FIRST OPERATION. Dollars. SECOND OPERATION. Dollars. 39. The city of New York owes 13,960,856 dollars, and Boston owes 7,779,855 dollars ; how much more does New York owe than Boston ? Ans. 6,181,001 dollars. 40. From five hundred eighty-one thousand take three thousand and ninety-six. Ans. 577,904. 41. It was ascertained by a transit of Venus, June 3, 1769, that the mean distance of the earth from the sun is ninety-five millions one hundred seventy-three thousand one hundred twentyseven miles, and that the mean distance of Mars from the sun is one hundred forty-five millions fourteen thousand one hundred forty-eight miles. Required the difference of the distances. Ans. 49,841,021 miles. 33. To subtract when there are two or more subtrahends. Ex. 1. A man owing 767 dollars, paid at one time 190 dollars, at another time 131 dollars, at another time 155 dollars ; how much did he then owe? Ans. 291 dollars. In the first opera tion, the several subMinuend 7 67 Minuend 7 67 trahends, having been properly placed, are 131 131 added for a single sub190 Subtrahends 190 trahend, to be taken from the minuend. 1 5 5 1 5 5 In the second, the subSubtrahend 47 6 Remaind 291 trahends are substract ed, as they are added, Remainder 2 9 1 at one operation, thus: beginning with units, 5 and 1 equal 6, which from 7 units leaves 1 unit; passing to tens, 5 and 9 and 3 equal 17 tens; reserving the left-hand figure to add in with the figures of the subtrahends in the next column, the right-hand figure, 7, being larger than 6 tens of the minuend we add 10 to the 6, and, subtracting, have left 9 tens; and, passing to hundreds, we add in the left-hand figure 1 reserved from the 17 tens, and also add 1, equal 10 tens, to compensate for the 10 added to the minuend, and with the other figures, 1 and 1 and 1 equal 5 hundreds, which, taken from 7 hundreds, leave 2 hundreds; and 291 as the answer. 2. E. Webster owned 6,765 acres of land, and he gave to his oldest brother 2,196 acres, and his uncle Rollins 1,981 acres ; how much has he left ? Ans. 2,588 acres. 3. The real estate of James Dow is valued at 3,769 dollars, and his personal estate at 2,648 dollars ; he owes John Smith 1,728 dollars, and Job Tyler 1,161 dollars ; how much is he worth? Ans. 3,528 dollars. MULTIPLICATION. 34. WHEN any number is to be added to itself several times, the operation may be shortened by a process called Multiplication. Ex. 1. If a man can earn 8 dollars in 1 week, what will he earn in 4 weeks? ILLUSTRATION. - It is evident, since a man can earn 8 dollars in 1 week, that in 4 weeks he will earn 4 times as much, and the result may be obtained by addition ; thus, 8 +8+8+8 32 dollars ; or, by a more convenient process, by multiplying by 4, the number of times 8 dollars is to be taken; thus, 4 times 8 dollars are 32 dollars. Hence in 4 weeks he will earn 32 dollars. 10 times 1 are 10 10 2 64 20 10 3 30 10 4 40 10 66 5 50 10 6 60 10 7 70 10 8 80 10 9 90 10 10 11 times 7 are 77 8 66 88 11 66 9 99 11 66 10 11 66 11 11 12 66 110 66 121 6 132 11 times 1 are 11 2 22 3 33 4 44 11 5 55 11 12 times 3 are 36 12 48 12 66 60 12 72 12 84 12 8 96 12 9 12 10 12 11 12 12 66 108 6 120 66 132 66 144 6 100 12 times 1 are 12 66 66 34. How may the process of adding a number to itself several times be shortened ? 2. What cost 5 barrels of flour at 6 dollars per barrel ? ILLUSTRATION. - If i barrel of flour cost 6 dollars, 5 barrels will cost 5 times as much ; 5 times 6 dollars are 30 dollars. Therefore 5 barrels of flour at 6 dollars per barrel will cost 30 dollars. 3. What cost 6 bushels of beans at 2 dollars per bushel ? 4. What cost 5 quarts of cherries at 7 cents per quart ? 5. What will 7 quarts of vinegar cost at 12 cents per quart? 6. What cost 9 acres of land at 10 dollars per acre. 7. If a pint of currants cost 4 cents, what cost 9 pints ? 8. If in one penny there are 4 farthings, how many in 9 pence? In 7 pence? In 8 pence? In 4 pence? In 3 pence? 9. If 12 pence make a shilling, how many pence in 3 shillings? In 5 shillings? In seven shillings? In 9 shillings? 10. If one pound of raisins cost 6 cents, what cost 4 pounds ? 5 pounds ? 6 pounds ? 7 pounds ? 8 pounds ? 9 pounds ? 10 pounds ? 12 pounds ? 11. In one acre there are four roods; how many roods in 2 acres ? In 3 acres ? In 4 acres ? In 5 acres? In 6 acres ? In 9 acres ? 12. A good pair of boots is worth 5 dollars ; what must I give for 5 pairs ? For 6 pairs ? For 7 pairs ? For 8 pairs ? For 9 pairs ? 13. A cord of good walnut wood may be obtained for 8 dollars ; what must I give for 4 cords ? For 6 cords ? For 9 cords? 14. What cost 4 quarts of milk at 5 cents a quart, and 8 gallons of vinegar at 10 cents a gallon ? 15. If a man earn 7 dollars a week, how much will he earn in 3 weeks ? In 4 weeks ? In 5 weeks? In 6 weeks? In 7 weeks? In 9 weeks? 16. If 1 thousand feet of boards cost 12 dollars, what cost 4 thousand ? 5 thousand? 6 thousand ? 7 thousand ? 9 thousand ? 12 thousand ? 17. If 3 pairs of shoes buy one pair of boots, how many pairs of shoes will it take to buy 7 pairs of boots ? 18. If 5 bushels of apples buy 1 barrel of flour, how many bushels of apples are equal in value to 12 barrels of four ? 19. If one yard of canvas cost 25 cents, what will 12 yards cost ? ILLUSTRATION. 25 is composed of 2 tens and 5 units ; 12 times 2 tens are 24 tens; and 12 times 5 units are 60 units, or 6 tens, 24 tens added to 6 tens make 30 tens, or 300. Therefore, 12 yards will cost 300 cents. 20. In 1 pound there are 20 shillings; how many shillings in 3 pounds? In 4 pounds ? In 6 pounds ? 21. A gallon of molasses is worth 25 cents; what is the value of 2 gallons ? Of 4 gallons? Of 9 gallons ? 22. If 1 man earn 12 dollars in 16 days, how much would 10 men earn in the same time? 23. If a steam-engine runs 28 miles in 1 hour, how far will it run in 4 hours ? In 6 hours ? In 9 hours ? 24. If the earth. turns on its axis 15 degrees in 1 hour, how far will it turn in 7 hours? In 11 hours? In 12 hours ? 25. In a certain regiment there are 8 companies, in each company 6 platoons, and in each platoon 12 soldiers; how many soldiers are there in the regiment? 26. If 1 man walk 7 miles in 1 hour, how far will he walk in 8 hours ? In 11 hours ? In 20 hours ? In 30 hours ? 35. Multiplication is the process of taking a number as many times as there are units in another number. In multiplication three terms are employed, called the Multiplicand, the Multiplier, and the Product. The Multiplicand is the number to be multiplied or taken. The Multiplier is the number by which we multiply, and denotes the number of times the multiplicand is to be taken. The Product is the result, or number produced by the multiplication. The multiplicand and multiplier are often called Factors. Note. The multiplicand may be either an abstract or concrete number, but the multiplier must always be regarded as an abstract number. The product is of the same kind as the multiplicand. The Sign of Multiplication is formed by two short lines crossing each other obliquely; thus, X. It shows that the numbers between which it is placed are to be multiplied together; thus, 7 X 5 = 35 is read, 7 multiplied by 5 is equal to 35. 35. What is multiplication? What three terms are employed? What is the multiplicand? The multiplier ? The product? What are the multiplicand and multiplier often called? The sign of multiplication? What does it show? 36. When the multiplier does not exceed 12. OPERATION. Ex. 1. Let it be required to multiply 175 by 7. Ans. 1225. Having written the multiplier under the Multiplicand 175 unit figure of the multiplicand, we multiply Multiplier 7 the 5 units by 7, obtaining 35 units, or 3 tens and 5 units, and set down the 5 units Product 1 2 2 5 under the 7, and reserve the 3 tens for the tens' column. We then multiply the 7 iens by 7, obtaining 49 tens, and, adding the 3 tens which were reserved, we have 52 tens, or 5 hundreds and 2 tens. Writing down the 2 tens, and reserving the '5 hundreds, we multiply the 1 hundred by ?; and, adding the reserved 5 hundreds, we have twelve hundreds, which we write down in full; and the product is 1225. 9. Multiply 767853 by 9. Ans. 6910677. 10. Multiply 876538765 by 8. Ans. 7012310120. 11. Multiply 7654328 by 7. Ans. 53580296. 12. Multiply 4976387 by 5. Ans. 24881935. 13. Multiply 8765448 by 12. Ans. 105185376. 14. Multiply 4567839 by 11. Ans. 50246229. 15. What cost 8675 barrels of flour at 7 dollars per barrel ? Ans. 60725 dollars. 36. How must numbers be written for multiplication? At which hand do you begin to multiply? Why? Where do you write the first or right-hand figure of the product of each figure in the multiplicand ? Why? What is done with the tens or left-hand figure of each product? How, then, do you proceed when the multiplier does not exceed 12 ? |