More Exercises for the Slate. 2. What will 36 hundred weight of sugar cost, at 29 dollars a hundred? A. 1044 dollars. 3. Multiply 3065428 by 35. 4. Multiply 4078945 by 96. 5. Multiply 18934 by 108. 6. Multiply 45678 by 144. A. 107289980. A. 391578720. A. 2044872. A. 6577632. SIMPLE DIVISION. ¶ XV. 1. If you divide 12 apples equally between two boys, how many will each have? How many times 2 in 12, then? Why? A. Because 2 times 6 are 12. 2. How many oranges, at 8 cents apiece, can you buy for 48 cents? For 96 cents? How many times 8 in 48? 8 in 96? Why? 3. A man bought 8 lemons for 80 cents; how much did he give apiece? How many times 8 in 80? Why, or proof? 4. How many gallons of brandy, at 3 dollars a gallon, can be bought for 36 dollars? For 60 dollars? For 90 dollars? For 300 dollars? Why? 5. Four boys found a bag containing 48 silver dollars; how many will they have apiece, if it be divided equally? 6. When oranges are 2 cents apiece, how many will 8 cents buy? Will 16 cents buy? Will 32 cents? Will 36 cents? Will 48 cents? Will 100 cents? 7. If you pay 9 cents for one pound of sugar, how many pounds can you buy for 45 cents? For 54 cents? For 108 cents? 8. How much is one half (2) of 4? Of 8? Of 16? Of 20? Of 24? Of 30? Of 100? Of 200? 9. Harry had 16 apples, and gave one half (2) of them to Thomas; how many did Thomas receive? 10. How much is one third (3) of 6? Of 24? Of 30? Of 36 ? 11. How much is one half () of 8? One One fourth (4) of 16? One fifth (†) of 35? third (†) of 24 ? One sixth (†) of 24? One seventh (4) of 35? One eighth (). of 56? One ninth () of 108? One twelfth (1) of 144? 12. How many times 4 in 40? 3 in 60? 5 in 100? 6 in 1200? 8 in 480? Q. What is this method of finding how many times one number is contained in another, or of dividing a number into equal parts, called? A. Division. Q. What is the method of finding how many times one number is contained in another of only one name, or denomination, called. A. Simple Division. Q. What is the number given to divide by called? A. The Divisor. Q. What is the number to be divided called? A. The Dividend. Q. What is the number of times that the divisor is contained in the dividend called? A. The Quotient. Q. What is that which is sometimes left after dividing, or after the operation is performed, called? A. The Remainder, which must always be less than the Divisor. Q. Of what name, or denomination, is the remainder ? A. The same as the dividend. Q. If your dividend, for instance, be ounces, remainder be? A. Ounces. Q. How many times 4 in 40? and why? what will your Q. From this example what does division appear to be the opposite of? A. Multiplication. 3d time he had 0 left. Q. James, having 12 oranges, was desirous of dividing them equally among his 4 little sisters, and, in order to do this, he handed them at first one apiece; how many had he left? Q. When he handed them another apiece, how many had he left? Q. When he handed them one more apiece, how many had he left? Q. From these illustrations how does it appear that a number may be divided into equal parts? A. By Subtraction. Q. How many times did James give to each of his sisters an orange apiece? Q. How many times did you subtract? A. Three times. Q. By this we see that the quotient represents the number of subtractions: now, if the quotient were 4000, how many times would it be necessary to take the divisor from the dividend to perform division by subtraction A. 4000 times. Q. What, then, is Divisi a quick way of performing? A. Many subtractions. SHORT DIVISION. 1 XVI. Q. What is SHORT DIVISION? A. When the divisor is 12, or less. 1. How many oranges, at 3 cents apiece, may be bought for 657 cents? OPERATION. Dividend. Divisor, 3)657 cents. Quotient, 219 oranges, Ans. How do you obtain the 2 (hundreds) in the quotient? A. I begin on the left of the dividend, and say, 3, the divisor, is contained in 6 (hundreds) 2 (hundreds) times, that is, 200 times, writing the 2 (hundreds) down under the 6 (hundreds). How do you get the 1 (ten)? A. 3 in 5 (tens) 1 time, and 2 (tens) left. What do you do with the 2 which is left? A. I join, or carry it as 2 tens, that is, 20 units, to the 7 units, making 27. How do you proceed to get the 9, then? A. 3 in 27, 9 times. PROOF. Quotient, 219 Dividend, 657 How many times 6 in 30? and why? How, then, would you proceed to prove the foregoing example? A. I would multiply 219 (the quotient) by 3 (the divisor), making 657 (the dividend)-therefore right. From the illustrations now given we derive the following RULE. I. At which hand of the dividend do you place the divisor? A At the left. II. How many figures do you take first? A. Enough to contain the divisor once, or more. III. What do you set down underneath? A. The quotient. IV. If there should be a remainder, how would you pro ceed? A. I join, or carry it to the next figure of the dividend, as so many tens. For example, suppose 3 remain, and the next figure be 8, how would you say? A. I would say, 3 (to carry) to 8, makes 38. V. How do you proceed if the divisor be not contained in the next figure of the dividend? 4. Write a cipher in the quotient, and join this figure to the figure next to it, as so many tens. PROOF.-Which terms do you multiply together to prove the operation? A. The divisor and quotient. What is to be done with the remainder, if there be any? A Add it to this product. What must the amount be like? A. The dividend. More Exercises for the Slate. 2. Rufus divided 42 oranges equally between his two little brothers; how many had they apiece? A. 21. 3. If 3 bushels of apples cost 360 cents, how much is that a bushel? A. 120 cents. 4. How many months are there in 452 weeks, there being 4 weeks in each month? A. 113 months. 5. A man, having 416 dollars, laid them all out in cider, at 4 dollars a barrel; how much cider did he buy? A. 104 barrels. 6. A man bought 6 oxen for 318 dollars; how much did he pay a head? A. 53 dollars. 7. How much flour, at 7 dollars a barrel, can be bought for 1512 dollars? 9. 216 barrels. 8. At 8 cents apiece, how many oranges will 8896 cents buy? A. 1112 oranges. 9. At 10 dollars a barrel, how many barrels of flour may be bought for 1720 dollars? A. 172 barrels. 10. 12 men by contract are to receive 1500 dollars for a job of work; how many dollars will be each man's part, if they be divided equally among them? A. 125 dollars. 11. 2 men, trading in company, gained 2450 dollars; how much was each man's part? A. 1225 dollars. 12. At 3 dollars a barrel, how many barrels of pork can be bought for 5463 dollars? A. 1821 bbls. Note. The total remainder is found by adding together what remains after each operation. 13. Divide 256587 by 2. A. 128293-1 rem. A. 94155-3 rem. The operation, thus far, has been carried on partly in the mind, and partly by writing the numbers down; but oftentimes the divisor will be too large to be thus performed. When, therefore, we write the operation out at length, what is the process called? A. Long Division. LONG DIVISION. ¶ XVII. 1. A man, dying, left 957 dollars to be divided equally among his 4 sons; what was each son's part? Long Division. OPERATION. Dividend. Quotient. Divisor, 4) 957 (239 8 15 12 37 36 Short Division. OPERATION. 4) 957 2391 Quotient. As Long and Short Division are exactly alike, except in Short Division the whole operation is not written down, to begin, then, in Short Division, we should say, 4 in 9, 2 times, and 1 over. This we discover by saying in the mind, 4 times 2 are 8, and 8 from 9 leaves 1. Now, to express in figures this operation, we may write the numbers where we please: where, then, for the sake of convenience, may the 2 (times, the quotient figure) be placed? A. At the right hand of the dividend. We are next to say, 4 times 2 are 8: this 8, you know, must be subtracted from 9: where would it be convenient to place the 8? A. Under the 9. By taking 8 from 9, we have 1 remaining, which we should, in Short Division, carry or join to 5, the next figure of the dividend; how can we do this now? A. By joining or bringing down the 5 to the right hand of the 1, making 15. How do you get the 3 in the quotient? A. I say, 4 in 15, 3 times. How do you proceed next? A. I say, 3 times 4 are 12; and 12 from 15 leaves 3. What do you do with the 3? A. I bring down 7 of the dividend to the right hand of the 3, making 37. How do you get the 9 in the quotient? A. I say, 4 times 9 are 36, and subtracting 36 from 37 leaves 1, remainder. |