cents, coursequently the ,45 hundredths are cents, of which it takes 100 to make a unit, and so of mills it takes 1000 to make a unit. Such being the nature of Federal Money, it is plain that tenths represent dimes; hundredths, cents; and thousandths, mills; but we commonly express the decimals where the unit is a dollar, in cents and mills; or taken together they represent thousandths of a dollar. Considering decimals in this light, the student must understand them, because he can see and handle the inferiour denominations, of a dollar, that is the decimal parts of a dollar. Tens. Units. Tenths. Ten Thousandths. Hundred Thousandths. 678 3 6, 0 0 5 6, 0 0 0 0 3 4 8 7 3 7 05, 0 0 0 0 0 1 ... 705, and 1 Millionth. 36, and 5 Thousandths. Thirty six, and eight tenths 36% 36,8 the decimal expression. In like manner write the following sums: Fifteen, and three tenths.. Eighteen, and seventy five hundredths. One, and one millionth. Seventy five, and seventy five hundredths. Eleven, and seven thousandths. Eleven millionths. "Three, and seven tenths. Four, and four hundredths. ADDITION OF DECIMALS. RULE. Place the given numbers according to the value of their places, whether mixed or pure decimals; so that tenths may stand under tenths, and hundredths under hundredths, &c. Add the same as in whole numbers, and point off as many places for decimals, at the right hand, as shall equal the greatest number of decimal places in any of the given numbers. Or set the decimal point in the amount, exactly under those in the given numbers. EXAMPLES. 1. Add ,3 dimes and ,9 dimes together, or 3 tenths and 9 tenths. Ans. $1,2di. or 1,2. di. ,9 ,3 or ,9 $1,2 Ans. 1,2 Ans. DEM. It is plain, that 3 tenths and 9 tenths make 12 tenths, or one unit and two tenths, because ten tenths are equal to a unit or one, and the reason of our pointing off directly below the given points, is manifest, because whenever our tenths exceed 9 tenths, they must equal something in units; and the number of figures must increase, and that increase must be units; for when we suppose these decimals to be dimes, (which correspond with tenths,) ter of which make a dollar or unit, we then have $1 or 1 unit, and 2 dimes, or two tenths. 2. Add 1,7, 3,45, 6,75, 1,705, ,50,05 together. 1,7 3,45 6,75 1,705 ,50 ,05 Ans. 14,155 DEMONSTRATION. Ans. 14,155. -The learner will per ceive, that we add the same as in whole numbers, and the reason is plain, for as the parts diminish in a tenfold proportion from the left to the right, so they must increase in a tenfold proportion from the right to the left, which the student may perceive from Federal Money, in which it takes ten mills, which occupy the place of thousandths, to make a cent; ten cents, which occupy the place of hundredths, to make a dime; and ten dimes, which occupy the place of tenths, to make a dollar. 7. Add three hundredths, five tenths, forty five hundredths, eleven thousandths, three ten-thousandths, and four millionths together. Ans. ,991304. 8. Add together the following sums, viz: forty five thousandths, four tenths, four hundredths, and five thousandths. Ans. ,49. 9. Add six tenths, six hundrendths, six thousandths, six ten-thousandths, five hundrenths, and eleven thousandths. Ans. ,7276. 10. Add 105,7 19,4 1119,05 648,006 and 19,041 together. Ans. 1911,197. 11. Add one thousand and one thousandth, three hundred and eleven thousandths. Ans. 1300,012. SUBTRACTION OF DECIMALS. RULE. Place the given numbers the same as in addition of decimals, with the less under the greater, and subtract the same as in whole numbers. Set the decimal point in the remainder directly under those in the given numbers. EXAMPLES. 1. From ,65 take ,32, or from 6 dimes and 5 cents, take 3 dimes and 2 cents. Ans. ,33, or 3 dimes and 3 cents. ,65 ,32 or dimes. o cents. ,6 ,3 2 ,33 Ans. ,3 3 Ans. DEM.-It is plain, if we take 2 hundredths from 5 hundredths, 3 hundredths will remain; and if we take 3 tenths from 6 tenths, 3 tenths must remain, consequently the difference between the given numbers, is 3 tenths, and 3 hundredths, or as we express it, 33 hundredths. When we consider the deci mal to be Federal Money, with which it agrees exactly in principle, it will appear still more plain to the young student; for if we take 2 cents from 5 cents, 3 cents will remain; and if we take 3 dimes from 6 dimes, 3 dimes will remain; the difference then between the given numbers, is 3 dimes, and 3 cents, or as we commonly express the decimal of a dollar, it stands 33 cents. 9. From sixty five and eight tenths, take forty nine and seventy five hundredths. 10. From one, take one millionth. 11. From one hundred, take one tenth. Ans. 16,05. Ans. ,999999. Ans. 99,9. 12. From nine tenths, take seventy five hundredths. Ans. ,15. Practical Exercises in Addition and Subtraction. 1. A man bought cloth at several times as follows, viz: 20,3 yards, 41,5 yards, 21,7 yards, 9,025 yards and 5 yards; how much did he buy in all? Ans. 97,525 yards. 2. From 4 dollars, take 3 mills. Ans. $3,99,7. 3. From five dollars and twenty five cents, take eight cents and four mills. Ans. $5,16,6. 4. From ninety yards and three tenths, take forty seven yards and eight tenths... Ans. 42,5. 5. A man pays eleven dollars and eighty cents, on his note of forty five dollars; what remains due ? Ans. $33,20 cents. 6. If you take three tenths from one and eight tenths ; how much will remain? Ans. 1 and 5 tenths. 7. What will be the amount if one millionth be added to one thousandth? Ans. ,001001. 8. From 9 tenths of a gallon, take 25 hundredths of a gallon. Ans. ,65 of a gallon. 9. If from 3 yards, you take 75 hundredths of a yard; what will remain ? Ans. 2,25 yards. 10. If from a tenth you take a thousandth; what will remain ? Ans. ,099. MULTIPLICTION OF DECIMALS. RULE.-Multiply the same as in whole numbers, and point off from the right hand of the product as many figures for decimals as there are decimal places in both the factors; if there are not figures enough in the product, supply the deficiency by annexing ciphers to the left hand of the product. EXAMPLES. 1. Multiply 20,8 by,5. 20,8 ,5 Ans. 10,40 Product, or Ans. 10,4. DEM.-Here we multiply by 5 tenths, which is equal to, consequently we are taking one half of the multiplicand; because when our multiplier is 1, we take the multiplicand once; when it is 2, we take the multiplicand twice, &c.; when it is, we take one fourth part of the multiplicand; when, we take one third part of the multiplicand. So it will be perceived that multiplying by a pure decimal, is only taking a part of the multiplicand, as plainly appears from the example, for our multiplier is only the half of a unit, and our product is only one half of the multiplicand. NOTE.-In the multiplication of decimals, the student should keep this truth constantly in mind; that when he multiplies by a pure decimal, that is, by something less than a unit, the product must be less than the multiplicand; and that the product bears the same proportion to the multiplicand, that the multiplier bears to a unit; as may be seen from our first example, when thus arranged: as 10,40 is to 20,8, so is 5 tenths to a unit or 1. 2. Multiply 4 by 5 tenths. 4 ,5 -Ans. 2,. DEM.-Our multiplier is one half of a unit, and it is plain, that we have taken one half of our multiplicand, because our product is 2, and our multiplicand 4. The 2,0 reason of our pointing off, is also evident, because our multiplier, 5 tenths, is only one tenth part the value it would be, standing in the place of units, consequently the product can have only 1 tenth part the value it would have, multiplied by 5 units; and we give it one tenth the value by pointing off one figure from the right of the product; for without pointing, the product would stand 20, but by pointing, it is only 2, which is one tenth part of 20; he same reasoning will hold good when we have any number of decimals. 3. Multiply 67,924 by,003. 4. Multiply ,0007 by ,003. 5. Multiply 44, by,4. 6. Multiply 10, by,1. 7. Multiply 100, by one tenth. Ans.,203772. Ans. ,0000021. Ans. 17,6. Ans. 1,. Ans. 10, 8. Multiply 7 dollars, 4 dimes, 6 cents and 3 mills by C. 9. Multiply 46,5 by 37,9. Ans. $44,77,8 Ans. 1762,35. |