· 5. A gentleman by his will divided an estate of $40,750 as follows: bis son to have $10,000; each of his three daughters $5,475, and the widow the remainder; what did the widow receive? Ans. $14,325. 6. Two brothers, A and B, bad each $25,000; A loaned B $7,500 and then borrowed of him $15,450, and lost so much of it in speculation that B had $7050 more than A ; how much did A lose ? Ans. $22,950. 7. Dr. Willard receives the following bill from George Jobson: 1 barrel of kerosene, $18.75; 2 barrels of flour, $19.75; 25 lb. of sugar, $3.62}; he presents at the same time his bill, as follows: foi medical services, $11.50; medicines, $3.45; what is the balance, and in whose favor ? Ans. $27.171 in Jobson's favor. 8. Hood & Martin bought of Chambers & Rogers the following goods: 1 piece blue anchor cloth, $215.16 ; 2 pieces black cloth, $91.80; 1 piece doeskin cassimere, $45; 2 pieces black ribbed cloth, $102.69. Chambers & Rogers bought of Hood & Martin as follows: 1 piece red twilled flannel, $24.30; 2 pieces blue twilled flannel, $48.95; 1 piece white domestic flannel, $10.08; what was the balance due Chambers & Rogers ? Ans. $371.32. 9. Required the balance of the following account in my bank book: Dr. 66 ;"1" Railroad stock, 2387 50 June 30, “ Balance, Co., MULTIPLICATION. 102. Multiplication is the process of finding the product of two numbers. 103. The Product of two numbers is the result obtained by taking one number as many times as there are units in another. 104. The Multiplicand is the number to be multiplied. 105. The Multiplier is the number by which we multiply. 106. The Sign of Multiplication is X, and is read multiplied by, times, or into. When placed between two numbers it denotes that one is to be multiplied by the other. Notes.-1. The Sign of Multiplication consists of two short lines of equal length bisecting each other at an angle of 45 degrees with the line of writing. 2. The symbol X was introduced by Wm. Oughtred, an English mathematician, born in 1574. PRINCIPLES. 1. The multiplier is always an abstract number. For, the multiplier shows the number of times that the multiplicand is taken; hence it cannot be yards or bushels, or any other concrete number. 2. The product is always similar to the multiplicand. Thus, 4 times 5 dollars are 20 dollars, and not 20 yards or 20 days, or anything else besides dollars. 3. The product of two numbers is the same whichever is made the multiplier. Thus, the 12 stars in the diagram inay be regarded as * * * * 3 fours, or as 4 threes; hence 3 times 4 equals 4 times 3, * * * * and the same may be shown of any other number. * * * * 4. If the multiplicand be multiplied by all the parts of the multiplier, the sum of all the partial products will be the true product. This is evident from the axiom that the whole is equal to the sum of all the parts. NOTE.-From Prin. 1 we see that such problems as “multiply 25 cents by 25 cents,” or “2 shillings and 6 pence by itself,” are impossible and absurd. PROBLEM. 107. To multiply one number by another. 1. Multiply 685 by 345. SOLUTION.- We write the multiplier under the OPERATION. multiplicand, and draw a line beneath. 5 times 5 685 units are 25 units, or 2 tens and 5 units; we write the 345 5 units and carry the 2 tens; 5 times 8 tens are 40 tens, plus 2 tens, are 42 tens, or 4 hundreds and 2 3425 tens; we write the 2 tens and carry the 4 hundreds; 2740 2055 5 times 6 hundreds are 30 hundreds, plus 4 hundreds, are 34 hundreds, which we write: in the same way we 236325 find 4 tens times 685 are 2740 tens, and 3 hundred times 685 are 2055 hundreds ; adding these partial products we have 5 units+4 tens+3 hundreds times 685, or 345 times 685, which is 236325. Rule.-1. Write the multiplier under the multiplicand, placing terms of the same order in the same column, and draw a line beneath. II. Begin at the right, and multiply the multiplicand by each term of the multiplier, writing the first term of each product under the term of the multiplier which produces it. III. Add the partial products, and their sum will be the entire product. Proof.—Multiply the multiplier by the multiplicand, and if the work is correct this product will equal the first product. NOTES.-1. When there are ciphers between the significant terms of the multiplier, pass over them and multiply by the significant terms alone. 2. We begin at the right to multiply, so that when any product exceeds nine, we may add the number expressed by the left hand figure to the next product. Multiply 2. 4276 by 394. Ans. 1684744. 3. 5364 by 207. Ans. 1110348. 4. 8075 by 3165. Ans. 25557375. 5. 2856 by 6124. Ans. 17490144. 6. 31876 by 3256. Ans. 103788256. 7. 48306 by 2078. Ans. 100379868. 8. 61357 by 3851. Ans. 236285807. 9. 86195 by 4287. Ans. 369517965. 10. 38056 by 3075. Ans. 117022200. 11. 91654 by 41652. Ans. 3817572408. 12. 43729 by 50706. Ans 2217322674. 13. 384675 by 65078. Ans. 25033879650. 14. 8765932 by 850704. Ans. 7457213410128. . 10. 1234567 by 4600407. Ans. 5679510668769. PRACTICAL PROBLEMS. 1. At 37 dollars a ton, what will be the cost of 796 tons of pig iron ? OPERATION. SOLUTION.-If 1 ton cost $37, 796 tons will cost 796 796 times $37, which we find, multiplying 796 by 37 for 37 convenience, to be $29452. $29452 2. What will be the cost of 691000 Philadelphia pressed brick at $33 a thousand? Ans. $22,803. 3. A grocer bought 46 barrels of superfine flour at $12.50 per barrel, and 75 barrels of corn meal at $4.50 per barrel; what was the whole cost? Ans. $912.50. 4. If a man spends 25 cents a day for cigars, how much will he spend in 40 years, allowing for ten leap years, there being 52 weeks and one day in a year, and 7 days in a week? Ans. $3652.50. 5. In a freight car there are 9 boxes of books, each weighing 465 pounds; 12 barrels of pork, each weighing 200 pounds; and 45 boxes of shoes, each weighing 156 pounds; what was the weight of all? Ans. 13605 pounds. 6. A wholesale druggist sold 5 kegs of baking soda containing 112 lbs. each, at 6 cents a lb.; 7 casks of washing soda, containing 300 lbs. each, at 3 cents a lb., 15 boxes of French Castile soap, containing 30 lbs. each, at 16 cents a lb.; and 50 dozen cakes of perfumed toilet soap at 75 cents a dozen; what did the whole amount to? Ans. $206.10. 7. A lady preparing to go to housekeeping purchased at Van Harlingen and Arrison's 1 piece of linen sheeting, containing 38 yards, at 45 cents a yard; 3 pieces of muslin sheeting, containing 43 yards each, at 28 cents a yard; 10 yards of damask table linen at $1.25 a yard; 15 yards of better quality of table linen at $2 a yard; 1 dozen napkins at $2.50 a dozen, and 1 dozen doilies, at $2 a dozen; what was her bill ? Ans. $109.22. CONTRACTIONS IN MULTIPLICATION. 108. Contractions in Multiplication are abbreviated methods of multiplying. 109. A Composite Number is the product of two or more numbers, each greater than a unit, called factors. Thus, 24 is a composite number, whose factors are 4 and 6, or 3 and 8, or 2, 3, and 4. 110. Continued Multiplication is the process of finding the product of three or more numbers by multiplying the first and second, this result by the third, etc. The result is the Continued Product. PRINCIPLES. 1. The product of two numbers is equal to the continued product of one number by the factors of the other. Thus, 27 multiplied by 24 is equal to 27 multiplied by 6 and that product by 4. 2. The continued product of several factors will be the same in whatever order the factors are taken. Thus, 2x3X 4 is equal to 2X4X3, or 3X 4X2, or to the product in whatever other order the numbers may be taken. CASE 1. 111. When the multiplier is a composite number. 1. Multiply 256 by 24. OPERATION. SOLUTION.—24 equals 4 times 6, hence 24 times 256 256 equals 4 times 6 times 256; 6 times 256 equals 1536, and 4 times 1536 equals 6144; therefore 256 1536 multiplied by 24 equals 6144. Hence the 6144 Rule.- Multiply the multiplicand by one factor, this product by another factor, and thus continue until all the factors have been used; the last product will be the result required. Multiply 2. 367 by 28. Ans. 10276. 6. 2057 by 63. Ans. 129591. 3.764 by 35. Ans. 26740. 7. 3895 by 72. Ans. 280440. 4. 596 by 42. Ans. 25032.8. 4168 by 108. Ans. 450144. 5. 4783 by 56. Ans. 267848. 9. 4796 by 144. Ans. 690624. |