21. A man sold his farm for 6756 dollars, and divided the amount equally between his wife and 5 children: how much did each receive? 22. There are 576 persons in a train of 12 cars: how many are there in each car? 23. If a township of land containing 2304 acres be equally divided among 8 persons, how many acres will each have? 24. If it takes 5 bushels of wheat to make a barral of flour, how many barrels can be made from 65890 bushels? 25. Twelve things make a dozen: how many dozens are there in 2167284? 26. Eleven persons are all of the same age, and the sum of their ages is 968 years: what is the age of each? 27. How many barrels of flour at 7 dollars a barrel can be bought for 609463 dollars? 28. An estate worth 2943 dollars, is to be divided equally among a father, mother, 3 daughters and 4 sons: what is the portion of each? 29. A county contains 207360 acres of land lying in 9 townships of equal extent: how many acres in a township? 30. If 11 cities contain an equal number of inhabitants, and the whole number is equal to 3800247: how many will there be in each ? EQUAL PARTS OF NUMBERS. 64. 1. If any number or thing be divided into two equal parts, one of the parts is called one-half: one half of a single thing is written thus; 2. 2. If any number is divided into three equal parts, one of the parts is called one-third, which is written thus; ; two of the parts are called two-thirds: which are written thus; . 2 3. If any number is divided into four equal parts, one of the parts is called one-fourth, which is written thus ; ; two of the parts are called two-fourths, and are written thus; three of them are called three-fourths, and written; and similar names are given to the equal parts into which any number may be divided. 63. What is short division? How is it generally performed? Give the rule? How do you prove short division? 4. If a number is divided into five equal parts, what is one of the parts called? Two of them? Three of them? Four of them? 5. If a number is divided into 7 equal parts, what is one of the parts called? What is one of the parts called when it is divided into 8 equal parts? When it is divided into 9 equal parts? When it is divided into 10? When it is divided into 11? When it is divided into 12? of 8 7 of 6. What is one-half of 2? of 4? of 6? of 8? of 10? of 12? of 14? of 16? of 18? 7. What is two-thirds of 3? ANALYSIS-Two-thirds of three are two times one-third of three. One-third of three is 1, therefore, two-thirds of three are two times 1, or 2. Let every question be analyzed in the same manner. What is one-third of 6? 2 thirds of 6? One-third of 9? 2 thirds of 9? One-third of 12? two-thirds of 12? 8. What is one-fourth of 4? 2 fourths of 4? 3 fourths of 4 ? What is one-fourth of 8? 2 fourths of 8? 3 fourths of 8? What is one-fourth of 12? 2 fourths of 12? 3 fourths of 12? fourth of 16? 2 fourths of 16? 3 fourths? One 9. What is one-seventh of 7? What is 2 sevenths of 7? 5 sevenths? 6 sevenths? What is one-seventh of 14? 3 sevenths? 5 sevenths? 6 sevenths? What is one-seventh of 21 ? of 28? of 35 ? 10. What is one-eighth of 8? of 16? of 24? of 32? of 40? of 56? 11. What is one-ninth of 9? 2 ninths? 7 ninths? 6 ninths? 5 ninths? 4 ninths? What is one-ninth of 18? of 27? of 54? of 72? of 90? of 108? 12. How many halves of 1 are there in 2? ANALYSIS-There are twice as many halves in 2 as there are in 1. There are two halves in 1; therefore, there are 2 times 2 halves in 2, or 4 halves. 13. How many halves of 1 are there in 3? In 4? In 5? In 6? in 8? In 10? In 12? 14. How many thirds are there in 1? How many thirds of 1 in 2? In 3? In 4? In 5? In 6? In 9? In 12? 15. How many fourths are there in 1? of 1 in 2? In 4? In 6? In 10? In 12? How many fourths 35 16. How many fifths are there in 1? How many fifths of 1 are there in 2? In 3? In 6? In 7? In 11? In 12? 17. How many sixths are there in 2 and one-sixth? In 3 and 4 sixths? In 5 and 2 sixths? In 8 and 5 sixths? 18. How many sevenths of 1 are there in 2? In 4 and 3 sevenths how many? How many in 5 and 5 sevenths? In 5 and 6 sevenths? 19. How many eighths of 1 are there in 2? How many in 2 and 3 eighths? In 2 and 5 eighths? In 2 and 7 eighths? In 3? In 3 and 4 eighths? In 9? In 9 and 5 eighths? In 10? In 10 and 7 eighths? 20. How many twelfths of 1 are there in 2? In 2 and 4 twelfths how many? How many in 4 and 9 twelfths? How many in 5 and 10 twelfths? In 6 and 9 twelfths? In 10 and 11 twelfths? 21. What is the product of 12 multiplied by 3 and one half, (which is written 34)? ANALYSIS.-Twelve is to be taken 3 and one-half times (Art 45). Twelve taken times is 6; and 12 taken three times is 36; therefore, 12 taken 3 times is 42. 22. What is the product of 10 multiplied by 5? 23. What is the product of 12 multiplied by 3? 24. What is the product of 8 multiplied by 4? 25. What will 9 barrels of sugar cost at 2 dollars a barrel? ANALYSIS.-Nine barrels of sugar will cost nine times as much as 1 barrel. If one barrel of sugar costs 23 dollars, 9 barrels will cost 9 times 2 dollars, which are 24 dollars. For, 2 thirds taken 9 times gives 18 thirds, which are equal to 6; then 9 times 2 are 18, and 6 added gives 24 dollars. 26. What will 6 yards of cloth cost at 5 dollars a yard? 27. What will 12' sheep cost at 4 dollars apiece? 28. What will 10 yards of calico cost at 93 cents a yard? 29. What will 8 yards of broadcloth cost at 75 dollars a yard? 30. What will 9 tons of hay cost at 93 dollars a ton? 31. How many times is 2 contained in 10? ANALYSIS.-Two and one-half is equal to 5 halves; and 10 is equal to 20 halves; then 5 halves is contained in 20 halves 4 times: hence, In all similar questions change the divisor and dividend to the same fractional unit. (Art. 144). 32. How many yards of cloth, at 3 dollars a yard, can you buy for 14 dollars? how many for 21 dollars? 33. If oranges are 3 cents apiece, how many can you buy for 20 cents? 34. If 1 yard of ribbon costs 23 cents, how many yards can you buy for 12 cents? 35. If 1 yard of broadcloth costs 33 dollars, how many yards can be bought for 33 dollars? 36. If 1 pound of sugar costs 4 cents, how many pounds can be bought for 36 cents? 37. How many times is 5 contained in 44? 38. How many times is 23 contained in 24 ? 39. How many lemons, at 2 cents apiece, can you buy for 32 cents? 40. How many yards of ribbon, at 1 cents a yard, can you buy for 12 cents? 21 LONG DIVISION. 65. LONG DIVISION is the operation of finding the quotient of one number divided by another, and embraces the case of Short Division, treated in Art. 63. 1. Let it be required to divide 7059 by 13. ANALYSIS.-The divisor, 13, is not contained in 7 thousands; therefore, there are no thousands in the quotient. We then consider the 0 to be annexed to the 7, making 70 hundreds, and call this a partial dividend. The divisor, 13, is contained in 70 hundreds, 5 hundreds times and something over. To find how much over, multiply 13 by 5 hundreds and subtract the product 65 from 70, and there will remain 5 hundreds, to which bring down the 5 tens and consider the 55 tens a new partial dividend. OPERATION. Thous. Hunds. 13) Tens. Units. Hunds. →Tens. Units. 65 55 39 39 65. What is long division? Does it embrace the case of short division? What is a partial dividend? Then, 13 is contained in 55 tens, 4 tens times and something over. Multiply 13 by 4 tens and subtract the product, 52, from 55, and to the remainder 3 tens bring down the 9 units, and consider the 39 units a new partial dividend. Then, 13 is contained in 39, 3 times. Multiply 13 by 3, and subtract the product 39 from 39, and we find that nothing remains. 66. PROOF.-Each product that has arisen from multiplying the divisor by a figure of the quotient, is a partial product, and the sum of these products is the product of the divisor and quotient (Art. 51, NOTE). Each product has been taken, separately, from the dividend, and nothing remains. But, taking each product away in succession, leaves the same remainder as would be left if their sum were taken away at once. Hence, the number 543, when multiplied by the divisor, gives a product equal to the dividend: therefore, 543 is the quotient (Art. 61): hence, to prove division, Multiply the divisor by the quotient and add in the remainder, if any. If the work is right, the result will be the same as the dividend. 67. Let it be required to divide 2756 by 26. We first say, 26 in 27 once, and place 1 in the quotient. Multiplying by 1, subtracting, and bringing down the 5, we have 15 for the first partial dividend. We then say, 26 in 15, 0 times, and place the 0 in the quotient. We then bring down the 6, and find that the divisor is contained in 156, 6 times. OPERATION. 26)2756(106 26 156 156 If any one of the partial dividends is less than the divisor, write O for the quotient figure, and then bring down the next figure, forming a new partial dividend. Hence, for Long Division, we have the following RULE.-I. Write the divisor on the left of the dividend. II. Note the fewest figures of the dividend, at the left, that will contain the divisor, and set the quotient figure at the right. 66. What is a partial product? What is the sum of all the partial products equal to? How do you prove division? 67. What do you do if any partial dividend is less than the divisor? What is the rule for long division? |