plied by .2, the product will be 1 tenth as large, or .8. If it be multiplied by .02, the product will be 1 hundredth as large, or .08. If multiplied by .002, the product will be .008. Again, if .1 be multiplied by 1, the product will be .1; if multiplied by .1, the product will be 1 tenth as large, or .01. If multiplied by .01 it will be 1 tenth as large as when multiplied by .1, or 001. Again, .01 multiplied by 1 is .01; if multiplied by .1, it is 1 tenth as large, or .001. All these multiplications may be thus expressed: Multiplicands, 4 4 4 4 .1 Multipliers, 2 .2 .02 .002 .1 .1 .01 1 .1 .01 .01 1 .1 .1 .01 .001 .01 .001 RULE FOR MULTIPLICATION OF DECIMALS. Multiply as in whole numbers, and point off in the product as many places for decimals as there are decimal places in both the factors. If the product does not contain so many figures, prefix as many naughts as are necessary to make the required number. 11. How many are 307 .5? 8065 ×.08? (12.) 910.8 X6? 50.471 × .09? (13.) 3.0169 × .007? (14.) Multiply 350.006 by .009; 860.1049 by .0004; .8401 by .007. 28. A number that is composed of two or more factors, as 14=2 × 7; or 30=2 × 3 × 5, is called a Composite number. A Prime number is one that has no factors except itself or unity; as, 1, 2, 3, 5, 7, 11. 1. Write all the prime numbers from 1 to 100. 2. What are the factors of 4? 6? 8? 9? 10? 12? 14? 16? 18? 20? 21? 22? 24? 25? 26? 27? 28? 30? 32? 36? 42? 45? 48? 54? 56? 64? 72? 100? RULE. TO MULTIPLY BY A COMPOSITE NUMBER. When the multiplier is a composite number larger than 12, first multiply by one of its factors, and then that product by another, and so on till all have been used. If 1 bushel of corn is worth 65 cents, how much are 28 bushels worth? As 7 times 4 are 28, we may find the price of 4 bushels, and then of 28 bushels, which is 7 times the price of 4 bushels. $0.65 4 Price of 4 bushels, $2.60 7 3* Price of 28 bushels, $18.20 In the same manner multiply the following: (3.) 19 × 21; 64.7 × 36. (4.) (5.) 674 by 72; 8.041 by 96. 4? 6. How many are 10 times 4? 6.17 × 42; 3016.8 × 45. 100 times 4? 1000 times 7. How many are 10 times .4? 100 times .04? .004 X 1000? RULE. When the multiplier is 10, 100, 1000, &c., the multiplication is performed by simply annexing the naughts to the right of the multiplicand; or, in decimals, by removing the decimal point in the multiplicand as many places to the right as there are naughts in the multiplier. The reason for this is, that by removing the decimal point one place to the right, tenths become units, units become tens, tens become hundreds, &c. So that the value of each figure is increased tenfold. Thus, 10 times 354 are 3540; 100 times 516 are 51600; 31.85 X 10=318.5; 418.06 X 100=41806; 5180.46 X 10000=51804600. (8.) How much is 840 × 10? 7916 × 100? (9.) 8451 X 10000? 31.04 X 10? (10.) 40.168 × 100? 31.6008 X 1000? (11.) 308.09 × 1000? $49.75 × 1000? 12. How many are 817 X 20? Multiply first by 2, and then that product by 10. See rule for multiplying by composite numbers. 817 20 16340 13. Multiply 84.17 by 30. Solution. 3 times 84.17 are 252.51, and 10 times 252.51 are 2525.1. 14. Multiply 3618 by 400. 100. (15.) Multiply 381679 by 50000. Multiply first by 4, then by 700; 37019 by 9000; 45 by 16. Multiply 801.7 by 4000. 4 times 801.7 are 3206.8, and by removing the decimal point three places to the right, 1000 times 3206.8 are 3206800. 510.78 X× 9000? 17. How much is 875.4 X 400? tiply the 375 by 8, and annex the naughts afterwards. many are 7 times 156000? Mul How RULE. If there are naughts at the right of either of the factors, they may be omitted in the multiplication, and annexed afterwards. 29. Multiply 1875 by 5037. In this example the multiplier is 5000+30 +7. We may therefore multiply by each of these numbers and add their products together; thus. But the work is more easily performed in this manner. 1875 5037 It will be seen that this is the same as the other, with the omission of the naughts at the right of the 2d and 3d products. In multiplying by the tens the first figure of the product is written in the tens' place, and in multiplying by the thousands the first figure is written in the thousands' place; because the product of units by tens is tens, and the product of units by thousands is thousands. RULE. When there are more than one significant figure in the mul tiplier, multiply by each figure separately, writing the first figure of each product-under the figure by which you are multiplying. The sum of the several products will be the product required. If there are decimals in either of the factors, point off the decimals in the product as before directed. (37.) NOTE. If naughts occur in the multiplier between the other figures, pass them over, and multiply by the next significant figure; placing the first figure of the product as the rule directs. PROOF OF MULTIPLICATION. Make the multiplicand the multiplier, and the multiplier the multiplicand, and repeat the operation. The product should be the same as before. QUESTIONS. What is multiplication? What do we find by it? What terms are used in multiplication? What is the multiplicand? The multiplier? The product? What are called factors of the product? Give an example. What is the sign of multiplication? Give an example. What is the rule for multiplication when the multiplier has but one significant figure? Repeat the rule for the multiplication of decimals. What is a composite number? A prime number? How may multiplication be performed when the multiplier is a composite number? Give an example. How may you multiply by 10, 100, 1000, &c.? What is the reason for this? How may you perform the multiplication when there are naughts at the right of either factor? What is the rule for multiplication when there are more than one significant figure in the multiplier? What is to be done if naughts occur in the multiplier between other figures? How may multiplication be proved? Miscellaneous Exercises in Addition, Subtraction, and Multiplication. 30. 1. What is the sum of 27+5+7+9+3+5+8+7 +12? 2. Add 15+7+8+9+7+6+4+8+15. 125+7+8 10+7+9+3+5. 3. How much is 45-23? 386 143? 59-36? 84-13? 168-87? 4. How much is 35 — 17? (24.) 38—19? 44 — 27? 53—25 ? 65-46? 117-38? 257- - 38? 324-138? 647-359? 845-257? 705-187? 9? 917-358? 5. How many are 36 X 4? 48 X5? 57X6? 75 X 8? 95 X 178 X 4? 6. Repeat the table, Art. 49. 7. How many shillings in 1 pound? £3 15s.? 8. How many pence in 1 shilling? 5s.? 4s. 8d.? 5s. 9d.? 7s. 8d.? £1 3s.? £1 3s. 8d.? £2 3s. 6d. ? 10s. 4d.? 9. How many farthings in 1 penny? 8d.? 6d. 1qr. 1s. 3d.? 1s. 3d. 2qr.? 2s. 2d. 2qr.? 8d. 3qr.? £1 12s. 4d. 2qr.? 10. Repeat the table, Art. 51. 1d. 2qr.? 4d. 3qr.? 3s. 5d. 2qr.? 7s. 11. How many ounces in 1 lb. Troy? 5lb.? 2 lb. 3 oz.? 8 oz. 8 lb. 10 oz.? 7 lb. 12. How many pennyweights in 1 oz.? 8 oz.? 5 oz. 3 dwt.? 9 oz. 10 dwt.? 13. How many grains in 1 dwt.? 4 dwt.? 3 dwt. 5 gr.? 8 dwt. • 12 gr.? 14. Repeat the table, Art. 53. 15. How many quarters in 3 cwt.? 3 cwt. 3 qr.? 5 cwt. 1 qr.? 15 cwt. 3 qr.? 16. How many pounds in 1 qr.? 1 qr. 15 lb. ? 2 qr. 23 lb. ? 1 cwt. 3 qr. 15 lb.? 3 qr. 17 lb. ? 17. How many ounces avoirdupois in 1 lb. 2 lb. 3 lb. 8 oz.? 5 lb. 7 oz.? 3 lb. 14 oz. ? 18. Repeat the table, Art. 54. 19. How many furlongs in 1 mile? 3 m.? 2 fur.? 15 m. 3 fur.? 20. How many rods in 1 furlong? 3 fur. 15 rd.? 2 fur.? 21. How many feet in 1 yd.? 3 yd.? 8 yd. 2 ft.? 22. How many inches in 1 foot? 3 ft.? 2 ft. 3 in.? 8 ft. 5 in.? 5 m. 7 fur.? 1 mile? 23. How many inches in 1 yd.? In 1 rd.? 1 fur.? league? 3 miles? 24. How many feet in 1 rd.? 1 m. 5yd. 3 ft.? 5 ft. 7 in.? 1 m. ? 1 5 rd.? 84 rd.? 25 rd.? In 1 25. How many yards in 8 rd.? In 9 rd.? In 1 fur.? In 1 mile? 3 miles? 26. How many drams in 1 lb.? In 3 lb.? In 71⁄2 lb.? In 1 qr.? In 1 cwt.? Exercises in Addition, Multiplication, and Division. 31. 1. What will 54 bushels of wheat come to at $1.12 a bushel ? 2. In an orchard there are 27 rows of trees, and 15 trees in each row. How many trees are there in the orchard? 3. Suppose each tree to yield 3 barrels of apples, how many barrels are there in the orchard, and what will they come to at $1.625 a barrel ? 4. The gathering and marketing of the above apples cost 17 cents a barrel. What was the net * value of the apples? 5. A man bought 28 acres of wood land at $27.25 per acre. How much did it come to? How many cords of wood were there on the land, supposing it to yield 37.5 cords per acre? 6. What would the wood come to at $1.875 a cord, after paying 45 cents a cord for cutting? 7. A man sold 75 bushels of wheat at 95 cents per bushel, and received in exchange 75 gallons of molasses at 28 cents a gallon; 48 pounds of sugar at 10 cents a pound; 3 bushels of grass seed at $2.125 a bushel; 150 pounds of salt fish at 4 cents a pound; 5 yards of broadcloth at $3.25 per yard, and the remainder in money. How much money did he receive? * Net value means the value after deducting all expenses. |