SECTION XXII. A. How much will 8 apples cost at 3 cents apiece? The above is a question in what is called Multiplication. Solving it as we have heretofore solved similar examples, we should say: If one apple costs 3 cents, 8 apples will cost 8 times 3 cents, or 24 cents. To examine the nature of the operation on the numbers, let us suppose that a person ignorant of all numerical processes, except that of counting, should be called upon to solve the same question. If he had a quantity of cents, he might lay 3 in one place, 3 more in another, 3 more in another, and so go on laying 3 in a place till he should have 8 piles of 3 cents each. Since the cents in each pile would buy 1 apple, the cents in all would buy 8 apples; he might then, by counting the cents in the 8 piles, find how much the apples would cost. If he should have no cents, he might still determine the result in a similar way, by using pebbles, sticks, marks, or anything else of a like character. After learning how to add, he might obtain the result by adding 8 threes together. If, after having obtained the result in some way similar to the above, he should remember it, he would ever after be able, without counting or adding, to give the answer to any question requiring the amount of 3 taken 8 times. If he should learn in a similar manner the several amounts of 10 and each number below 10, taken as many times as there are units in each successive number from 1 to 10, he would learn the common multiplication table as far as ten. If he should now learn how to apply this knowledge to the decimal system of numbers, he would be master of the process of multiplication. Multiplication, then, may be defined as a process by which we ascertain how much any given number will amount to, if taken as many times as there are units in some other given number. The number supposed to be taken is called the multiplicand, the number showing how many times the multiplicand is supposed to be taken is called the multiplier, and the result is called the product. The multiplier and multiplicand are called factors of the product. B. In the previous work, the pupil must have observed that it has made no difference with the product of two numbers, whether he has considered one or the other as the multiplier. Thus, 6 times 4 4 times 6, or 6, fours 5 times 3 = 3 times 5, or 5, threes = 4 sixes. 3 fives. The principle may be proved true for all numbers, by the following arrangement of dots. Considering the dots as being arranged in horizontal rows, there are 3 rows with 5 dots in each row; considering them as being arranged in vertical rows, there are 5 rows with 3 dots in each row; and reckoning in either way we include all the dots. Now, if these rows were extended in either direction, always being kept equal to each other, it is evident that the number of rows reckoned in one direction would always be equal to the number of dots that would be in a row were the rows reckoned in the other direction, and that all the dots would be reckoned in both instances. The number that represents the multiplicand when the rows are reckoned in one direction, will represent the multiplier when they are reckoned in the other, while the product or number of dots will be unaltered. Hence, it must always be true that it makes no difference with the product which of the two factors is taken for a multiplier, provided the other be taken as the multiplicand. It will generally be most convenient to consider the larger factor as the multiplicand, though not always so. C. When either factor is a large number it will be well to consider its denominations separately, and if we write the results as we obtain them, it will be well to begin with the lowest denomination. Suppose we wish to obtain the product of 75.69 multiplied by 7. The usual method is to write the numbers in some convenient way, as the following: 529.83 = Product. Explanation. 7 times 9 hundredths = 63 hundredths, or 6 tenths and 3 hundredths. Writing the 3 in the hundredths' place, we reserve the 6 tenths to add to the product of the tenths by 7. 7 times 6 tenths 42 tenths, and 6 tenths added 48 tenths 4 units and 8 tenths. Writing the 8 tenths, we reserve the 4 units to add to the product of the units by 7. 7 times 5 units = 35 units, and 4 units added 3 tens and 9 units. Writing the 9 units, we reserve the 3 tens 39 units to add to the product of the tens by 7. 7 times 7 tens = 49 tens, and 3 tens added 52 tens, which, being our last product, we write. The result, then, is 529.83. For further illustration take Example 2. What is the product of 3£, 9s. 7d. multiplied by 8? Explanation. Beginning with the lowest denomination, we have 8 times 7d. = 56d. = 4s. 8d. Writing the 8d. and reserving the 4s. to add to the shillings of the next product, we have 8 times 9 shillings 72 shillings, and 4 shillings from the former product added, are 76 shillings 3£, 16s. Writing the 16s. and reserving the 3£ to add with the pounds of the next product, we have 8 times 3£ 24£, and 3£ added = 27£, which, being the last product, we write. = D. We can test the truth of our result, by writing out by itself the product of the multiplication of each denomination, beginning either at the left or right, and afterwards adding these products together. The sum should equal the former product. Below is the written work of examples 1 and 2, as proved by beginning at the left and writing each denomination of the product separately. Another method of proof would be to consider the multiplicand as the multiplier, and see if we can get the same result as before. The figures being in this way presented in a different order, we shall not be liable to repeat any mistake we may have made in our first work. We can either reduce and add our partial products as we did in the first method of performing the example, or write them out in full as in the second. 75.69 7. 529.83 Another method of proof is, after having written out the work as at first performed, to begin at the left hand, thus; 7 times 7 tens are 49 tens; but as there are 52 tens in our product, 3 tens must have come from the product of the lower denominations. 3 tens - : 30 units, and adding to this the 9 units written in the units' place, we find there ought to be 39 units in the product. 7 times 5 units are only 35 units, hence, if our work be right, 4 units must have come from the product of the lower denominations. 4 units =- 40 tenths, and adding to this the 8 tenths written in the tenths' place of our product, we find that there ought to be 48 tenths in the product. 7 times 6 tenths are only 42 tenths, hence, if our product be right, 6 tenths must have gone from the product of the hundredths. 6 tenths = 60 hundredths, and adding to this the 3 hundredths written in the hundredths' place, we find that there ought to be 63 hundredths in the product, and as 7 times 9 hundredths are 63 hundredths, we may infer that our work is right. 24.£. 8 times 3£= But as in the written product there are 27£, 3£ must have come from the lower denominations. 3£: 60s., and adding to this the 16s. written in the shillings of the product, we find that there ought to be 76 shillings in the produet. 8 times 9s. = 72s.; therefore, if the work be right, 4 shillings must have come from the lower denominations. 4s. = 48d., and adding to this the 8d. already written, we find there ought to be 56d. in the product, and as 8 times 7d. = 56d. we infer that the work is correct. = If the pupil should find by multiplying numbers in one method a result different from that he has obtained by multiplying the same numbers in some other method, he may be sure that there is an error in one operation or the other, and he should examine his work patiently till he finds it. No person who is willing to allow an error to pass undetected can be a good arithmetician. E. 1. What is the product of 84687 X 4? 10. What will 7 acres of land cost at $94.839 per acre? 11. 1 pound Avoirdupois of distilled water contains 27.7015 cubic inches. How many cubic inches will 8 pounds contain? 12. 1 pound Troy of distilled water contains 22.794377 cubic inches. How many cubic inches will 8 pounds contain? 13. How many cubic inches are there in 7 cubic feet? in 8? in 9? in 6? 14. Bought the following articles, viz.: 7 yd. of cloth at $2.875 per yard, 4 yd. of silk at $1.725 per yard, 6 shawls at $7.628 apiece, and 8 pairs kid gloves at $1.1875 per pair. How much did they all cost? 15. What is the product of 22£, 18s. 8d. 2qr. by 7? 7 times 18s. = 7 times 22£ = = 4s. 11d. Write 11d. 126s., 4s. 130s. 6£, 10s. Write 10s. The product is, therefore, 160£, 10s. 11d. 2qr. NOTE. When any denomination to be multiplied is very near a unit of the next higher, the work may frequently be much shortened by considering it a unit of that higher denomination, and subtracting for its deficiency in value. For instance, in the example above, since |