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9 X 3 = =
27, +9=36, +7 x 5, or 35,=71. 71 hun. dreds = 7 thousands and 1 hundred.
9 X 8= 72, +7=79, +7 x 3, or 21, = 100. 100 thousands 10 ten-thousands and 0 thousands.
7 X 8= 56, + 10 = 66. 66 ten-thousands.
This gives for an answer 660124, as did the first method. The last process being understood, the work may be still further abbreviated by omitting to name the factors used.
Thus, 54 units = 5 tens and 4 units. 45+5=50, +42=92. 92 tens=9 hundreds and 2 tens.
27 +9=36, +35=71. 71 hundreds=7 hundreds and 1 thousand.
72+7=79, +21=100. 100 thousands = 10 ten-thou. sands and 0 thousands.
56 + 10 : 66. 66 ten-thousands. Answer, as before, 660124.
Finally, the work may be abbreviated so as to name only results :
54 units = 5 tens and 4 units. 45, 50, 92 tens = 9 hundreds and 3 tens. 27, 36, 71 hundreds 7 thousands and 1 hundred. 72, 79, 100 thousands = 10 ten-thousands and 0 thousands. 56, 66 ten-thousands.
Note. The above methods are much more expeditious than is the method of writing the product by each figure of the multiplier separately, and are no more liable to inaccuracy.
1. How much will 97 acres of land cost at $347 per acre ?
2. If a cubic yard of sand weighs 2537 lb., how much will 38 cubic yards weigh?
3. How many pounds are there in 18 T. 17 cwt. 1 qr. ?
4. If a ship sails 96 miles in one day, how far will she sail in 247 days ?
5. Bought 24 bundles of hay, each bundle containing 497 lb. How
many pounds were there in all ? 6. Bought 2947 gallons of oil at $.84 per gallon. How much did it cost ? Sold it for $.97 per gallon. How much was received for it? What was the gain on it ?
7. Mr. Russell bought 86 balls of twine, each ball containing
8794 ft., and Mr. Greene bought 5.7 times as much. How many feet of twine did Mr. Russell buy? How many did Mr. Greene buy ?
8. How much will 8.3 casks of old wine cost at $138.47 per cask ?
9. How much will .67 of a ton of lead cost at $139.48 per ton ?
10. Mr. Hovey bought 6247 feet of land, and Mr. Ewell bought .94 of the quantity. How many feet did Mr. Ewell buy?
11. How many pounds are there in 958 boxes of sugar, each box containing 743.67 lb. ?
Note. The products and sums employed in solving the above example are given below, but the pupil should be prepared to give a more full explanation.
56 hundredths 5 tenths and 6 hundredths. 5 + 48 + 35 = 88. 88 tenths - 8 units and 8 tenths.
8 + 24 + 30 + 63 = 125. 125 units = 12 tens and 5 units.
12 +32 +15+54= 113. 113 tens=11 hundreds and 3 tens.
11 +56 +20 +27=114. 114 hundreds = 11 thousands and 4 hundreds.
11 + 35 + 36 : :82. 82 thousands = 8 ten-thousands and 2 thousands.
8 + 63=71. 71 ten-thousands.
Note. By extending the principles before explained we can write the final product at once, as below.
849095685. In the following forms, 2 products are written:
= 8795 x 96543.
= 8795 X 96543. 13. If a cubic foot of iron weighs 486.25 lb., how much will 347 cubic ft. weigh?
14. If 1 cubic foot of walnut timber weighs 41.9375 lb., how much will 43 sticks of hewn walnut timber, each 16 ft. long, 11 ft. wide, and 1 ft. thick, weigh?
15. How many square feet are there in a rectangular lot, 4327 feet long and 249 ft. wide ?
16. How much will 354.87 acres of land cost at $83.968 per acre ?
17. What will 3.798 tons of hay cost at $14.278 per ton ?
18. I bought 287 bales of cloth, each bale containing 247.986 yards. How many yards did they all contain ?
19. If a horse trots 3.674 hours, at the average rate of 7.2968 miles per hour, how far will he trot?
20. How many square inches are there in a lot 247 ft. long and 187 ft. wide ?
21. What will 47.983 yards of cloth cost at $2.83 per yd. ? 22. What will 678.94 bbl. of flour cost at $6.37
bbl. ? 23. How many solid inches in 5 C. 6 Cd. ft. 12 cu. ft. 1437 cu. in. ?
24. How many sq. in. in 8 sq. rd. 21 sq. yd. 6 sq. ft. and 28
sq. in. ?
25. How many dr. in 18 T. 16 cwt. 1 qr. 14 lb. 6 oz. 11 dr. ?
26. A grain dealer sold 28.7 bushels of wheat at $1.294 per bushel, and 14.79 bushels at $1.267 per bu. What did he receive for it?
27. A city merchant went into the country to purchase flour. He was absent from the city 27 days, and his expenses while absent were $7.386 per day. He bought 175 bbl. of flour at $5.875 per bbl., 516 bbl. at $5.948 per bbl., 1386 bbl. at $6.11 per bbl., and 827 bbl. at $6.087 per bbl. It cost him $.634 per bbl. to get the flour transported to the city. He sold 697 bbl. of it at $7.114, 824 bbl. at $7.213 per bbl., and the rest at $6.978 per bbl. Did he gain or lose by the adventure, and how much?
28. A merchant bought 49.5 cases of cassimere, each case
containing 297 yd. at $1.1875 per yd. It cost him $.125 per case to have the cloth removed to his store, and $.045 per case to have it hoisted into his loft. One case of the cloth was stolen from him; he sold 23 cases at $1.423 per yd., and the remainder at $1.357 per yard, agreeing to deliver it at a railroad depot, 1 mile from his store. It cost him $.158 per case to have it carried to the depot. Did he gain or lose on the cloth, and how much?
A. 1. How many apples, at 3 cents apiece, can be bought for 24 cents ? This is a question in Division. It is obvious that, to determine its answer, we must find how many parts of 3 cents each there are in 24 cents; for each such part is the price of 1 apple.
We can find the number of these parts by counting 24 cents into piles of 3 cents each, and then counting the number of piles ; — by finding how many threes must be added to make 24; -or by finding how many times 3 is contained in 24, as we have done in previous examples of this character. This last method is called Division.
2. If 3 pounds of raisins cost 24 cents, how many cents will i lb. cost? This is also a question in Division. To determine its answer, we must find how many cents there will be in each part obtained by separating 24 cents into 3 equal parts.
We can do this by laying out 24 cents into 3 equal piles, and then counting the number of cents in each pile; or by finding } of 24 in the usual manner. This last would be called the process of division.
All questions in division require results similar to that required by one or the other of the above. We may, therefore, give the following definition :
Division is a process by which we ascertain the number of parts of a given size into which a given number may
separated, or by which we ascertain the number of units there will be in each part obtained by dividing a given number into a given number of equal parts.
The nature of the question is the only thing that will determine which of these results is required. The same arithmetical process may be employed to obtain the answer to a question of
either kind, but the reasoning process required in one is very different from that required in the other. The following solu. tions to questions 1 and 2 will illustrate the resemblances and differences above alluded to.
Ques. 1. Solution. If for 3 çents one apple can be bought, for 24 cents as many apples can be bought as there are times 3 cents in 24 cents, which are 8 times. Therefore, 8 apples can be bought for 24 cents, when one apple costs 3 cents.
Ques. 2. Solution. If 3 pounds of raisins cost 24 cents, one pound will cost one third of 24 cents, which is 8 cents. Therefore, one pound of raisins will cost 8 cents, when 3 pounds cost 24 cents.
The number supposed to be divided into parts is called the dividend. In the first class of questions, the number which shows the size of each part, and, in the second class, that which shows the number of parts, is called the divisor. The result is called the quotient.
We perform the division and make the necessary reductions in the same way, whether we write our work or not. If we write it, when our divisor is a small number, we usually write only the divisor, dividend, quotient, figure by figure as we obtain it, and the final remainder, if there be one; but when the divisor is a large number, we usually write the divisor, dividend, and quotient, as before, and also the products of the divisor by each figure of the quotient, with the remainder obtained by subtracting these products from the corresponding denominations of the dividend.
B. To illustrate the usual method of writing the work when the divisor is a small number, we give the following example :
How many barrels of flour, at 8 dollars per barrel, can be bought for 29576 dollars ?
By the usual reasoning process we determine that as many barrels can be bought as there are parts of 8 dollars each in 29576 dollars.
Quotient. 3697. = number of parts of $8 each number of barrels that can be bought.
In explaining the division, we should say, As in 2 ten-thousands there are not as many as ten thousand parts of 8 each,