38. If we wish to multiply one number by another we proceed as follows: 376 1504 For convenience in multiplying, we write the multiplier under the multiplicand. We then multiply, beginning with the units of the first order, as follows: 4 X 6 24. We write down the 4 in the units place, and reserve the 2 to be added to the next product obtained. This next product is 4 X 7 28. To it we add the 2 thus 282 The 0 is set down, and the 3 reserved to be added to the next product, which is 4 × 3 = 12. Adding the 3 we have 12+3 = 15 30. which, being the last product, is written down. We now have the total product or answer which is 1504. This is further illustrated by the following: 376 4 24 28 12 1504 By this we see that we multiply the 6 in the units place, putting the result below. We then multiply by the 7 and the product, being in the tens position, takes one place farther to the left, i. e., directly under the 7 itself. In like manner the 12 (the product of 4 and 3) is placed with its units place under the 3. We now add and obtain the answer 1504. We also see that this process is equivalent to adding 376 four times. 376 376 376 376 1504 39. If the multiplier consists of more than one figure we proceed in the following manner: Let a, b, and c represent respectively the units, tens, and hundreds positions. We first multiply by 4 as we did in the preceding example, and write the result in the units column, that is under a. This is the first partial product. We then obtain the second partial product by multiplying in the same way by 3, placing the result with its units figure under b, that is in the tens position. In like manner we multiply by 2 and write the third partial product in the hundreds column or under c. Adding these partial products we obtain the total product or answer which is 87,282. 40. If we have one 0 or more in an example, the procedure is practically the same. The following examples illustrate such It is customary to contract the work as in the second case. We perceive also by the second case, that (with the last partial product) the real operation is simply multiplying by 700 because we write the result in the hundreds position. Let us now multiply 624 by 700. 624 700 436800 We here perceive that we get the result in multiplying as we did above in the last partial product. Hence, zeros may be disregarded in multiplying, when they stand at the right hand of other figures, but there must be added to the product the same number of zeros as were disregarded. This applies to both multiplicand and multiplier, as is shown by the following examples: 41. Any number multiplied by zero alone equals 0. 42. RULES. (a) Write multiplier under multiplicand, units under units, tens under tens, etc. (b) Each figure of the multiplicand is multiplied by each significant figure of the multiplier, and the right-hand figure of each product is placed under the figure of the multiplier used to obtain it. (c) The sum of the partial products will be the entire product. NOTE. When there is a zero in the multiplier, multiply by the significant figures only, taking care to place the right-hand figure of each partial product under the figure used in obtaining it. Proof. Multiply multiplier by multiplicand. The product should be the same. 43. We must understand that the multiplier is always regarded as an abstract number. The multiplicand and product are like numbers, and may be either concrete or abstract; i.e., if we multiply 5 rivets by 6 we have 30 rivets. We cannot, however, multiply 5 rivets by 6 rivets. (17) There are 746 watts in a horse-power. How many watts are there in 20 horse-powers? Ans. 14,920 watts. (18) A piston of twelve inches diameter has an area approximately of 113 square inches. If the steam pressure is 47 pounds per square inch, what is the total pressure upon it? Ans. 5,311 lbs. (19) The end of a boiler about ten feet in diameter has an area of 11,310 square inches. If the pressure per square inch is 40 pounds, what is the total pressure? Ans. 452,400 lbs. (20) Two steamers are going in the same direction. One makes 15 miles per hour, the other 11 miles per hour. How far apart are they at the end of 7 hours? At the end of 17 hours? Ans. 28 miles after 7 hours; 68 miles after 17 hours. |