6. From 463,05 take 17,0613. 7. From 412,63402, take 17,1. 8. From 19, take 14673. 9. From sixty-five and eight tenths, seventy-five hundredths. 10. From one, take one millionth. 11. From one hundred, take one tenth. Ans. 445,9887. Ans. 395,53402. Ans. 18,85327. take forty-nine and Ans. 16,05. Ans. ,999999. Ans. 99,9. 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. 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. 2. From 4 dollars, take 3 mills. 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. MULTIPLICATION 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 2,0 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 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; the 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. 16. 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 6. 9. Multiply 46,5 by 37,9. Ans. $44,77,8. Ans. 1762,35, 10 Multiply 7,1 by 8,2. Ans. 58,22. 11. Multiply 4 dollars 30 cents by 12 cents. 1 Ans. $0,5160-51 cents 6 mills. 12. Multiply 10 dollars by 10 cents. 13. Multiply 100 by 1 cent. Ans. $1. Ans. $1. NOTE. It may, at first, seem strange to the learner, that multiply ing a number should diminish it, but the difficulty vanishes when he considers the value of our multiplier; because when our multiplier is 1, the product is exactly the same as the multiplicand; now the student must understand that a less multiplier must produce a less product than a greater multiplier, and by noticing the 13th example, where we multiply 100 dollars by one cent, our product is only 1 dollar, because our multiplier is only the one hundredth part of a unit or dollar, consequently our product, I dollar, is only the one hundredth part of the multiplicand, 100 dollars; if multiplying by a decimal did not give a product less than the multiplicand, a less multiplier would give a product equal to a greater multiplier, which is absurd. 14. What cost 6,5 yards of cloth, at 2 dollars, 35 cents, 2 mills per yard? 15. What will 20,75 yards cost at 16. What is the worth of 10,5 cents per gallon? 17. If rice is worth 5 cents per of 8,75 pounds? Ans. $15,28cts. 8 mills. $3,50 cents per yard? Ans. $72,62 cts. 5 mills, gallons of whiskey, at 25 Ans. $2,62 cts. 5 mills. pound; what is the worth Ans. $043 cts. 7,5 mills. Ans. ,0154. 18. Multiply 15,4 by one thousandth. To multiply by 10, 100, 1000, &c. Ans. 53,4. Ans. 2,04, Remove the separatrix as many places to the right hand as the multiplier has ciphers, thus ,256 multiplied by ten stands 2,56, because removing the separatrix one figure to the right gives it ten times the value; and to multiply by 100, remove the separatrix two figures to the right, thus,256 multiplied by 100 becomes 25,6, &c. DIVISION OF DECIMALS. Division of decimals is exactly the reverse of multiplication, and they consequently prove each other. Multiplying by a decimal, is taking a part of the multiplicand as many times as the multiplier contains like portions of an integer, and consequently the product must be less than the multiplicand. When we divide by a pure decimal which is less than an integer or one, the quotient must be greater thar the dividend, because the quotient shows how many times the divisor may be subtracted from the dividend; and a pure decimal, which is less than a unit, can evidently be subtracted oftener than a unit. RULE.-Divide as in whole numbers, and from the right hand of the quotient, point off as many figures for decimals as the decimal places in the dividend exceed those in the divisor; if there be not enough in the quotient, prefix ciphers to the left hand of the quotient, to supply the deficiency. EXAMPLES. 1. Divide 10,5 by 5 tenths. 10 ,5 5 10.5 Proof. 5 0 Ans. 21. DEM. It is evident, if we should divide the dividend by a unit or 1, that the quotient would be the same as the dividend; then when we divide by 5 tenths, which is of a unit, it is evident that our quotient must be double our dividend, because the dividend must contain a half or 5 tenths twice as often as a whole; the quotient then shows that five tenths, or one half a unit, can be subtracted from the dividend, 10,5, twenty-one times.-The reason of pointing off in the quotient is evident from the principles of multiplication, because in multiplication we point off as many places for decimals in the product as there are in both the factors; the dividend in division is that product, and the divisor and quotient are the factors; consequently what the divisor falls short must be made up from the quotient, by counting from the right to the left. Again it is evident, because our divisor is only 1 tenth part of five units, and consequently is contained in the dividend ten times oftener than five units, and our quotient, by pointing off according to the rule, has ten times the value it would have; divided by five units, because our quotient is now 21, and if our divisor had been five units, it is evident, that our quotient must have been two and one tenth. NOTE. When the dividend has not as many decimal places as the divisor, or will not contain the divisor, annex ciphers to the right hand of the dividend to supply the deficiency, and if there be a remainder after all the figures of the dividend are brought down, annex ciphers to the remainders till your quotient shall at least contain two or three decimal places, and these ciphers which you annex to the remainders are counted as belonging to the dividend. 2. Divide 5 by 25 hundredths. ,25)5,00(20, Ans. 20. DEM. It is plain, that or 25 hundredths can be subtracted from the dividend four times as often as a unit; for it is plain, that we can subtract four times as many quarters as whole ones; and as our quotient shows, it is plain that or 25 hundredths is contained in five, twenty times, because in five there are 20 quarters. 4. Divide 206,79 by 2,46. 5. Divide 6,344 by,54 6. Divide 44 by 2 tenths. 7. Divide 15 by 25 hundredths. 0 Ans. 84,06. Ans. 11,748 Ans. 220. *Ans. 60. Ans. 100. Ans. 50. Ans. 1000. Ans.,5. DEM. The learner may, at first thought, think it impossible to divide a less number or quantity by a greater; but how soon the difficulty vanishes when he learns the nature of notation; the divisor, 40, we see is not contained în 20, but when we annex a cipher to our dividend it is then reduced to tenths, and then the divisor is contained in 200 tenths 5 times, and the 5 is tenths, because the dividend that produced it is tenths. The principle is the same as if our dividend had been dollars; we know that the divisor, considered as $40, is not contained in 20 dollars; but if we annex to the $20 a cipher, the dividend is then dimes, and the divisor, $40, is contained in the dimes 5 times, and the quotient is dimes, because the dividend which produced it is dimes, or tenths of a unit. 12. Divide 25 by 100. 13. Divide one by one thousand. 14. Divide 25 by 1000. Ans. 25. Ans. ,001. Ans. ,025. NOTE. When decimals or whole numbers are to be divided by 10, 100, 1000, &c., that is, unity with any number of ciphers; it is performed by placing the decimal point in the dividend as many places towards the left hand as there are ciphers in the divisor. |