250 DECIMAL FRACTIONS. LII. Q. When such fractions as these occur, viz. fo To0, 1000, how is a unit supposed to be divided? A. Into 10 equal parts, called tenths; and each tenth into 10 other equal parts, called hundredths; and each hundredth into 10 more equal parts, called thousandths, &c. Q. How is it customary to write such expressions? A. By taking away the denominator, and placing a comma before the numerator. 65 Let me see you write down, in this manner, fo, foo, 100, 1000. 525 Q. What name do you give to fractions written in this manner? A. Decimal Fractions. Q. Why called decimal? A. From the Latin word decem, signifying ten; because they increase and decrease in a tenfold proportion, like whole numbers. Q. What are all other fractions called? A. Vulgar, or com mon fractions. Q. In whole numbers, we are accustomed to call the right hand figure, units, from which we begin to reckon, or numerate; hence it was found convenient to make the same place a starting point in decimals; and, to do this, we make use of a comma; what, then, is the use of this comma? A. It merely shows where the units' place is. Q. What are the figures on the left of the comma called? A. Whole numbers. Q. What are the figures on the right of the comma called? A. Decimals. Q. What, then, may the comma properly be called? A. Separatrix. Q. Why? A. Because it separates the decimals from the whole numbers. Q. What is the first figure at the right of the separatrix called? A. 10ths. Q. What is the second, third, fourth, &c.? A. The second is hundredths, the third thousandths, the fourth ten thousandths, and so on, as in the numeration of whole numbers. Let me see you write down again fo 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 T8o. A. ,05. 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 Too? A.,002. Q. Why do you write 2 down with 2 ciphers before it? A. Be cause in Too, the 2 is thousandths; consequently, the 2 must be thousandths when written down in decimals. Q. What does,5 signify? A. . Q. What does,05 signify? A. TOO. Q. Now, as to, and as multiplying 180 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 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 a 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; 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, 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, Indredths, 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. 2536 hundredths. Q. How is 5,125 read? A. 5125 thousandths. Q. What would 5125 thousandths be, written in the form of a vulgar or common fraction? A. 5188. This is evident from the fact, that 188 (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 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 2 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. 60, 12, 62, 21%, 3180, 262T000, 321880, 2100,000, 451008000, 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—2—24 in its lowest terms. 1. 45,5 7. 6,28 A. 675 9. 3,00025 A. 3.000 A. 45 2. 9,25 A. 91 8. 6,005 3. 23,75 A 232 4. 11,8 A. 11 10. 6,08 A. 6 5. 19,9 A. 19 11. 9,2 A. 9 6. 25,255 A. 25. 12. 7,000005 A. 7200000 2 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 89 1. A merchant bought 5 barrels of rice at one time for $272, at another of a barrel for $4,255, at another f of a barrel for $10, and at another of a barrel for $2183; how many barrels did he buy in all? and what did they cost him? As we have seen that decimals correspond with the denominations of Federal Money, hence we may write the decimals down, placing 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. 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 ewt. 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 1677%; what is the sum of all their ages? A. 46,5 years. 4. William expended for a chaise $255, for a wagon $372, 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 72265 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 221000 miles, and the fifth 291 miles; how far did he travel in 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. 8. What is the amount of fo, 2450, 61000, 24510000, 1100000, 1000, 427100000, 410, 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 hundredths? A. 47,009. SUBTRACTION OF DECIMALS. ¶ LIV. 1. A merchant, owing $270,42, paid $192,625; how much did he then owe? |