63. One Part of the Multiplier a Factor of Another. 8693 328 69544 278176 2851304 1. What is the product of 8693 × 328? = SOLUTION.-328 32 tens + 8 units, and 32 tens = 4 tens times 8. Hence, to find the required product, we first multiply by 8, then multiply that product by 4 tens, and add the two results together. 2. What is the product of 864727 × 25175? 864727 21618175000 SOLUTION.-25175 = 25000 + 175, and 175 is 7 times 25. Hence, to find the required product, we first multiply by 25, writing the product as thousands, and then multiply this product by 7, writing the result as units, and add the two results together. (c.) What is the product of— 3. 9783 X 427 ? 4. 6324 X 186? 7. 54533 × 96486? 8. 37496 × 927 ? 9. 47938 × 19899? 10. 27645 × 54279? 64. Abbreviated Method for Long Division. (a.) The labor of writing the successive products in Long Division may be avoided by subtracting each product when we obtain it. 1. What is the quotient of 54786 ÷ 217? 217)54786(25210 1138 536 SOLUTION.-217 is contained in 547 twice. 2 times 7 are 14, which taken from 17 leaves 3. 2 times 1 are 2 and 1 are 3, which taken from 4 leaves 1. 2 times 2 are 4, which taken from 5 leaves 1. The first remainder is therefore 113, to which annexing the 8 gives 1138 as the number to be divided. 217 is contained in 1138 5 times. 5 times 7 are 35, which taken from 38 leaves 3. 5 times 1 are 5, and 3 are 8, which taken from 13 leaves 5, etc. 102 Rem. (b.) Find the following quotients: 2. 96348 27. 3. 2695398. 4. 82564973. 5. 468251 196. 6. 833443783. 65. To Divide by any convenient Fractional Part of 10, 100, 1000, etc. (a.) Since 100 = 4 times 25, or 8 times 12, it follows that the quotient of a number divided by 25 must be 4 times the quotient of the same number divided by 100; the quotient of a number divided by 12 must be 8 times the quotient of the same number divided by 100. A similar thing will be true of any other fractional part of 10, 100, 1000, etc. 1. What is the quotient of 397463 ÷ 125? SOLUTION.- Since 1000 = = 8 times 125, the quotient of a number divided by 125 must be 8 times the quotient of the same number divided by 1000. But 3974631000 397.463, and 8 times this result is 3179.704, which equals the quotient of 397463 ÷ 125. NOTE. The above form gives the complete quotient. The entire part of the quotient is found at the left of the decimal point: the remainder may be found by dividing the figures at the right of the decimal point by the number by which we multiplied the quotient of the division by 100 or 1000. Thus, in the above example, the entire part of the quotient is 3179, and the remainder is of 704, or 88. (b.) Find the following quotients: (a.) When one person sells merchandise to another, it is customary for him to write out for the purchaser a statement of the articles sold, with their prices. Such a statement is called a BILL. NOTE. There are few persons, in any walk of life, who are not at some time or other called on to pay a bill, or to make out or receipt one. Indeed, the bill is the most common business paper. The student, then, will appreciate the importance of thoroughly mastering the following explanations. 1. On the 10th day of July, 1857, J. G. Lester, of Providence, Bold to F. L. Gay, for cash (i. e. for ready money), 16 pounds of sugar, at 14 cents per pound; 3 pounds of tea at 68 cents per pound; and 1 barrel of flour for $9.50. Mr. Lester made the following bill: (b.) In considering the above bill, we observe - 1st, the DATE, “Providence, July 10, 1857," which tells when and where the transactions took place. 2d, the HEADING, "F. L. Gay bought of J. G. Lester," which shows the names of the parties and the nature of the transaction. 3d, the STATEMENT of the articles bought, with their prices and amount. 4th, the RECEIPT, consisting of the SIGNATURE "J. G. Lester," written after the words "Received Payment." REMARKS.-1. The date is sometimes placed at the bottom of the bill. 2. The heading "F. L. Gay to J. G. Lester, Dr." is much used instead of the preceding, and means that F. L. Gay is DEBTOR to J. G. Lester, i. e. that he owes him for the articles mentioned in the bill. 3. The words "Received Payment" would have no value unless Mr. Lester had written his name under them, or authorized some one to do it for him. 4. If D. S. Stone had received the money for Mr. Lester, he would have receipted the bill by writing after the words "Received Payment" some form similar to the following: "D. S. Stone for J. G. Lester," or "J. G. Lester by D. S. Stone." (c.) The following bills furnish further illustrations of this subject. Let the class find the amount of each. Town of Dedham 2. Dedham, July 1, 1857. To C. A. Brigham, For 3 months' services as Grammar-School in District No. Received Payment, C. A. Brigham, Dr. $225 by T. Baker. 8. Providence, July 24, 1857 Mr. Francis Peavey To G T. Gladding & Co. June 3. For 3 yd. Broadcloth @ $5.29 Dr. "17. Cassimere 66 1.62 NOTE. Balance due G. T. G. & Co. The date of the last bill, July 24, 1857, shows when it was made out. The dates at the left show when each transaction took place. The "CREDIT ENTRY," "July 20. By Cash $18.00," shows that on the 20th of July Mr. Peavey paid $18.00 towards the bill. Opposite the words "Balance due G. F. G. & Co." should be written the sum still due on the bill. If Mr. Peavey should pay this balance Aug. 1st, the receipt would be written thus: "Aug. 1, 1857. Received Payment, "G. F. Gladding & Co." NOTE.-When the bill is receipted, Mr. Peavey should keep it as evidence that he has paid it in full. (e.) Make out bills for each of following transactions: 4. Suppose that you are a grocer, and sell to some person in your neighborhood, for cash, 7 gallons of molasses at 39 cents per gallon; 1 box soap containing 28 lb. at $.09 per lb.; 1 cheese weighing 18 lb. at $.09 per lb.; and 31 lb. butter at 21 cents per lb. |