65. Examples with Decimals. 34. A person having 205.6 acres of land, sold 10.75 acres. How many acres had he left ? 35. What is the difference between 0.7 and 0.385 ? 36. How many thousandths must you add to 0.485 to make 1. ? (37.) 86.67 -9.8 = ? (40.) 641.34 – 56.345 = ? (38.) 7561.2 - 9.6456 = ? (41.) 101.1 - 90.014 = ? (39.) 961.62 -- 54.645 = ? (42.) 970.2 - 86.37 = ? 66. Miscellaneous Examples. 43. James Fry has in his possession $ 172; he owes $ 28 to A, $36 to B, and $ 19 to C. After paying his debts, what will remain ? 44. In a certain mill 2415 persons were employed, of whom 581 were natives, 1119 were foreigners, and the rest unknown. How many were unknown ? 45. I have $462 in the savings-bank, and $ 2180 in government bonds. How much more must I have that I may purchase a house worth $4700 ? 46. A man gave to his son $ 3575, to his daughter $4680, and to his nephew $ 2495 less than to his daughter. His whole property was worth $30500; what sum remained ? 47. Two persons who are 250 miles apart, travel towards each other, one 36 miles, the other 52 miles a day. How far apart will they be at the end of one day? 48. If the same persons travel away from each other, how far apart will they be at the end of one day ? 49. From 9460 subtract 5466; from the result subtract 1284; to this add 3989, and from this subtract 5987. 50. A man bought a lot of land for $ 1296, and built upon it a house costing $7364. If he sold the property for $ 10000, how much did he make ? SECTION IV. MULTIPLICATION. 67. Unite three 7's into one number. 7 This may be done by adding them together thus : 7, 14, 21. 7 By this process we find that three 7's are 21. In the same way 7 we can find that seven 6's are 42, eight 9's are 72, eight 7's are 56, and in fact all the results which we commit to memory 21 when we learn the Multiplication Table. 68. The process of unitiny two or more equal numbers into one number is multiplication. 69. One of the equal numbers to be united is the multiplicand. 70. The number that tells how many equal numbers are to be united is the multiplier. 71. The result obtained by multiplication is the product. 72. The multiplicand and multiplier are called factors (makers) of the product. In the example "three 7's are 21,” which is the multiplicand ? the multiplier ? the product ? Name two factors of 21. 73. The multiplication of numbers is indicated by the sign x. Thus, the expression 50 x 4 = 200 means that four 50's are 200; and is read“ 50 multiplied by 4 equals 200.” 74. Oral Exercises. Turn to page 58, and multiply the numbers expressed a. In column h by 3. e. In column q by 7. b. In column j by 4. f. In column r by 8. c. In column k by 5. g. In column v by 9. d. In column o by 6. b. In column v by 12. 75. Compare the product of five 4's with that of four 5's: are they equal or unequal ? Compare the products 3x4x6, 4 x 3 x6, and 6x4x3: are they equal or unequal ? From examples like these we learn this general principle : The product of two or more factors is the same, whatever the order in which the factors are taken. To multiply mentally Numbers greater than 10. [At the option of the Teacher.) 76. ILLUSTRATIVE EXAMPLE. At $ 34 each, what will + cows cost ? Solution. - At $34 each, 4 cows will cost 4 times $34. Four 30's are 120, and four 4's are 16, which, added to 120, make 13. Ans. $ 136. i. If 9 men can build a wall in 25 days, how long would it take 1 man to do it? j. How many gallons of water in 5 hogsheads of 67 gallons each? k. At $8 a month, what is the amount of a soldier's pen: sion for 1 year? for 9 years? 1. How many are three 27's ? four 16's ? eight times 84 ? For additional practice, multiply each number expressed in A, page 58, by such numbers from 1 to 12 as the teacher may select. See also oral exercises in multiplication, pages 59 and 63. Examples for the Slate. 77. ILLUSTRATIVE EXAMPLE I. If a steamship goes 258 miles each day, how far does she go in 6 days? Explanation. - If the steamship goes 258 miles each day, in 6 days she will go 6 times 258 miles. Multiplicand, 258 We have then to multiply 258 by 6. Writing the Multiplier, 6 multiplicand and the multiplier as in the margin, Product, 1548 we multiply the units, tens, and hundreds separately, beginning with the units. WRITTEN WORK. Six 8's are 48. The 48 units are equal to 4 tens and 8 units. We write 8 under the line in the units' place, and carry 4 tens to the product of tens. Six 5's are 30. The 30 tens with the 4 tens carried are 34 tens, or 3 hundreds and 4 tens. We write 4 under the line in the tens' place, and carry 3 hundreds to the product of hundreds. Six 2's are 12. The 12 hundreds with the 3 hundreds carried are 15 hundreds, or 1 thousand and 5 hundreds. We write, under the line, 5 in the hundreds' place and 1 in the thousands' place. The entire product is 1548. Ans. 1548 miles. For the sake of rapid working, use as few words as possible. Thus, in the example above say “forty-eight; thirty, thirtyfour; twelve, fifteen": while saying “forty-eight," write 8; while saying “thirty-four," write 4; and while saying "fifteen," write 5 and 1. 1. How many pounds of flour are there in 5 barrels, each containing 196 pounds ? 2. How many pounds of cheese are there in 6 cheeses of 172 pounds each ? 3. If a person earns $313 every year for 7 years, how many dollars does he earn ? 4. What will 9 pianos cost at $475 each ? 5. From Chicago to Peoria is 160 miles; how far does a man travel who goes from Chicago to Peoria and back 8 times ? 6. If a person by working 11 hours a day can do a piece of work in 37 days, how many days will it take him if he works 1 hour a day? 7. There are 5280 feet in a mile. How many feet long is a telegraph-wire that connects Boston with Reading, 12 miles distant? 78. ILLUSTRATIVE EXAMPLE II. If 1 barrel of flour costs $ 8, what will 427 barrels cost? Solution. — If i barrel of flour costs $ 8, 427 barrels will cost 427 times $ 8. But 427 times $8 is the same as 8 times $ 427 (Art. 75), which is $ 3416. Ans. $ 3416. 8. What will 732 quarts of milk cost at 7 cents a quart? 9. What must I pay for 324 sheep at $ 9 apiece ? 10. When coal is $6 a ton, what must I pay for 476 tons ? 11. At 4 cents a mile, what must I pay for riding 1289 miles ? 12. If 294 persons gave $8 apiece for a charitable object, how much did all give ? 13. What must I pay for 626 car-fares at 5 cents apiece, and for 87 car-fares at 9 cents apiece ? 14. Multiply 267 by 2; by 3; by 4; and add the products. 15. Multiply 628 by 5; by 6; by 7; and add the products. 16. Multiply 3401 by 8; by 9; and add the products. 17. Multiply 90021 by 10; by 11; and add the products. 18. Multiply 66285 by 12; by 8; and add the products. 19. Multiply 89079 by 7; by 12; and add the products. For additional examples in multiplication by one term only, see pages 59 and 63. WRITTEN WORK. 79. ILLUSTRATIVE EXAMPLE III. Multiply 12 by 10; 12 by 100; 12 by 1000. Explanation. - 10 twelves equal 12 12 12 12 tens (Art. 75), or 120 ; 100 twelves 10 100 1000 equal 12 hundreds, or 1200 ; and 1000 twelves equal 12 thousands, or 12000. 120 1200 12000 In multiplying by 10, 100, 1000, etc., the written work may be omitted, and the product immediately found by annexing to the multiplicand as many zeros as there are in the multiplier. 20. What will 10 bushels of potatoes cost at 65 cents a bushel ? 21. At $100 a share, what will 100 shares in a whip com pany cost ? 22. Multiply $ 75 by 10; by 100; and add the products. |