The adding of the left hand figure in the product of any figure, to the next product, is carrying one for every ten, as in Addition. In the first example, we might have begun by saying 3 times 0 is 0, and then said 3 times 7 is 21, &c. But this amounts to the same thing as merely placing a 0) on the right, and beginning the multiplication with 3 times 7. In the second example, 243 multiplied by 3 produces 729. But since the true multiplier is 30, which is 10 times 3, the true product must be 10 times 729; and this tenfold value is assigned to 729 by the 0 on the right, which removes each of its figures one place farther toward the left from units (16). We thus find that for each 0 omitted in the right of either factor, a 0 must be placed in the right of the product. The Operation Proved. § 44. Multiplication may be verified or proved, by multiplying the multiplicand by the multiplier minus 1, and adding the multiplicand to the product thus obtained; the sum must be equal to the product found with the entire multiplier. EXERCISES. 1. Mary bought two books, at 31 cents apiece. How many cents did she pay for them? The 2 books cost twice 31 cents; or 31 cents+31 cents (§ 38). Ans. 62 cents. 2. A farmer sold 3 horses at 125 dollars apiece. What sum did he receive for them? Ans. 375 dollars. 3. What would be the value of 20 shares of road stock, at 95 dollars for each share? 4. What would be the weight of 30 ing 450 pounds to each bale? 5. How many pounds of flour are there there being 196 pounds in each barrel? 6. If a steamboat can run 305 miles miles could it run in 4 days? Ans. 1900 dollars. bales of cotton, allowAns. 13500 pounds. in 10 barrels of flour, Ans. 1960 pounds. in a day, how many Ans. 1220 miles. 7. There being 1760 yards in a mile, what number of yards is there in 5 miles? Ans. 8800 yards. 8. A farmer sold 100 acres of land, at 43 dollars per acre, What sum did he receive for the land? Ans. 4300 dollars. 9. There being 660 feet in a furlong, how many feet are there in a mile, which is 8 furlongs ? Ans. 5280 feet. 10. A merchant bought 60 sacks of coffee, at 13 dollars a sack. What did the whole amount to? Ans. 780 dollars. 11. A butcher bought 7 fat oxen, at an average of 32 dollars a head. What sum did he pay for them? Ans. 224 dollars. 12. Find the sum that should be paid for 9 horses at 130 dollars apiece, and 10 cows at 21 dollars apiece. Ans. 1380 dollars. 13. Bought at one time 5 coils of rope, each containing 139 yards; and at another, 7 coils, each containing 150 yards. What was the whole number of yards bought? Ans. 1745 yards. 14. A farmer bought 5 horses, at 95 dollars each; 12 mules, at 60 dollars each; and 40 head of cattle, at 27 dollars a head. What did he pay for all the stock purchased? Ans. 2275 dollars. 15. Sold to A 7 pieces of cotton, to B 11 pieces, and to C 12 pieces, each piece containing 31 yards. What was the number of pieces, and what the number of yards that was sold? Ans. 30 pieces; and 930 yards. 16. A merchant bought 213 beaver hats, at 4 dollars apiece. What did the whole amount to? The hats amounted to 213 times 4 dollars; but 4 times 213 is the same number as 213 times 4. (§ 39). Ans. 852 dollars. 17. Bought 175 cords of wood, at 3 dollars per cord. What did the whole amount to? Ans. 525 dollars. 18. A planter sold 325 bales of cotton, at the average sum of 30 dollars per bale. What did the sale amount to? Ans. 9750 dollars. 19. If a steam ship can run at the rate of 300 miles per day, how far would she run in 31 days. Ans. 9300 miles. 20. A drover sold 573 head of cattle, at the average price of 20 dollars a head. What did the sale amount to? Ans. 11460 dollars. 21. A farmer filled, at one time, 83 sacks of corn; and at another time 112 sacks;-each sack containing 5 bushels. What quantity of corn was put into all the sacks? it? Ans. 975 bushels. 22. Bought of A, 137 acres of land; of B, 89 acres; and of C, 384 acres;-at the average price of 40 dollars per acre. What quantity of land was bought, and what sum was paid for Ans. 610 acres; and 24400 dollars. 23. Shipped to New Orleans, at one time, 120 barrels of apples; at another, 75 barrels; at another, 100 barrels; and at another, 9 barrels;-each barrel containing 3 bushels. Required the number of barrels, and the number of bushels shipped. Ans. 304 barrels; and 912 bushels. 24. A merchant bought 275 yards of cloth, at 6 dollars per yard, of which he has sold 133 yards, at 10 dollars per yard. What would he gain on the whole, by selling the remainder at 9 dollars per yard? Ans. 958 dollars. 25. If 10 masons can build a wall, in 34 days; in how many days ought one mason to build the same wall! It would take 1 mason 10 times as long as it would 10 masons, to build the wall; that is, 10 times 34 days. Ans. 340 days. 26. How long ought one man to subsist on a stock of provisions which would support 7 men for 29 days! Ans. 203 days. 27. If 9 men could mow a certain meadow, in 19 days; in how many days ought one man to mow the meadow ? Ans. 171 days. 28. If 115 bushels of oats will feed one horse for 3 months; what quantity of oats would feed 12 horses, the same time? Ans. 1380 bushels. 29. Allowing 20 pieces of artillery to demolish a fortress in 48 hours, in what time ought one piece to demolish the fortress? Ans. 960 hours. 30. A lent B 8 yoke of working oxen, for 13 days. How long ought B to lend A 1 yoke, to requite the favor? Ans. 104 days. 31. C exchanges silk, at 1 dollar per yard, with D, for broadcloth, at 7 dollars per yard. How many yards of silk should be given for 125 yards of broadcloth ? Ans. 875 yards. 32. Two merchants barter as follows: A gives 75 yards of cloth, at 9 dollars per yard, to B, for 109 beaver hats, at 7 dollars apiece; the deficiency on either side being made good in money. Which of them receives money, and how much? Ans. B receives 88 dollars. 33. A gave B 3 horses, and 30 head of cattle, for 50 barrels of flour and 200 barrels of corn. A sold the flour at 6 dollars, and the corn at 3 dollars a barrel; while B sold his horses at 120 dollars, and his cattle at 25 dollars, each? Which of the two gained by the trade, and how much? Ans. B gained 210 dollars. 34. Find the Product 10X24738730. 35. Find the Product 200X3076801. 36. Find the Product 1200×706840. 37. Find the Product 90X10707800. 38. Find the Product 300×7060000. 39. Find the Product 1100X870003. 40. Find the Product 12000X70000. Ans. 247387300. RULE VI. § 45. To Multiply by any Number exceeding 12, and containing two or more SIGNIFICANT Figures. 1. Multiply by each significant figure, separately, of the multiplier, placing the several rows of products one under another, with the first figure of each product under the multiplying figure, and, in that order, add the several products together for the entire product. 2. Ciphers in the right of either or both of the factors, are omitted in multiplying; but as many Os must be placed in the right of the product. EXAMPLES. 1. To Multiply 8072 by 39; that is, to find 39 times 8072. 8072 72648 24216 314808 Multiplying first by 9, we say 9 times 2 is 18, and set the first product figure 8 under the multiplying figure 9; then, 9 times 7 is 63, and 1 makes 64; 9 times 0 is 0, and 6 makes 6, &c. Multiplying next by 3, we say 3 times 2 is 6, and set the 6 under the multiplying figure 3; 3 times 7 is 21; 3 times 0 is 0, and 2 is 2, &c. Adding the two rows of product figures together, in the order in which they stand, we say 8 is 8; 6 and 4 are 10, &c. The entire Product is 314808. 2. To Multiply 8420 by 30900; or to find 30900 times 8420; 8 420 7578 260178000 Omitting Os in the right of both factors, we multiply 842 by 309, setting the first figures, 8 and 6, of the products under the multiplying figures, 9 and 3, respectively. To the product thus found, we annex three Os, for the Os omitted in multiplying. When the multiplying figure is tens, or hundreds, &c., the first product figure is set under tens, or hundreds, &c., respectively, to increase the product in the same degree in which the multiplying figure is increased in value, by distance from the unit's place (§ 16). In the first example, 2 multiplied by 3 produces 6. But the 3 is 3 tens, or 10 times the simple 3; the 6 must therefore be 10 times the simple 6; and this tenfold value is assigned to 6 by setting it under the 3 tens, and adding it in the ten's column. In the second example, we multiplied 842 by 309. In 309 the 3 is 3 hundreds, or 100 times the simple 3. The first product figure 6 must therefore be 100 times the simple 6; and this hundredfold value is assigned to 6 by setting it under the 3 hundreds, and adding it in the hundreds column. The first product figure being set in its proper place, the second, third, &c., fall in their proper places, in ascending orders of units towards the left. The sum of the partial products is the entire product (§ 23). As already shown, in connexion with Rule V, for each 0 omitted in the right of either factor, a 0 must be placed in the right of the product. 50. The number of yards in a mile being 1760, how many yards are there in 15 miles ? The number of yards in 15 miles is 15 times 1760 yards. 51. There are 24 hours in one day. are there in a year of 365 days? Ans. 26400 yards. How many hours then The number of hours in a year is 365 times 24 hours; but 24 times 365 will produce the same number, and it is more convenient to make the less number the multiplier. Ans. 8760 hours. 52. A hogshead of wine or brandy contains 63 gallons. How many gallons would there be in 250 hogsheads? Ans. 15750 gallons. 53. What sum should be paid for a plantation containing 765 acres, at 43 dollars per acre? Ans. 32895 dollars. |