of the lowest denomination, or right hand figure, to the whole. Thus, '6853 (the lowest denomination, or right hand figure, being ten-thousandths) is read, 6853 ten-thousandths. Any whole number may evidently be reduced to decimal parts, that is, to tenths, hundredths, thousandths, &c. by annexing ciphers Thus, 25 is 250 tenths, 2500 hundredths, 25000 thousandths, &c. Consequently, any mixed number may be read together, giving it the name of the lowest denomination or right hand figure. Thus, 25'63 may be read 2563 hundredths, and the whole may be expressed in the form of a common fraction, thus, 2563. The denominations in federal money are made to correspond to the decimal divisions of a unit now described, dollars being units or whole numbers, dimes tenths, cents hundredths, and mills thousandths of a dollar; consequently the expression of any sum in dollars, cents, and mills, is simply the expression of a mixed number in decimal fractions. Forty-six and seven-tenths =4670 =46'7. Write the following numbers in the same manner: Fifty-two and six hundredths. Nineteen and four hundred eighty-seven thousandths. One and five thousandths. 135 and 3784 teh-thousandths. 9000 and 342 ten-thousandths. 10000 and 15 ten-thousandths. 974 and 102 millionths. 320 and 3 tenths, 4 hundredths and 2 thousandths. 500 and 5 hundred-thousandths. 47 millionths. Four hundred and twenty-three thousandths. ADDITION AND SUBTRACTION OF DECIMAL FRACTIONS. ¶ 70. As the value of the parts in decimal fractions increases in the same proportion as units, tens, hundreds, &c., and may be read together, in the same manner as whole numbers, so, it is evident that all the operations on decimal fractions may be performed in the same manner as on whole numbers. The only difficulty, if any, that can arise, must be in finding where to place the decimal point in the result. This, in addition and subtraction, is determined by the same rule; consequently, they may be exhibited together. 1. A man bought a barrel of flour for $8, a firkin of butter for $3'50, 7 pounds of sugar for 83 cents, an ounce of pepper for 6 cents; what did he give for the whole ? As the denominations of federal money correspond with the parts of decimal fractions, so the rules for adding and subtracting decimals are exactly the same as for the same operations in federal money. (See ¶ 28.) 2. A man, owing $375, paid $175'75; how much did he then owe? The operation is evidently the same as in subtraction of federal money. Wherefore, in the addition and subtraction of decimal fractions, RULE: Write the numbers under each other, tenths under tenths, hundredths under hundredths, according to the value of their places, and point off in the results as many places for decimals as are equal to the greatest number of decimal places in any of the given numbers. EXAMPLES FOR PRACTICE. 3. A man sold wheat at several times as follows, viz. 13'25 bushels; 8'4 bushels; 23'051 bushels, 6 bushels, and '75 of a bushel; how much did he sell in the whole ? Ans. 51'451 bushels. 4. What is the amount of 429, 21, 37, 355000, 100, and 170? Ans. 808. 143, or 808'143. 5. What is the amount of 2 tenths, 80 hundredths, 89 thousandths, 6 thousandths, 9 tenths, and 5 thousandths? Ans. 2. 6. What is the amount of three hundred twenty-nine, and seven tenths; thirty-seven and one hundred sixty-two thousandths, and sixteen hundredths? 7. A man, owing $4316, paid $376'865; how much did he then owe? Ans. $3939'135. 8. From thirty-five thousand take thirty-five thousandths. Ans. 34999'965. Ans. 1'5507. Ans. 234'9925. 9. From 5'83 take 4'2793. 12. From 48 100 ake 2 Ans. 976'07307. Remainder, 198, or 1'98. 993, 315 13. What is the amount of 29, 3741008000, 9770 1.27 2d 10&? Ans. 942'957009, |