59 DIVISION OF DECIMAL OR FEDERAL MONEY. DIVISION OF DECIMAL OR FEDERAL MONEY. RULE. Divide the same as in whole numbers. And remember 1 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 costof 1 bushel? Ans. $1,25 cents. $ cts. Ans. $1, 2 5 cts. 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 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 cents; 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 ne receive per bushel ? Ans. 32 cts. 8. Bought 250 acres of land, for 1375 dollars; what is that per acre? Ans. $5,50 cts. 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 yards cost 40 dollars; what is the cost of one yard? Ans. $1,33 cts., 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 dol lars and fifty-six cents? what was paid a pound? Ans. 80 cts. Ans. 8 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 dividend 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 is your work called short division? A. When the divisor does not exceed twelve. If the divisor exceeds twelve, what s it called? A. Long division. Where do you generally place your quotient in short division? A. Directly under the dividend. In long division where is the quotient usually placed? A. At the right hand of the dividend. Where do you place your divisor? A. At the left of the dividend. How do you proceed in the work? A. Assume a sufficient number of figures at the left hand of the dividend to contain the divisor not exceeding nine times, and set the result in the quotient;' multiply the divisor by this quotient figure, and place the product under the assumed part of the dividend, and subtract it therefrom; and to the right of the remainder annex the next right hand figure of the dividend, and again seek how often the divisor is contained in this portion of the dividend, setting the result in the quotient, and so proceed till all the figures of the dividend have been brought down. If your remainder at any time be greater than your divisor, what must be done? A. Increase the quotient figure. If the product of the divisor and quotient figure exceeds that part of your dividend directly above it; what must be done? A. Lessen the quotient figure. How is divi sion proved? A. It may be proved by addition, by subtraction, by multiplication, and by division. How isdivision proved by addition? A. Add the divisor as many times as the quotient expresses a unit, and add the remainder, if there be any, to the amount; and if the whole amount equal the dividend, the work is right; because it is evident, if the divisor be added as often as it has been taken away from the dividend, it must again produce the dividend. Is there any other way of proving division by addition? A. Yes, by adding the remainder to the several lines of products in the order in which they stand, and if the amount equal the dividend, the work is right; this is on the principle of proving subtraction, adding the remainder and subtrahend, for obtaining the minuend; and it is plain, if these several subtrahends and the remainder be added, the amount must equal the minuend; because it is collecting in one sum the subtractions which destroyed the dividend. How is division proved by subtraction? A. By subtracting the divisor from the dividend as often as the quotient contains a unit, and if it reduce it to nothing the work is right. When there is a remainder in division the same remainder must be left in subtraction; this proof is evident, because division is only a short way of performing these several subtractions. How can division be proved by multiplication? A. By multiplying the divisor by the quotient, or the quotient by the divisor; and if the product equal the dividend the work is right; because it is plain, if 12 be divided by four, the quoticnt will be three; then if we repeat the 4 three times our product must be twelve; if there be a remainder you must add it to the product, because in subtracting the divisor there was something left of the dividend. How is division prov ed by itself? A. By dividing the dividend by the quotient, and if your last quotient equal the first divisor, the work is right; because it is as plain, that the quotient can be subtracted from the dividend as often. as the divisor contains a unit, as it is, that your divisor can be subtracted from the dividend as often as your quotient expresses a unit; thus, when we divide twelve by four, it is plain, our quotient must be three, and when we divide twelve by three, it is plain our quotient must be four, which is our first divisor. When you have the price of a quantity given, how do you obtain the price of one? A. By divid ing the price of the quantity, by the number that expresses the quan tity, and the quotient will be the price of one. Suppose the price of twelve yards to be 48 dollars; how would you find the price of one yard A. By dividing $48, the price of the quantity, by 12, the number expressing the quantity; and the quotient will be the price of one yard. Why should that give the price of one yard? A. Because one yard must be worth a twelfth part as much as twelve yards, and by dividing the price of twelve yards by 12, our quotient is a twelfth part of our dividend. How may you divide by 10, 100, 1000, &c.; A. I divide by 10, by cutting off one figure from the right of the dividend and the figures at the left will be the quotient; the one cut off, will be the remainder. Why should that give the true quotient? A. Because the figures at the left of the one cut off, have only a tenth part of the value which they possessed before; for the figure that before expressed tens, now only expresses units, and those at the left have diminished in the same proportion; and in dividing by 100, we cut off two of the right hand figures of the dividend, which gives those at the left only one hundredth part of the value which they possessed before, and the figures at the right are the remainder; because in dividing in the usual way, the ciphers would come under those figures, and consequently they would come down for a remainder; and so when we divide by 1000, the figure expressing thousands, after the operation of cutting off, only expresses units. When there are ciphers at the right hand of the divisor, what may be done? A. They may be cut off, and also an equal number of figures from the right of the dividend; and then the operation is performed with the remaining figures of both, the same as if those cut off belonged to neither; and the figures cut off from the right hand of the dividend, must be brought down to the right hand of the remainder, for the true remainder. Why does that appear to be correct? A. Because as often as the whole divisor is contained in the whole dividend, so, like portions of the divisor must be contained in like portions of the dividend; therefore it saves, by cutting off the ciphers and a like number of figures from the dividend, the needless repetition of ciphers. How do you proceed to divide by a composite number? A. First divide the dividend by one of the component parts of the divisor, and then that quotient by the other. How do you find the true remainder? A. Multiply the last remainder by the first divisor, and to the product add the first remainder, and it will give the true remainder. If your divisor be eight, what would you call the component parts? A. Two and four. Why? A. Because twice four are eight. Why should you obtain the same quotient, by first dividing by 4 and then dividing that quotient by 2, that you would by dividing your dividend directly by 8? A. Because it is plain, that the half of a fourth of any number, is the eighth of the whole. If you divide a number by sixteen and have no remainder, what part of your dividend is your quotient? A. The sixteenth part. How can you prove it? A. By repeating the quotient sixteen times, the produc will then equal the dividend. What is division of decimal or federal money? A. It is dividing a number that is composed of integers, and decimal parts of an integer, or simply the parts of an integer, How do you proceed in dividing. F A. The same as in whole numbers, only point off from the right of the quotient as many figures for decimals, as are equal to the decimal places in the dividend. In dividing dollars, if you have a remainder, what do you do, after all the figures are brought down from the dividend? A. Join a cipher to the remainder, and then seek how often the divisor is contained as before, and if something again remain, annex another cipher to the remainder, and divide as before. What will the two figures in the quotient next to dollars express? A. Cents. What does the next to cents express? A. Mills. Why do we divide decimals the same as whole numbers? A. Because they increase in a ten fold proportion from the right to the left. SUPPLEMENT TO MULTIPLICATION. Multiplying by a mixed number, as 6, 54, &c. is taking the multiplicand as many times as there are units in the mul tiplier; and likewise taking a part of the multiplicand, as many times as there are like portions of a unit in the fraction of the multiplier. RULE.-Multiply the multiplicand by the whole number of the multiplier; and take such parts of the multiplicand for the product of the fraction, as there are like parts of a unit contained in the fraction. NOTE.-Multiplying by 1, is taking the multiplicand once; multiplying by 2, is taking it twice; multiplying by 3, is taking the multiplicand 3 times, and so on. Multiplying by a fraction, is taking a part of the multiplicand, as many times, as there are like parts of a unit in the multiplier. Multiplying by is taking one half of the multiplicand; multiplying by, is taking one fourth of the multiplicand; multiplying by, is taking one sixth of the multiplicand; multiplying by, is taking one sixth of the multiplicand, twice. DEMONSTRATION.-The reason of this work appears plain from the preceding note; because in the first example we wish to repeat our multiplicand by a half, and the half of 36, is evidently obtained by dividing thirty-six by 2; and the |