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For example, suppose we wish to find the product of 7 by 5; we look for 7 at the top of the table, and for 5 at the left hand, and where the lines intersect is 35, the number sought; or, we may look for the 7 at the left hand, and the 5 at the top, and find, where the lines intersect, the same result.
55. The repeated addition of a number to itself is equivalent to a multiplication of that number. Thus, 7 -(- 7 7 -J7 is equivalent to 7 X 4, the sum of the former and the product of the latter being the same. Hence multiplication has sometimes been called a concise method of addition.
56. The product must be of the same kind or denomination as the multiplicand, since the taking of a quantity any number of times does not alter its nature. Thus: 5, an abstract number, X 3 = 15, an abstract number; and 9 yards X 7 = 63 yards.
57. The multiplier must always be considered as an abstract number. Thus, in finding the cost of 4 books at 9 dollars each, we cannot multiply books and dollars together, which would be absurd, but we can, by regarding the 4 as an abstract number, take the 9 dollars, or cost of 1 book 4 times, and the product, 36 dollars, will be the result required.
58. The product of two factors will be the same, whichever is taken as the multiplier. Thus, 8x6 = 6x8 = 48; and the cost of 5 hats at 2 dollars each gives the same product as 2 hats at 5 dollars each. Also, the product of any number of factors is the same, in whatever order they are multiplied. Thus, 2x3x5 = 3x5x2 = 5x2x3 = 30.
59. A Composite number is a number produced by multiplying together two or more numbers greater than 1. Thus, 10 is a composite number, since it is the product of 2 X 5; and 18 is a composite number, since it is the product of 2 X 3 X 3.
69. To multiply simple numbers.
Ex. 1. Let it be required to multiply 1538 by 9.
Operation. Having written the multiplier, 9, unMultiplicand 1 5 3 8 der the unit figure of the multiplicand, Multiplier 9 we multiply the 8 units by the 9, ob
taining 72 units = 7 tens and 2 units. 1 roduct 1^842 -\ye wrjte down the 2 units in the units' place, and reserve the 7 tens to add to the product of the tens. We then multiply the 3 tens by 9, obtaining 27 tens, and, adding the 7 tens which were reserved, we have 34 tens = 3 hundreds and 4 tens. We write down the 4 tens in the tens' place, and reserve the 3 hundreds to add to the product of the hundreds. We next multiply the 5 hundreds by 9, obtaining 45 hundreds, and, adding the 3 hundreds which were reserved, we have 48 hundreds = 4 thousands and 8 hundreds. We write down the 8 hundreds in the hundreds' place, and reserve the 4 thousands to add to the product of the thousands. By multiplying the 1 thousand by 9 we obtain 9 thousands, and, adding the 4 thousands reserved, we have 13 thousands, which we write down in full; — and the product is 13842.multiplicand, and add together its successive figures toward the right, till the sum obtained equals or exceeds the number nine. If it equals it, drop the nine, and begin to add again at this point, and proceed till you obtain a sum equal to, or greater than, nine. If it exceeds nine, drop the nine as before, and carry the excess to the next figure, and then continue the addition as before. Proceed in this way, till you have added all the figures in the multiplicand and rejected all the nines contained in it, and write the final excess at the right hand of the multiplicand.
2. Let it be required to multiply 2156 by 423.
Ans. 911988. Operation. In this example the multiplicand is Multiplicand 2 15 6 to be taken 423 times = 3 units times
rectly under its multiplier, that units of the same order may stand in the same column, for convenience in adding.
Rule. — Write the multiplier under the multiplicand, arranging units under units, tens under tens, Sfc.
Multiply each figure of the multiplicand by each figure of the multiplier, beginning with the right-hand figure, writing the right-hand figure of each product underneath, and adding the left-hand figure or figures, if any, to the next succeeding product.
If the multiplier consists of more than one figure, the right-hand figure of each partial product must be placed directly under the figure of the multiplier that produces it. The sum of the partial products will be the whole product required.
Note. — When there are ciphers between the significant figures of the multiplier, pass over them in the operation, and multiply by the significant figures only, remembering to set the first figure of the product directly under the figure of the multiplier that produces it.
61i First Method of Proof. — Multiply the multiplier by the multiplicand, and, if the result is like the first product, the work is supposed to be right. (Art. 58.)
62. Second Method of Proof.— Divide the product by the multiplier, and, if the work is right, the quotient will be like the multiplicand.
Note. — This is the common mode of proof in business; but, as it anticipates the principles of division, it cannot be employed without a previous knowledge of that process.
63. Third Method of Proof. — Begin at the left hand of the
Proceed in the same manner with the multiplier, and write the final excess under that of the multiplicand. Multiply these excesses together, and place the excess of nines in their product at the right.
Then proceed to find the excess of nines in the product obtained by the original operation; and, if the work is right, the excess thus found will be equal to the excess contained in the product of the above excesses of the multiplicand and multiplier.
Note. — This method of proof, though perhaps sufficiently sure for common purposes, is not always a test of the correctness of an operation. If two or more figures in the work should be transposed, or the value of one figure be just as much too great as another is too small, or if a nine be set down in the place of a cipher, or the contrary, the excess of nines will be the same, and still the work may not be correct. Such a balance of errors will not, however, be likely to occur.
3. Multiply 7325 by 3612.
Multiplicand 7 3 2 5
Multiplier 3 6 12
Product 2 6 45 7 9 0 0
4. Required the product of 82907 by 652.
PROOF BV MULTIPLICATION
3 6 12 7325
1 80 60 7 2 2 4 1 08 3 6 25 28 4
Product 2 6 457900
17. What will 365 acres of land cost at 73 dollars per acre? Ans. $26645.
18. What will 97 tons of iron cost at 57 dollars a ton?
19. What will 397 yards of cloth cost at 7 dollars per yard? ' Ans. $2779.
20. What will 569 hogsheads of molasses cost at 37 dollars per hogshead? Ans. $21053.
21. If a man travel 37 miles in one day, how far will he travel in 365 days? Ans. 13505 miles.
22. If a vessel sails 169 miles in one day, how far will she sail in 144 days?
23. What will 698 barrels of flour cost at 7 dollars a barrel?
24. What will 376 1bs. of sugar cost at 13 cents a pound?
Ans. 4888 cts.
25. What will 97 1bs. of tea cost at 93 cents a pound?
Ans. 9021 cts.