methods, together with their explanations, are too complex for insertion here. The living teacher can best present such processes. Unless the student is made familiar with them, they are of no practical importance. ART. 12. When the multiplier is 14, 15, 16, etc. Operation. 3425 × 15 51375 Product. Remark. It is not necessary to put down any part of the operation. The result may be written at once by the following RULE. Multiply by the unit's figure, adding, after the unit's place, the figures of the multiplicand. Examples. 2. Multiply 34809 by 13. 8. Multiply 369403 by 14. ART. 13. When the multiplier is 31, 41, 51, etc. Ex. 1. Multiply 3425 by 51. Operation. 13700 140425 Product. RULE. Multiply by the ten's figure and add the product to the proper orders of the multiplicand. Examples. 2. Multiply 3486 by 71. 3. Multiply 864 by 51. 4. Multiply 86047 by 41. 5. Multiply 38967 by 91. ART. 14 When the multiplier consists of two figures, the product may be written at once. Ex. 1. Multiply 675 by 56. 675 x 56-37800. Explanation. The process is based upon the fact that units multiplied by units give units, tens by units tens, tens by tens hundreds, hundreds by units hundreds, hundreds by tens thousands, etc. We first multiply 5 units by 6 units=30 units=3 tens and O units. Write O units. Multiply 7 tens by 6 units=42 tens, and add 3 tens, (received from the units,)=45 tens, and to this add 5 tens by 5 units=25 tens, which gives 70 tens=7 hundreds and O tens. Write O tens. Multiply 6 hundreds by 6 units=36 hundreds, and add the 7 hundred, received from the tens, which gives 43 hundreds. Then multiply the 5 tens by 7 tens=35 hundreds, and add 43 hundreds-78 hundreds-7 thousands and 8 hundreds. Write 8 hundreds. Multiply 6 hundreds by 5 tens 30 thousands, and add the 7 thousands received from hundreds=37 thousands. Write 37 thousand. The product is 37800. ART. 15. When the multiplier is a convenient part of 10, 100, 1000, etc. RULE. Multiply by 10, 100, 1000, etc. (by annexing ciphers) of which the multiplier is a part, and take the same part of the product. Ex. 1. Multiply 357 by 331. Operation. 3)35700 11900 Product. Explanation.-Since 33 is one third of 100, 33 times 357 must equal of 100 times 357. Note. The following are some of the convenient parts often occurring of 10-21, 31; of 100-12, 16, 25, 331, 50; of 1000-125, 166, 250, 3331, 500, 666. etc. ART. 16. To divide by a convenient part of 10, 100, 1000, RULE. Multiply by the quotient, found by dividing 10, 100, 1000, etc., (as the case may be) by the given divisor, and divide the result by 10, 100, 1000, etc. Ex. 1. Divide 850 by 163. Operation. 850 6 51.00 Ans. 51. Explanation.-Since 100 is 6 times 163, 100 is contained as many times in 6 times a given number as 163 is in the number itself. Remark.-Dimes and eagles are not mentioned in ordinary business transactions. In writing dollars and cents together, a point, called the separatrix (.), is placed between the dollars and cents; and, since cents occupy two places, the first figure at the right of cents is mills. It is not customary to separate cents and mills. Examples. 1. How many mills in 28 cents? In 37 cents? 2. How many cents in 15 dimes? In 16 dimes? 3. Reduce $12.50 to mills. 4. Change $90 to mills. 5. How many cents in 2 eagles, 5 dollars, and 8 dimes? 6. Reduce 4360 cents to dollars. 7. Add the following: $9.60, $12.70, $45.371, $.06, $1.50, $4.98, $68.33, $8.39, $60, and $.80. 8. Sold a carriage for $120.75, a horse for $90.60, a harness for $15.60, and a saddle for $13.12; amount received ? 9. From $108 take 12 cents. what was the Ans. $240.075. Ans. $107.875. 10. Bought a barrel of flour for $6.37, and sold it for $5.87; what did I lose? Ans. $.50. Paid $40 for a 11. Bought a house and lot for $1500. front fence, $110.90 for painting house, $9.75 for fruit trees, and $15 for other improvements. I then sold the property for $1800. What did I gain? 12. What will be the cost of 45 barrels of flour at $5.80 per barrel? 13. What will 80 bushels of coal cost at 15 cents per bushel ? 14. What will be the cost of 60 bushels of wheat at $1.12 per bushel; 146 bushels of corn at 663 cents a bushel; and 45 bushels of oats at 25 cents a bushel? 15. How many bushels of coal at 12 cents a bushel can be bought for $125 ? Suggestion. The dividend and divisor must be reduced to the same denomination. Change both to mills. 125.000÷ .125-1000. Ans. 1000 bushels. 16. How many pounds of butter at 16 cents per pound must be given for 15 barrels of flour at $8 per barrel ? 17. How many barrels of flour at $5.62 per barrel can be bought for $225 ? Ans. 40 barrels. 18. How many half-dimes would it take to pay for 16 cows at $16.37 per head ? 19. A drover bought 105 head of cattle at $57 per head. He paid for their pasturage one month $250, and then sold them at $60 per head. What did he gain by the transaction? Ans. $65. BILLS. ART. 18. A Bill of Goods, or simply a Bill, is a written statement of goods sold and their prices. It contains the time and place of the transaction and the names of the parties. A bill is drawn against the purchaser, and in favor of the merchant or seller. A bill is receipted by writing the words Received payment at the bottom and affixing the seller's name. A bill may be receipted by a clerk, agent, or any authorized person, as in bills 2 and 3. When sales are made at different times, the dates of the several transactions may be written at the left. |