Q. We have seen that ,5 is 10 times as much in value as ,05, or To what effect, then, does a cipher have placed at the left of decimals? A. It decreases their value in a tenfold pro portion. Q. Since decimals decrease from the left to the right in a tenfold proportion, how, then, must they increase from the right to the left? A. In the same proportion. Q. Since it was shown, that ,5; 25; what, then, will always be the denominator of any decimal expression? A. The figure 1, with as many ciphers placed at the right of it as there are decimal places. Let me see you write down the following decimals on your slate, and change them into a common or vulgar fraction, by placing their proper denominators under each, viz.,5,05,005,62,0225 ̊,37. Q.,25 is 10, and ,5 is = 1; which, then, is the most in value, ,25 or ,5? Q. By what, then, is the value of any decimal figures determined? A. By their distance from the units' place, or separatrix. Q. When a whole number and decimal are joined together, thus, 2,5, what is the expression called? A. A mixed number. Q. As any whole number may be reduced to tenths, hundredths, thousandths, &c. by annexing ciphers, (for multiplying by 10, 100, &c.) thus, 5 is 50 tenths, 500 hundredths, &c.; how, then, may any mixed number be read, as 25,4 ? A. 254 tenths, giving the name of the decimal to all the figures. Q. How is 25,36 read? Q. How is 5,125 read? A. 5125 thousandths. Q. What would 5125 thousandths be, written in the form of a vul gar or common fraction? This is evident from the fact, that (an improper fraction), reduced to a mixed number again, is equal to 5,125. The pupil may learn the names of any decimal expression, as far as ten-millionths, also how to read or write decimals, from the following Table: 310000000 =. read 5 Tenths. read 6 Hundredths. read 25 Thousandths. read 1328 Ten-Thousandthis. .6,000009 read 7, and 8 Tenths. read 6, and 9 Millionths. 26.26,25.. ⚫ read 26, and 25 Hundredths. 3,0000008 read 3, and 8 Ten-Millionths. 365 365,000 00 00 read 365. Exercises for the Slate. Write in decimal form 7 tenths, 42 hundredths, 62 and 25 hundredths, 7 and 426 thousandths, 24 thousandths, 3 ten-thou sandths, 4 hundredths, 2 ten-thousandths, 3 millionths. Write the fractional part of the following numbers in the form of decimals, viz. 670, 100, 6210, 21%, 3150, 2621000, 321880, 2100 000, 45100000, 710000000, 510000. Write the following decimal numbers in the form of vulgar or common fractions, then reduce them to their lowest terms by ¶ XXXVII.; thus, 2,5=215%=24 in its lowest terms. Q. What money is adapted to decimal rules? A. Federal money. Q. What is the money unit? A. The dollar. Q. How is it so adapted? A. As 10 dimes make a dollar, and 10 cents a dime, &c., dimes are 10ths of a dollar, cents are 100ths, and mills are 1000ths of a dollar. Q. How are 3 dollars 2 dimes 4 cents and 5 mills written? A. $3,245. ADDITION OF DECIMALS. LIII. Q. As we have seen that decimals increase from right to left in the same proportion as units, tens, hundreds, &e., how, then, may a. the operations of decimals be performed ? A. As in whole numbers. Note. The only difficulty which ever arises, consists in determining where the decimal point ought to be placed. This will be noticed in its proper place. 1. A merchant bought 5 278 2 89 barrels of rice at one time for $27, at another of a barrel for $4,255, at another 1000 of a barrel for $10, and at another of a barrel for $21000; how many barrels did he buy in all? and what did they cost him? 627 As we have seen that decimals cor 27,825 respond with the de 4,255 nominations of Fed ,278 0,72 89 2,627 cimals down, placing Ans. 6,988 barrels, for $35,427 dimes under dimes, cents under cents, &c., that is, tenths under tenths, hundredths under hundredths, &c., and add them up as in Addition of Federal Money. From these illustrations we derive the following RULE. Q. How are the numbers to be written down? A. Tenths under tenths, hundredths under hundredths, and so on. Q. How do you proceed to add? A. As in Simple Addition. A. Directly under the separating points above. More Exercises for the Slate. 2. James bought 2,5 cwt. of sugar, 23,265 cwt. of hay, and 4,2657 cwt. of rice; now much did he buy in all? A. 30,0307 cwt. 3. James is 14 years old, Rufus 150, and Thomas 1617; what is the sum of all their ages? A. 46,5 years. 255 4. William expended for a chaise $255, for a wagon $37, for a bridle $10, and for a saddle $111; what did these amount to? A. $304,455. 2657 5. A merchant bought 4 hhds. of molasses; the first contained 62 gallons, the second 720 gallons, the third 50 gallons, and the fourth 550 gallons; how many gallons did he buy in the whole ? A. 240,6157 gallons. 6. James travelled to a certain place in 5 days; the first day he went 40 miles, the second 28 miles, the third 421 miles, the fourth 221000 miles, and the fifth 29100 how far did he travel in all? A. 162,0792 miles. miles; 7. A grocer, in one year, at different times, purchased the following quantity of articles, viz. 427,2623 cwt., 2789,00065 ewt., 42,000009 cwt., 1,3 cwt., 7567,126783 cwt., and 897,62 cwt.; how much did he purchase in the whole year? A. 11724,309742 ewt. 89 62 75 8. What is the amount of, 2451, 6100, 24510000 100000 1000, 4271002‰00, 410, 10000, and 1925? A. 2854,492472. 9. What is the amount of one, and five tenths; forty-five, and three hundred and forty-nine thousandths; and sixteen nundredths? A. 47,009. SUBTRACTION OF DECIMALS. ¶ LIV. 1. A merchant, owing $270,42, paid $192,625; how much did he then owe? OPERATION. $270,42 Ans. $77,795 For the reasons shown in Addition, we proceed to subtract, and point off as in Subtraction of Federal Money Hence we derive the following RULE. Q. How do you write the numbers down? A. As in Simple Subtraction. More Exercises for the Slate. 1. Bought a hogshead of molasses, containing 60,72 gallons, how much can I sell from it, and save 19,999 gallons for my own use? A. 40,721 gallons. 2. James rode from Boston to Charlestown in 4,75 minutes, Rufus rode the same distance in 6,25 minutes; what was the difference in the time? A. 1,5 min. 3. A merchant, having resided in Boston 6,2678 years, stated his age to be 72,625 yrs. How old was he when he emigrated to that place? A. 66,3572 yrs. Note. The pupil must bear in mind, that, in order to obtain the answer, the figures annexed to each question, are first to be pointed off, supplying ciphers, if necessary, then added together as in Addition of Decimals. 4. From ,65 of a barrel take 125 of a barrel-525; take 2 of a barrel-45; take ,45 of a barrel-2; take ,6 of a barrel-5; take 12567 of a barrel-52433; take 26 of a barrel-39. A. 2,13933 barrels. 5. From 420,9 pipes take 126,45 pipes-29445; take,625 of a pipe-420275; take 20,12 pipes-40078; take 1,62 pipes41928; take 419,89 pipes-101; take 419,8999 pipes-10001 A. 1536,7951 pipes. MULTIPLICATION OF DECIMALS. ¶ LV. 1. How many yards of cloth in 3 pieces, each piece containing 20 yards? OPERATION. 20,75 3 Ans. 62,25 yds. the multiplier also we In this example, since multiplication is a short way of performing addition, it is plain that we must point off as in addition, viz. directly under the separating points in the multiplicand; and, as either factor may be made the multiplicand, had there been two decimals in must have pointed off two more places |