26. If a regiment of soldiers consists of 1128 men, how many men are there in an army of 53 regiments ? Ans. 59784. 27. What is the product of 75432 X 47. Ans. 3545304. 28. What is the product of 76785316 X 7615. Ans. 584720181340. 29. What is the product of 67853 X 8765. Ans. 594731545. 30. What is the product of 3812345 x 31243. Ans. 119109094835. 31. What is the product of 40670007 X 10002. Ans. 406781410014. 32. What is the product of 31235678 X 10203. Ans. 318697622634. 33. What is the product of 76786321 X 3007. Ans. 230896467247. 34. What is the product of 6176777 X 22222. Ans. 137260338494. 35. What is the product of 7060504 x 30204. Ans. 213255462816. 36. Multiply 88888 by 4444. Ans. 395018272. 37. Multiply 7008005 by 10008. Ans. 70136114040. 38. Multiply 987648 by 481007. Ans. 475065601536. 39. Multiply 101010101 by 202020202. Ans. 20406081008060402. 40. Multiply 304050607 by 3011101. Ans. 915527086788307, 41. Multiply 908007004 by 500123. Ans. 454115186861492. 42. Multiply 2003007001 by 6007023. Ans. 12032109124168023. 43. Multiply 9000006 by 9000006. Ans. 81000108000036. 44. A full-grown elm will, it is computed, yearly, on an average, produce three hundred twenty-nine thousand three hundred and seventy-five seeds. How many seeds will three such trees produce in fifty-three years. Ans. 52370625. 45. John Alden can plant 3 plats of corn, containing 11 rows of 67 hills each, in 1 day, and Loring Blanchard can plant twice as much in the same time. How many hills can Blanchard plant in a month of 26 working days ? Ans. 114972. 46. If the multiplicand be three hundred and seventy-five millions two hundred and ninety-six thousand three hundred and twenty-ene, and the multiplier seventy-nine thousand and twenty-four, what will be the product ? Ans. 29657416470704. 64. When the multiplier is a composite number. Ex. 1. What cost 35 acres of land at 316 dollars an acre ? Ans. 11060 dollars. OPERATION. The factors of 35 are 31 6 dollars, cost of 1 acre. 7 and 5. Now, if we 7 multiply the price of one 2 2 1 2 dollars, cost of 7 acres. acre by 7, we get the cost of 7 acres; and, then, by 5 multiplying the cost of 7 110 60 dollars, cost of 35 acres. acres by the factor 5, it is evident, we obtain the cost of 5 times 7 acres, or 35 acres. Hence, when the multiplier is a composite number, we may Multiply the multiplicand by one of the factors of the multiplier, and the product thus obtained multiply by another, and so on until each of the factors has been used as a multiplier ; and the last product will be the one sought. EXAMPLES. 2. Multiply 3121 by 81, using its factors. 4. A certain house contains 87 windows, and each window has 32 panes of glass. How many panes in the whole house? Ans. 2784. 5. What is the product of 47134987 by 56 ? Ans. 2639559272. 6. If a garrison consume 6231 pounds of bread in 1 day, liow many pounds will the same consume in 144 days ? Ans. 897264 pounds. 65. When there are ciphers on the right in the multiplier or multiplicand, or both. OPERATION. Ex. 1. In 1 yard there are 36 inches; how many inches in 10 yards? In 100 yards? Ans. 360, 3600. We annex one cipher to the Multiplicand 3 6 36 multiplicand to multiply it by Multiplier 10 100 10, and two ciphers to multiply it by 100; since annexing one Product 3 60 3 6 0 0 cipher removes each figure of Or thus : 3 6 0, 3 6 0 0. the multiplicand one place to the left, and thus increases it 10 times; annexing two ciphers removes each figure two places to the left, and increases it 100 tiines ; and so on, each additional cipher having the effect to increase its value 10 times (Art. 30). 2. What will 700 bales of cotton cost at 40 dollars per bale? Ans. 28000 dollars. OPERATION. The multiplicand we resolve into the Multiplicand 700 factors 7 and 100, and the multiplier Multiplier 40 into the factors 4 and 10. Now, it is evident (Art. 58), that, if these several Product 28 0 0 0 factors be multiplied together, they will produce the same product as the original factors, 700 and 40. Thus 7 X 4 = 28, and 28 X 100 2800, and 2800 X 10 28000, the same result as in the operation. Hence, when there are ciphers, one or more, on the right of the multiplier, or multiplicand, or both, we may, for the required product, Multiply the significant figures together, and to their product'annex as many ciphers as there are on the right in both multiplicand and multiplier. EXAMPLES 3. Multiply 1819 by 10. Ans. 70000000. 6. Multiply 123000 by 78000. Ans. 9594000000. 7. Multiply 70000 by 10000. Ans. 700000000. 8. Multiply 900900 by 70070. Ans. 63126063000. 9. What must be the distance sailed by a steamship, whose average rate is 310 miles a day, in making a voyage from New York to Liverpool, in 12 days? 10. The annual salary of a member of Congress being 3,000 dollars, how much do 296 members receive ? Ans. 888,000 dollars. 11. The salary of the President of the United States is 25,000 dollars a year; how much will it amount to in 82 years? Ans. 2,050,000 dollars. 12. The earth is 95,000,000 of miles from the sun, and the planet Neptune is 30 times as far. How far is Neptune from the Sun ? Ans. 2,850,000,000 miles. DIVISION. 66. Division is the process of finding how many times one number is contained in another; or the process of separating a number into a proposed number of equal parts. In division there are three principal terms: the Dividend, the Divisor, and the Quotient. The dividend is the number to be divided. The quotient is the result, or number produced by the division, and denotes the number of times the divisor is contained in the dividend, or one of the equal parts into which the dividend is divided. When the dividend does not contain the divisor an exact number of times, the excess is called a remainder, and may be regarded as a fourth term in the division. When the dividend consists of a simple number, the process is termed Division of Simple Numbers. 67. Division is frequently indicated by writing the dividend above a short horizontal line and the divisor below; thus, . The expression = 3 is read, 6 divided by 2 is equal to 3. Another method of indicating division, is by a curved line placed between the divisor and dividend. Thus, the expression 6) 12 shows that 12 is to be divided by 6. When a number is divided into two equal parts, one of the parts is called one half ; when divided into three equal parts, one of the parts is called one third, two of the parts two thirds ; when divided into four equal parts, one of the 68. parts is called one fourth, two of the parts two fourths, three of the parts three fourths ; etc. Such equal parts are called FRACTIONS, since they are fractured or broken numbers. They are expressed by figures, in a form of division; thus, one half is written }; one third, }; two thirds, }; one fourth, £; two fourths, is ; three fourths, j; and may also be read, one divided by two, one divided by three, and so on. In any fraction, expressed in the manner now explained, the number above the line is called the numerator, and that below the line its denominator. Thus, in 1, 1 is the numerator, and 2 the denominator. 69. When the divisor and dividend are of the same kind or denomination, the quotient will denote the number of times the divisor is contained in the dividend, and will be an abstract number. Thus, to find how many pencils at 6 cents each can be bought for 24 cents, we inquire how many times 6 cents are contained in 24 cents, which are 4 times. Hence, 4 pencils, at 6 cents each, can be bought for 24 cents. 70. When the divisor and dividend are not of the same kind or denomination, the divisor must be regarded as an abstract number, and will denote the number of equal parts into which the dividend is divided, and the quotient will denote the number of units in each part, and will be of the same kind or denomination as the dividend. Thus, to find the cost of 1 pencil, when 4 pencils cost 24 cents, we separate or divide the 24 cents into 4 equal parts, of which one part is 6 cents. Hence, 1 pencil costs 6 cents, when 4 pencils cost 24 cents. 71. The remainder will always be of the same kind or denomination as the dividend, since it is a part of the dividend. 72. Division is the reverse of multiplication. The dividend answers to the product, and the divisor and quotient to the factors, of multiplication. In multiplication, two factors are given, to find their product; and in division, one of two factors and their product are given, to find the other factor. 73. To divide simple numbers. Ex. 1. How many yards of cloth, at 4 dollars a yard, can be bought for 948 dollars ? Ans. 237 yards. |