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Let the pupil, in doing sums, explain them as below.
14 24 is contained in 69 units, 2 times; in 69 hundreds, 2 hundred times. 2 hundred times 24 iš 48 hundred, which subtracted from 69 leaves 21 hundred.
21 hundreds are 210 tens, and the 9 tens of the dividend brought down, make 219 tens.
24 is contained in 219, 9 times; in 219 tens, 9 tens of times. 24 multiplied by 9 tens, is 216 tens, which subtracted from 219 tens leaves 3 tens.
3 tens are 30 units, and the 8 units of the dividend brought down make 38 units. 24 is contained in 38 units once, and 14 over, which is 14 of another time.
The dividend then contains the divisor 2 hundreds of times, 9 tens of times, 1 unit of times, and 14 of another time, or 291 times and if of another time.
Thus it appears, that in Long Division, each quotient figure, when set down, does not show the exact number of times the divisor is contained in the order which is divided ; but it shows, that the divisor is contained so many times as the quotient figure expresses, and then, a process fol. lows for discovering how many more times it is contained.
Let the pupil do the following sums, and explain them as above, until perfectly familiar with the mode. Divide 2479 by 14 Divide 3568 by 16 1954 " 18
5896 66 23 36964 " 17
38907 66 21 29006 « 28
46032 " 36
Explain the remainder of the process. In the second sum what is done first? second ? third ? fourth ? Explain the whole process. In Long Division what does each quotient figure not show? What does it show? What process follows ?
RULE FOR LONG DIVISION.
Place the divisor at the left of the dividend, and draw a line between. Take as many of the highest orders as would, if units, contain the divisor once, and not more than 9 times. Divide the orders so taken, as if they were units. Place the quotient figure at the right of the dividend, and draw a line between. Multiply the quotient and the divisor together, and subtract them from the part of the dividend already divided. To the remainder, add as many of the next undivided orders of the dividend as would enable it
, if units, to contain the divisor once, and not more than 9 times, and then divide 'as before.
If it is needful to add more than one order of the dividend to any remainder, (to enable it to contain the divisor) put one cipher in the quotient for every additional order. If any remains after dividing the unit order, put the divisor under it for a fraction.
EXAMPLES. Divide 2649 by
3294 by 14 2468 15 64329
16 1234 17
18 56789 19
36 35673 59 76542
41 45678 256
848 912345 481
59624 562 678122 984
541 34568 639
856 543219 656
1459862 942 678912 9481
724368 2586 9876533 6002
EXAMPLES FOR MENTAL EXERCISES. 1. Bought 12 pounds of raisins for 3 shillings a pound,
dollars did they cost? State the process thus. If one pound cost 3 shillings, 12 pounds cost 12 times as much, or 36 shillings. As there are 6 shillings in a dollar, they cost as many
dollars as there are sixes in 36.
What is the rule for Long Division ?
Let the following sums be stated in the same manner.
2. Bought 5 bushels of peaches at 4 shillings a bushel, how many
dollars did they cost ? 3. How many peaches at 4 cents each must you give for 9 oranges at 5 cents apiece ?
State the last sum thus. If one orange cost 5 cents, 9 cost 9 times as much, or 45 cents. As each peach is worth 4 cents, you must give as many peaches as there are fours in 45.
4. If you buy 10 yards of cotton, at 5 shillings a yard, and pay for it with butter at 2 shillings a pound, how many pounds will pay for it?
5. How many apples at 4 cents each, must you give for 3 pine apples at 12 cents each?
6. If you buy 48 bushels of coal for 12 cents per bush. el, and pay for it with cheese at 10 cents per lb. how many pounds do you give ?
7. How much rye at 5 shillings a bushel must you give for 12 bushels of wheat at 8 shillings a bushel ?
8. How much cloth worth 9 shillings a yard must you give for a firkin of butter worth 12 dollars?
(Change the dollars to shillings.)
9. How many dozen of eggs at 9 cents per dozen must be given, for 3 yards of cotton worth 20 cents per yard?
10. If you have 8 pine apples worth 9 cents each, and your companion has 9 quarts of strawberries worth 8 cents a quart, which he gives to buy the same worth of pine ap. ples, how many pine apples must you give him?
Divide £4,, 8s. by 2.
lf 2 dresses contain 24 yds. 2 qrs. how much in each dress ?
If 3 silver cups weigh 9 lbs. 6 oz. what is the weight of each?
In division we find how often one number is contained in another, and thus what part of one number is another.
Thus if we divide 8 lbs, 16 oz. by 4, we can either say how many times 4 is contained in 8 and in 16, or we can say what is one fourth of 8 lbs. and 16 oz.
If there is any remainder in dividing one order, it must be changed to units of the next lower order and added to it and then divide again. In doing the sum we place the figures thus :
3)4 “ 18
1 6 12 6 11 We proceed thus in explaining the process.
A third of £4 is £1 which is set under that order, and there is £l remaining which is changed to 20 shillings and added to the 18, making 38. A third of 38 shillings is 12 shillings, which are set under that order. 2 shillings remain, which are changed to 24 pence and added to the 9 pence, making 33 pence; a third of 33 pence is 11 pence, which are set in that order.
Let the following sums be performed and explained as above.
Divide 22£ 11s. 6d. by 6.
If 9 silver cups weigh 3 lbs. 6 oz. 8 pwt. 3 grs. what is the weight of each?
If 8 dresses contain 59 yds. 3 qrs. 2 n. how much in each dress?
If the divisor exceeds 12 and is a composite number, di. vide the sum by one of the factors as above and the an. swer by the other.
EXAMPLES. Divide £2 - 8s. “ 11d.“ 4 qr. by 44.
If 18 gal. “6 qr.“ 4 g. of brandy be divided equally into 28 bottles, how much does each contain ?
If 24 coats contain 62 yds. 3 qrs. 4 na. how much does each contain ?
If 32 teams be loaded with 40 T. 16 cwt. 3 qrs. how much is that for each team?
If the divisor exceeds 12 and is not a composite number, the following method is used.
Let the figures be placed thus.
d. t d. 139)461 " 11 " 11(3 “6 5
695 We first divide the pound order and 4 is the quotient figure, which is of the pound order because the dividend is pounds. This is put in the quotient with the £ put over it to indicate its order.
In order to find the remainder we subtract the product of the quotient and divisor from the 461.
The remainder is 44£. This must be changed to shillings, which is done by multiplying it by 20 and then the 11 shillings of the dividend are added.
This sum is then divided by 139 and the quotient figure is 6, which is of the shilling order and must be put in the quotient under that sign. Proceed as before till the orders are all thus divided.
Let the following examples be performed and explained as above.
Divide 239£ “ 16s. “ 4d. “ 3qr. by 123.
If 239 yds, of cloth cost 49£ 19s. Ild. what was that per yard?
Note.-Change the pounds to shillings first.
If 349 cwt. 3 qrs. 12 lbs. are contained in 264 barrels, how much is in each barrel.
If 42 cwt. of tobacco cost 826£ 18s. 9d. what is that