Write down Too, T80, TỔσ• Q. How would you write down in decimals Too? A. By placing 2 ciphers at the right of the separatrix, that is, before the 7. Let me see you write it down? A. ,007. Let me see you write down Too? A.,002. Q. Why do you write 2 down with 2 ciphers before it? A. Be cause in To, the 2 is thousandths; consequently, the 2 must be thousandths when written down in decimals. 50 Q. What does,5 signify? A. Q. What does,05 signify? A. TO• Q. Now, as, and as multiplying To by 10 produces To which is also equal to, how much less in value is ,05 than ,5? A. Ten times. Q. Why? A. Because the parts in T are ten times smaller than in ; and, as the numerator is the same in both expres sions, consequently, the value is lessened 10 times. Q. How, then, do decimal figures decrease in value from the left towards the right? A. In a tenfold proportion. Q. What does,50 mean. A. 5 tenths, and no hundredths. Q. What, then, is the value of a cipher at the right of deci mals? A. No value. 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 proportion. 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 =1, and,5 is =; 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 repa ratrix. Q. When a whole number and decimal are joined together, thus; 2,5, what is the expression called? A. Å mixed number Q. As any whole number may be reduced to tenths, hun dredths, 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? tenths, giving the name of the decimal to all the figures. Q. How is 25,36 read? A. 2536 hundredths. Q. How is 5,125 read? A. 5125 thousandths. A. 254 Q. What would 5125 thousandths be, written in the form of a vulgar or common fraction? A. 5135. This is evident from the fact, that 35 (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 2625.26,25 read 26, and 25 Hundredths. 310000000..3,0000008 read 3, and 8 Ten-Millionths. 365 365, 0000 000 read 365. Exercises for the Slate. Write in decimal form 7 tenths, 42 hundredths, 62 and 25 Bundredths, 7 and 426 thousandths, 24 thousandths, 3 ten-thou andths, 4 hundredths, 2 ten-thousandths, 3 millionths. Write the fractional part of the following numbers in th 62 of decimals, viz. 60, 42, 6225, 2, 3150, 262105, 321830, 2100 000, 451008000, 710000000, 510050. Write the following decimal numbers in the form of ulgar or common fractions, then reduce them to their lowest terms by ¶ XXXVII; thus, 2,5—2=21 in its lowest terms. A. 19% 11. 9,2 A. 9 12. 7,000005 A. 700000 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, &c., how, then, may all 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. 278 825 1 A merchant bought 5 barrels of rice at one time for $27, at another of a barrel for $4,255, at another 1000 oo of a barrel for $10, and at another of a barrel for $2,62%; how many barrels did he buy in all? and what did they cost him? From these illustrations we derive the following RULE. 1. How are the numbers to be written down? A. Tenths under tenths, hundredths under hundredths, and so on. II. How do you proceed to add? A. As in Simple Addition. III. Where do you place the separatrix? 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; how much did he buy in all? A. 30,0307 cwt. 3. James is 14 years old, Rufus 15, and Thomas 16705%; what is the sum of all their ages? A. 46,5 years. 25 4. William expended for a chaise $255, for a wagon $370, for a bridle $10%, and for a saddle $11; what did these amount to? A. $304,455. 5. A merchant bought 4 hhds. of molasses; the first contained 62 gallons, the second 72,265 gallons, the third 50 gallons, and the fourth 55 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 2200 miles, and the fifth 29 miles ; how far did he travel all? A. 162,0792 miles. 42 7. A grocer, in one year, at different times, purchased the following quantity of articles, viz. 427,2623 cwt., 2789,00065 cwt., 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 cwt. 89 8. What is the amount of, 2450, 610‰0, 245100, 1108800, 1000, 427100000, 40, Too00, 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 hundredths? 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 $192,625 Ans. $77,795 For the reasons shown in Addition, we proceed to subtract, and point off, as in Sub traction of Federal Money. Hence we derive the following RULE. 1 How do you write the numbers down? A. As in Additicz of Decimals. II. How do you subtract? A. As in Simple Subtraction. III. How do you place the separatrix? A. As in Addition of Decimals. 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 in the parentheses 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,12587 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 pipes; (41928) take 419,89 pipes; (101) take 419,8999 pipes; (10001). Ans. 1536,7951 pipes. MULTIPLICATION OF DECIMALS. 1 LV. 1. How many yards of cloth 'n 3 pieces, each piece containing 2017 yards? OPERATION 20,75 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 Ans. 62,25 yds. made the multiplicand, had there been twe 3 |