MULTIPLICATION. 52. 1. There are 7 days in 1 week; how many days in 4 weeks? This example may be solved by addition; thus, 7+7+7 +7=28; or, more briefly, by multiplication; thus, 4 times 7 are 28, Ans. 53 Multiplication is the process of finding how many units there are in any number of times a given number. 54. The Multiplicand is the number to be repeated. The Multiplier is the number which shows how many times the multiplicand is to be taken. The Product is the result of the multiplication. The Multiplicand and Multiplier are called Factors. 55. The sign of multiplication, ×, signifies that the two numbers between which it stands are to be multiplied together; thus, 6 x 530, that is, six multiplied by five equals thirty; or, five times six are thirty. A Parenthesis, (), or Vinculum,, indicates that all the quantities included are to be considered as a single quantity. Thus, 27 (84) x 2, or 27 8 + 4 × 2, means that two times the sum of 8 and 4, or 24, is to be subtracted from 27. The effect of a sign of multiplication (or division) does not pass over the signs + or Thus, 184 × 3 = 18 12. It is not 22 × 3; but (18 + 4) × 3 = 22 × 3. 56. The multiplier is always an abstract number. The product is of the same kind as the multiplicand. In the example in Art. 52 the multiplier is 4, not 4 weeks. We cannot take 7 days 4 weeks times; but we take 7 days 4 times, that is, as many times as there are units in the number of weeks, and the answer is days, 4 times 7 days = 28 days. 57. The pupil before advancing further must learn the following 58. Oral Exercises. 2. How many are 8 times 3? 3 times 8? 6 times 4? 4 times 6? 7 times 7 5 times 9? 3. How many are 9 times 7? 9 times 11? 8 times 6; 6 times 12 12 times 6? 9 times 8? 4. If I deposit $10 a month in a savings bank, how many dollars shall I deposit in 4 months? In 7 months? In 5 months? In 12 months? 5. When wood is worth $6 a cord, what shall I pay for 3 cords? 5 cords? 8 cords? 11 cords? 6. In one year there are 12 months, how many months in 2 years? 4 years? 7 years? 12 years? 7. If I study 11 hours in a day, how many hours shall I study in 3 days? 5 days? 8 days? 11 days? 8. What are 7 tons of coal worth at $6 a ton? 9. If salt beef is worth 12 cents a pound, what must I 9 pounds? 10. What will 7 melons cost at 8 cents each? pay for 11. At 9 cents a yard, what will 6 yards of ribbon cost? 12. At $7 a cord, what will 11 cords of wood cost? 13. If flour is worth $9 a barrel, what must I pay for 12 barrels ? 59. Exercises for Written Work. 14. In one bushel there are 32 quarts; how many quarts in 6 bushels? 32 32 Sum, 192 In 6 bushels there are 6 times as many quarts as in 1 bushel, and the number of quarts in 6 bushels may be obtained by adding, as in the margin; or, more briefly, by multiplying; thus, 6 1 ten and 2 units; write times 2 units are 12 units = the 2 units in the units' place, and then say 6 times 3 tens are 18 tens, which, increased by the 1 ten previously obtained, make 19 tens 1 hundred and 9 tens, and these, written in the place of hundreds and tens respectively, give the true product. Multiplicand, 32 192 128 First multiply by 6, as though 6 were the only figure in the multiplier; then multiply by 4, and place the first figure of this product in the place of tens; for, multiplying by the 4 tens is the same as multiplying by 40, and 40 times 2 units are 80 units: = 8 tens; that is, the product of units by tens is tens. Having multiplied by each figure in the multiplier, find the sum of the partial products and this will be the true product. Product, 1472 NOTE. So much of the product as is obtained by multiplying the whole multiplicand. by one figure of the multiplier is called a partial product; thus, in the 25th example, 192 is the first partial product and 128 tens is the second. 60. Multiplication with Decimals in the Multiplicand. 26. Multiply 783.24 by 7. 783.24 7 Ans. 5482.68 = 2 7 times 4 hundredths are 28 hundredths tenths and 8 hundredths; we write the 8 hundredths below the 7; then 7 times 2 tenths are 14 tenths, and the 2 tenths from the 28 hundredths added to it give 16 tenths 1 unit and 6 tenths; we write the 6 tenths in its place and add the 1 unit to the product of 7 and 3 units, and so on, exactly as in Ex. 25, remembering also to place the decimal point between the units and tenths. This explanation shows that there will be the same number of decimal places in the product as in the multiplicand. 27. Multiply 18.783 by 75. 18.783 75 93915 131481 Ans. 1408.725 61. From the preceding examples we derive for multiplication the following Rule. 1. Write the multiplier under the multiplicand and draw a line beneath. 2. Beginning at the right hand of the multiplicand, multiply the multiplicand by each figure in the multiplier, writing the first figure of each partial product directly under the figure of the multiplier which produces it. 3. Add these partial products and point off as many decimal places in the product as there are in the multiplicand, and the result will be the true product. 62. PROOF. Multiply the multiplier by the multiplicand, and, if correct, the product will be the same as the first product. NOTE 1. This proof rests on the principle, that the order of the factors is immaterial; thus, 3 × 4 = 4 × 3; 5 × 2 × 7 = 2 × 7 × 5. NOTE 2. For multiplication with decimals in the multiplier, see Art. 161. |