sandths, the fourth ten thousandths, and so on, as in the numeration of whole numbers. Let me see you write down again in the form of a decimal. Q. As the first figure at the right of the separatrix is tenths, in writing down To, then, where must a cipher be placed? A. In the tenths' place. Let me see you write down in the form of a decimal 180- Write down Too, T80, TOO. Q. How would you write down in decimals Tooo? 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 To‰0. Q. Why do you write 2 down with 2 ciphers before it? A. Because in T, the 2 is thousandths; consequently, the 2 must be thousandths when written down in decimals. Q. Now, as, and as multiplying To by 10 produces Too, which is also equal to, how much less in value is,05 than,5? A. Ten times. Q. Why? A. Because the parts in 18 are ten times smaller than in ; and, as the numerater is the same in both expressions, 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 cipher at the right of decimals ↑ 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 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 =; 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? A. 5125 thousandths. Q. What would 5125 thousandths be, written in the form of a vui gar or common fraction? A. $138. 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: 61000000 པ་ 6,000009. read 6, and 9 Millionths. 26.26,25. 310000000 3,0000008 read 3, and 8 Ten-Millionths. 3653 65,0 0 0 0 0 0 0 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. 42 Write the fractional part of the following numbers in the form of decimals, viz. 670, 100, 62155, 21%, 3150, 2621000, 321880, 2100 000, 451000000, 710000000, 5100oo. 202 Write the following decimal numbers in the form of vulgar or common fractions, then reduce them to their lowest terms by ¶ XXXVII.; thus, 2,522 in its lowest terms. Q. What money is adapted to decimal rules? Q. What is the money unit? 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. 1. A merchant bought 5 $27, at another 278 825 barrels of rice at one time for of a barrel for $4,255, at another 89 of a barrel for $, 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 5,2 27,825 respond with the de nominations of Fed 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. 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 150%, and Thomas 1610; what is the sum of all their ages? A. 46,5 years. 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 725 gallons, the third 50 gallons, and the fourth 55100 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 42 miles, the fourth 220 miles, and the fifth 29000 miles; how far did he travel in all? A. 162,0792 miles. 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 ewt. 89 62 8. What is the amount of fo, 2451%, 61000, 2451000 1100000 1000, 427100000, 40, 1000000, 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 |