14. The product of a number is 2336, and the multiplicand 146; what is the multiplier ? CASE, III.-To divide by 10, 100, 1000, &c. Ans. 16. RULE.-Cut off as many figures from the right hand of your dividend, as there are ciphers in the divisor. The figures so cut off at the right, will be the remainder, and the other figures of your dividend at the left, will be the quotient. EXAMPLES. 1. Divide 36784 by 10. Quotient. Rem. 3 6 7 8 4 DEM. In multiplication we have proved that the joining of a cipher to the multiplicand, increases it ten times; and division being the reverse of multi 10) 3 6 7 8 4 (3 678 plication we, instead of joining a figure, must cut one off, which diminishes the dividend, so that the figures in the quotient, have only a tenth part of the value which they have in the dividend; because the figure which expresses tens in the dividend, only expresses units in the quotient; and it is plain, if we should divide by 10 according to the usual mode of dividing numbers, the cipher of our and in subtracting, leave 4 a re 2. Divide 36048 by 100. Ans. 360, Rem. 48. Ans. 461, Rem. 340. Ans. 63. Rem. 4210. NOTE.-If we cut off one figure from the right of our dividend, the left hand figures which we call the quotient, express one tenth of the dividend, excluding the remainder; and the figure cut off at the right, shows what is left of the dividend after dividing it into ten equal parts. When we cut off two figures, the left hand figures express one hundredth part of our dividend; and when we cut off three, the left hand figures express only one thousandth part of the dividend; because the figure which before expressed thousands, after the operation, only expresses units; so that the effect may be easily discovered by the laws of notation. CASE, IV.-When the divisor is a composite number; that is, when it can be produced by multiplying any two figures in the table together. RULE.-Divide the dividend first by one of the figures, and then that quotient by the other, and the last quotient will be the answer. NOTE. The total remainder is found by multiplying the last re mainder by the first divisor, and to the product add the first remainder. Or multiply the whole divisor by the last quotient; and subtract the product from the dividend; and the difference will be the true remainder. EXAMPLES. 1. Divide 4 6 8 7 7 by 1 2. Dividend. Divisor, 4) 4 6 8 7 7 Three and four we take for the component parts of 12; because 3 times 4 are 12; or we might take 2 390 6,1X4=4+1=5 rem. and 6; because 2 Divisor, 3) 1 17 1 9-1 Rem. Quotient, 12 times 6 are 12. Or according to our rule, we may obtain our re mainder by subtracting the pro duct of our quotient and divisor from the dividend thus, DEM. The reason of this rule is plain, because it is evident, as we find in our example, that the 3d part of the 4th of any number, is the 12th of the whole. 2. Divide 146738 by 8. 3. Divide 167834 by 16. 4. Divide 56732 oy 24. 5. Divide 937387 by 54. 6. Divide 634679 by 64. Ans. 18342 Rem. 2. Ans. 10489 Rem. 10. Ans. 2363 Rem. 20. 7. Suppose a privateer takes a prize which is to be divided among 81 seamen; of each? worth $68526, what is the share Ans. $846. 8. A man bought ninety six horses for shipping, for which he gave 6720 dollars; what did they cost him on an averAns. $70. age? CASE, V.-When there are ciphers at the right of the divisor. RULE.-Cut off the ciphers from the right hand of the divisor, and a like number of figures from the right of the dividend. Divide the remaining figures without regard to the figures cut off from either; and to the right hand of the remainder, annex the figures cut off from the right of the dividend, which will give the true remainder. EXAMPLES 1, Divide 86348634 by 67000. 67193 NOTE.--The student, at first view, will perhaps look on our remainder, and wonder at its being so great; but his surprise vanishes, when he takes into consideration that our true divisor is not 67, but 67000. DEM.-Cutting off the ciphers from the divisor, and an equal number of figures from the right of our dividend, is dividing them both by 10, 100, 1000, &c.; and divisor is contained in the whole it is evident, as often as the whole dividend, so often must a like portion of the divisor, be contained in a like portion of the dividend; so it saves the needless repetition of the ciphers as they are repeated in the common way; as the following example shows. Divisor. Dividend. Quotient. 67000 193486 134000 594863 536000 588634 536000 52634 Rem. Proof. 86348634 2. Divide 6467321 by 460. 3. Divide 76173 by 320. 4. Divide 4673625 by 21400. 5. Divide 149596478 by 120000. 6. Divide 46646300 by 670000. 7. Divide 6346211 by 20000. 8. Divide 64613214 by 4000. NOTE. This last work shows, that the ciphers of the divisor, come under the figures cut off from the dividend, and coming at the right hand, the figures cut off from the dividend, come in the remainder of course. Ans. 14059 Rem. 181. Ans. 238 Rem. 13. Ans. 218 Rem. 8425. Ans. 1246 Rem. 76478. Ans. 69 Rem. 416300. Ans. 317 Rem. 6211. Ans. 16153 Rem. 1214. 9. A gentleman sold 300 acres of land for 2100 dollars at what rate did he sell it per acre? Ans. $7. 10. A merchant sold 300 pipes of wine for $56100 at what rate did he sell it per pipe? Ans. $187. 58 DIVISION OF DECIMAL OR FEDERAL MONEY. DIVISION OF DECIMAL OR FEDERAL MONEY. RULE.-Divide the same as in whole numbers. And remember to point off from the right hand of the quotient, as many figures for decimals, as you have decimal places in your dividend; and should you add ciphers to your remainder for the sake of continuing your division, you will count them as decimals belonging to your dividend. EXAMPLES. 1. If 6 bushels of wheat cost $7,50 cents; what is the cost of 1 bushel? Ans. $1,25 cents. $, cts. 6) 7, 50 Ans. 1, 2 5 cts. 6 Proof. $7,50 - Here we say 6 in 7 once, and 1 over; then 6 in 15 twice, and 3 over; then 6 in 30, five times. DEM. When we divide the price of 6 bushels by 6, our quotient is one sixth part of our dividend; and and it is evident, that one bushel should cost only a sixth part as much as six bushels. From this example we may lay down this general rule-That where the price of a quantity is given we obtain the price of one, by dividing the price of the quantity, by the number expressing the quantity; and the quotient will be the price of 1. The reason of our pointing off for decimals, is evident from the principles of subtraction; since this is only a concise method of performing subtraction. NOTE. When the quotient contains two or three decimal places, the remainder, if any, will not be noticed in the answer or quotient. 2. Divide $52,24 cents equally among 24 persons; what will each receive? Ans. $2,17 cts. 6 mills. 3. A merchant bought 235 yards of calico for $58,75 cts., what did he pay per yard? Ans. 25 cts. 4. A merchant paid $1615 for 380 yards of broadcloth; what did it cost him per yard? Ans. $4,25 cts. NOTE. If the price is given in dollars, and you have a remainder after dividing; annex a cipher and seek how often the divisor is contained, and if you again have a remainder, annex another cipher, and so continue your division, till you have annexed three ciphers to your dividend, which will reduce your dividend to mills; consequently you will have as many figures to point off from the right of your quotient as the decimals which you have joined to your dividend. 5. If 40 yards of calico cost twelve dollars; what is that per yard? Ans. 30 cts. 6. A man paid fifty six dollars for 50 sheep; what did he pay per head? Ans. $1,12 cts. 7. A man sold 200 bushels of oats for 64 dollars; what did he receive per bushel ? Ans. 32 cts. 8. Bought 250 acres of land, for 1375 dollars; what is per acre? Ans. $5,50 cts. that 9. A merchant bought 30 barrels of flour for $112,50 cts.; what does it stand him in per barrel? Ans. $3,75 cts. 10. If 30 yds. cost 40 dollars; what is the cost of one yard? Ans. $1,33cts., 3 mills. 11. Bought fifty bushels of barley for twenty four dollars; what was paid per bushel ? Ans. ,48 cts. 12. Paid for a fine cheese weighing 32 pounds, two dollars and fifty six cents; what was paid a pound? Ans. 8 cts. Ans. 80 cts. 13. A man laboured fifty days, and received for his labour forty dollars; what was that per day? 14. A man paid six dollars for 20 bushels of apples; what did they cost him a bushel? Ans. 30 cts. QUESTIONS ON SIMPLE DIVISION. NOTE.-The teacher should continue to question the student until he becomes perfectly familiar with all the answers given to the questions in this, and the other rules. Perhaps no method of instruction can be introduced, that has so good a tendency to exercise the faculties of the student, as that of asking him questions and explaining the principles on which the rules are founded; it brightens the faculties and facilitates thought. What is division? A: A short way of performing subtraction.What are the two given numbers called? A. Divisor and dividend. What is the number called that arises from the operation of the work? A. Quotient or answer. Do you sometimes have an uncertain number? A. Yes. What is it called? A. Remainder. What is the dividend? A. The number to be divided. What is the divisor? A. The number by which the dividend is to be divided. What does the quotient show? A. It shows how many times the divisor is contained in the dividend, or how often it may be subtracted; or when the divi. dend is divided into as many parts as the divisor expresses units, the quotient shows the quantity or number contained in each part. What do you understand by the remainder? A. That there are so many left after the divisor is contained as many times as the quotient expresses a unit, or after subtracting the divisor as often as the quotient expresses a unit there are so many left; but not enough to contain the divisor once more. When your work called short division? A. When the divisor does not exceed twelve. If the divisor exceeds twelve what is it called? A. Long division. Where do you generally place your quotient in short division? A. Directly under the dividend. In long |