CHAPTER VI. MULTIPLICATION. 110. A ST. ANDREW's cross X between two numbers means that one of the numbers is to be repeated as many times as is indicated by the other number. The number to be repeated is called the multiplicand; the number which shows how many times the multiplicand is to be repeated is called the multiplier; and the result is called the product. The sign is read times, or multiplied by, according as the multiplier precedes or follows the multiplicand. Thus, 5×4 cents=20 cents is read, five times four cents equals twenty cents; but, 4 cents x 5=20 cents is read: four cents multiplied by five equals twenty cents. 111. The multiplier and multiplicand are often called factors of the product. The product of two or more factors is the same in whatever order they are taken. Thus, 3 x 4 = 4 x 3. The dots in the margin, read horizontally, make 3 fours; read vertically, make 4 threes. 112. The sign x cannot extend its power, forward or backward, beyond a + or -, without the aid of a parenthesis. To illustrate: 2+3x4-1= 13; (2+3)x4-1= 19; 2+3 × (4 −1) = 11; (2+3) × (4 − 1) = 15. 113. The products, in all cases in which neither factor exceeds ten, should be thoroughly committed to memory. They will be found in the following table: 123 4 5 6 MULTIPLICATION TABLE. 2 3 4 5 6 7 8 4 6 69 9 10 11 12 8 10 12 14 16 18 20 22 24 9 12 15 18 21 24 27 30 33 36 8 12 16 20 24 28 32 36 40 44 48 40 45 50 5 10 15 20 25 30 35 114. In the above table, take the multiplier in the upper line, the multiplicand in the left-hand column; the products will be found directly under the multiplier, and opposite the multiplicand; as, 12 x 7 is 84.* 2.236068 4 115. To multiply any multiplicand by a multiplier less than 13, the work may be written as in the margin. Beginning at the right, 4 × 8 is 32; the 2 is written, and the 3 carried mentally and added to 4 × 6, making 27; and the process is thus continued to the left. 12x8 96; 12 x 672, and 9 makes 81; 12 x 672; 12 x 336, and 7 makes 43; 12 x 224, and 4 makes 28; 12 x 2 = 24, and 2 makes 26. 8.944272 2.236068 12 26.832816 *The table should be learned, not by lines but by squares; that is, first learn 2×2, 2 X 3 and 3 × 2, 3 X 3; next learn 2 × 4, 3 X 4, 4 × 2, 4 × 3, 4X4; thirdly, 2×5, 3×5, 4X 5, 5 × 2, 5X 3, 5 X 4, 5X5; fourthly, all products under 36, etc. The cards referred to in the footnote to 243 may be advantageously used for practice in multiplying two digits. Shuffle them, and pass them in couples from one hand to the other, naming the two factors, while the pupil names the products. 116. Multiply 111 by 5; 123 by 3; 231 by 2; 114 by 3; 421 by 4; 512 by 5; 4328 by 4; 1187 by 6; 1782 by 8; 8.287 by 7; 9.6198 by 3; 62.818 by 7; 9.2758 by 8; 52.134 by 9. 117. Multiply .5235988 by 6; .7853982 by 4; 3.14159265 by 5, and the product by 5. 118. Multiply 3.1416 by 11; by 12; by 10 and by 3, and add the two results; by 10 and by 4, and add the results; by 9 and by 6, and add the results. Multiply 2.236068 by 11; by 6 and by 7, and add the results; by 8 and by 9, and add the results; by 10 and by 7, and add the results (compare the sum of these two products with the sum of the last two products); by 10 and by 8, and add the results; by 12 and by 7, and add the results. 119. To multiply by 10, 100, 1000, etc., it is enough to move the decimal point of the multiplicand* as many places to the right as there are naughts after the one; annexing naughts to the multiplicand, if necessary. Thus, 10 x 15.4323=154.323; 10,000 x 15.43=154,300. 120. How much is 10 times 3.14159265? 100 times? a million times? What will 10 barrels of apples cost, at $3.75 a barrel? at $2.17? at $5.875? How much will 100 barrels cost at each of these prices, and at $3.375? at $5.125? 121. It is plain, that to multiply by .1, .01, .001, etc., it is necessary only to move the point of the multiplicand as many places to the left as there are decimal places in the * When the multiplicand is a whole number, the decimal point is not written, but understood, after the units' figure. multiplier, prefixing naughts to the multiplicand, if neces sary. Thus, .1 x 15.43.1.543; .001 x 15.43.01543. 122. What is a tenth of 2.36? a hundredth of 2.36? a thousandth of .63? Write the second members of the following equations, and then read them: 123. It is evident that 30 times is 3 times as much as 10 times, or 10 times as much as 3 times. Hence (§§ 115, 119), Find the cost of 30 barrels of flour, at $3.27 a barrel; of 70 barrels, at $4.58; of 90 barrels, at $6.76; of 100 barrels, at $7.84; of 120 barrels, at $8.57. 124. It is evident that .03 = 3 × .01. Hence (§ 121), Find the cost of .03 of a barrel of oil, at $27.875 a barrel; of.7; of .009; of .17*; of .019; of .13; of .8; of .83; of .014 of a barrel. It is plain from the above examples that, The decimal places in the product are as many as the sum of the decimal places in the multiplicand and the multiplier. 125. What is the numerical value of the expressions: 30 x 8.75? .07 × 6.975? 700 x 7.81? 8000 × 65.432? 300 x 7.85? .0009 × 10356.78? 126. Ex. 1. Multiply 6957 by 463. 463=400+60+3. If we wished to find the result by addition, it would be necessary to write 463 numbers, each equal to 6957, under each other, and find the sum of all these numbers. (The * Multiply first by .1, then by .07, and add the results. necessity of abridging such an operation has given rise to multiplication.) Or, we might make three additions, the first of 3 numbers, each equal to 6957; the second of 60 numbers, each equal to 6957; the third of 400 numbers, each equal to 6957; and then add together the three sums. In other words, we might multiply 6957, first by 3, then by 60, and then by 400, and find the sum of the three products obtained. Thus, Multiplicand, 6957 463 463 times the multiplicand = 3221091 It is evident that the zeros at the right of the second and third partial products do not affect the result of the addition; we may, then, omit them, if we observe to what place in respect to the next preceding product the righthand figure of each succeeding product belongs. Therefore, multiply by each digit of the multiplier, place the right-hand figure of each product under the multiplier used, and add the partial products. |