3. In what does the process of the other columns differ from these? REM. 1.-Let the class be thoroughly exercised in subtraction, especially in the matter of carrying from one order to another; `one example is sufficient to impress it on the mind, as the solution may be repeated as often as is necessary. REM. 2. The teacher can add other questions and larger examples if he think proper. PRACTICAL EXAMPLES. 1. A farmer has 7 cattle in one field and 8 in another; if he take 4 from the first field and put them in the second, how many will there then be in each field? 2. In one field there are 435 cattle and in another 657; if 320 be taken from the first field and put in the second field, how many cattle are then in each field? REM.-In subtraction the larger number is the minuend and the smaller number the subtrahend. 3. A merchant bought dry goods to the amount of 1263 dollars, groceries for 734 dollars, hardware for 231 dollars, and notions for 137 dollars; what is the amount of his purchases? 4. A farmer sold a horse for 175 dollars, cows for 97 dollars, and sheep for 51 dollars; what was the amount of his sales? 5. A gentleman owns five farms; the first is worth 10600 dollars, the second 4970 dollars, the third 5000 dollars, the fourth 6500 dollars, and the fifth 8500 dollars; the amount of his indebtedness is 7984 dollars; if the whole is disposed of at the above rates and the debt paid, what sum will he then have? 6. A man bought a horse for 125 dollars and sold it for 182 dollars; how much did he gain? 7. A man commenced business with 5000 dollars; the first year his profits were 720 dollars, the second year 500 dollars, the third year 1000 dollars, but the fourth year he lost 2000 dollars; what was then his capital? 8. A man purchased a lot for 900 dollars and erected a house on it at the cost of 3875 dollars for the carpenter's work, 550 dollars for masonry, and the painting cost 869 dollars; he then sold the property for 6000 dollars; did he gain or lose by the transaction, and how much? Ans. Lost $194. 9. A man bought a barrel of flour for 8 dollars, three barrels of pork for 35 dollars, salt for 16 dollars, and corn for 300 dollars; he sold the whole so as to gain 20 dollars. How much did he sell it for? 10. A merchant owns property worth 264956 dollars, and owes 89635 dollars; what is the net value of his property? 11. A farmer sold eight cords of wood for 144 dollars; he received in payment cloth valued at 60 dollars, and 48 dollars cash; how much was still owing him? 12. Bought 21693 yards of calico of one merchant, 560 yards of another, and 83946 yards of a third; sold 340 yards to one customer, and 69548 yards to another; how much is still on hand ? 13. Sold to one man 3246 acres of land at 6 dollars per acre, to another 4328 acres at 8 dollars per acre, to a third 9546 acres at 5 dollars per acre, and to a fourth 3261 acres at 9 dollars per acre. What was the amount of sales? 132 144 12 24 36 48 60 72 84 96 108 120 As a Multiplication Table, begin with the first line; thus, Once 1 is 1; twice 1 are 2; three times 1 are 3, etc. Second line, Once 2 are 2; twice 2 are 4; 3 times 2 are 6; 4 times 2 are 8, etc. Third line, Once 3 are 3; twice 3 are 6; 3 times 3 are 9; 4 times 3 are 12, etc. Recite each line similarly. REM. 4 times 3 are 12, and 3 times 4 are 12; hence, alternating the factors does not change the product. As a Division Table, begin with the first line; thus, 1 is contained in 1, once; in 2, twice; in 3, 3 times; in 4, 4 times, etc. Second line, 2 into 2 = 1; 2 into 42; 2 into 6 = 3; 2 into 8 = 4, etc. Third line, 3 into 3 = 1; 3 into 6 = 2, etc. REM.—As a Multiplication Table, it may also be read by the column, by which the factors are alternated, without changing the product. Any number is multiplied by 10 by adding a zero to it. As a Division Table, the first column has all the divisors, the first line all the quotients, and every number in each line is a dividend, which is always in the same line and the same column with the quotient and divisor. Any number having a zero in the units place is divided by 10 by removing the zero, THEOREM I. Any number is multiplied by 10 by annexing a zero to it. Since the product of any number multiplied by 1 is equal to the number itself, the product of any number multiplied by 2 is double the number, etc. For, as 240, 10 × 1 = 10, and 10 x 2 = 20, and 10 × 24 = and as alternating the factors does not change the product, hence, 1 × 10 10, and 2 × 10= 20, and 24 × 10 = 240. .. Any number is multiplied by 10 by annexing a zero to it. COR.-Any number is multiplied by 100 by annexing two zeros to it, and annexing three zeros multiplies it by 1000, etc. THEOREM II. The product of any two factors will have as many figures, or one less, than both factors. The products of the smaller figures of units will be but one figure until above 3, when there will be two figures, but never more, as 9 × 9 = 81, and every additional figure annexed to each or either factor, whether small or large, will make an increase of one figure and no more; therefore the product of any two factors will have as many figures, or one less than both factors. COR. 1.-The product of any two figures cannot be less than one figure, nor more than two. COR. 2. The product of units by units must be units, and when there are two figures, the left-hand figure will be tens. The product of tens by units must be tens, and when there are two figures, the left-hand figure will be hundreds; and if any order be multiplied by units, the right-hand figure of the product will be the same order as the multiplicand, and if there be two figures in the product, the left-hand figure will belong to the next higher order. COR. 3.-When the multiplier is tens, the product will. be ten times as great as if the multiplier were units; that is, each product will have one zero to the right of it, holding the units place, or the first figure of the product must be placed in the column of tens; when the multi |