§ 76.-1. How many pairs of boots can be bought for $24, at $3 a pair? (24÷3-how many?) 2. How many quarts of milk, at 4 cents a quart, can be bought for 16 cents? How many for 32 cents? for 24 cents? for 40 cents? for 48 cents? 3. How many pounds of cheese can be bought for 40 cents, at 8 cents a pound? for 32 cents? for 64 cents? for 56 cents? for 72 cents? for 96 cents? for 24 cents? 4. How long will it require to travel 49 miles at 7 miles an hour? 63 miles? 77 miles? 84 miles? 35 miles? 42 miles? 21 miles? 5. How long will 35 bushels of oats suffice for a horse, if he eat 5 bushels a week? How long will 45 bushels suffice? 60 bushels? 55 bushels? 30 bushels? 6. How long will it require to travel 72 miles at 9 miles an hour? how long to travel 108 miles? 99 miles? 54 miles? 45 miles? 63 miles? 7. If a barrel of molasses cost $11, how many barrels can be bought for $33? for $77? for $99? for $55? for $88 ? 8. If a man earn $8 in 1 day, in how many days will he earn $48? $64? $96? $40! $32? $72? 9. If a pound of sugar cost 9 cents, how many pounds can be bought for 81 cents? for 90 cents? for 45 cents? for 54 cents? for 63 cents? 10. If a man spend $12 a week, in how many weeks will he spend $96? $132? $72? $84? $60? $144? 11. At 4 cents a mile, how far can I ride for 48 cents? 32 cents? 16 cents? 20 cents? 40 cents? 44 cents? 12. If it cost 7 cents to ride 1 mile, how many miles can I ride for 63 cents? for 84 cents? for 56 cents? for 35 cents? for 77 cents? 13. How many shillings are there in 72 pence, 12 pence being required to make 1 shilling? in 96 pence? in 108 pence? in 132 pence? in 60 pence ? 14. If a pound of lard cost 11 cents, how many pounds may be bought for 110 cents? for 132 cents? for 88 cents? for 99 cents? for 55 cents? 15. Bought a sheep for $6. for $54 for $66? for $72 How many sheep can I buy for $48? for $36? for $42? 16. If 8 shillings inake 1 dollar, how many dollars are there in 64 shillings? in 88 shillings? in 56 shillings? in 96 shillings? in 72 shillings? 17. There are 12 ounces in 1 pound Troy. How many pounds in 72 ounces? in 48 ounces? in 84 ounces? in 60 ounces? in 36 ounces? § 77. It often happens that the dividing number is not an exact measure of the one to be divided; that is, after the division is performed, a portion of the divided number, less than the divisor, remains. This is called the remainder. The following is an illustration. Suppose 23 is to be divided by 5. The divisor 5 is not contained in 23 so many as 5 times, since 5 times 5 are 25, a number larger than 23. It is, however, contained in 23, 4 times, since 4 times 5 are 20; a number less than 23 by 3. Hence 5 is contained in 23, 4 times and 3 remainder. 1. How many times is 6 contained in 39, and what is the remainder? 2. How many times is 4 contained in 18, and what is the remainder? 3. A man having $37, bought cloth at $5 a yard. How many yards could he buy, and how many dollars would he have left? 4. A boy having 47 cents, bought melons at 6 cents each. How many melons could he buy, and how many cents have left? 5. A scholar has 51 sums to perform. During how many hours can he perform 7 per hour, and how many will remain? 6. If a pound of raisins cost 7 cents, how many pounds can I buy for 60 cents, and how many cents left ? 7. If a pound of tamarinds cost 6 cents, how many pounds can be bought for 70 cents, and how many cents left? for 49 cents? for 59 cents? 8. If a pound of dates cost 12 cents, how many pounds can be bought for 140 cents, and how many cents left ? 9. If a barrel of flour cost $9, how many barrels can be bought for $70? for $88 ? for $49? for $57 ? for $68? 77. What often happens? Illustrate? 10. How many times 4 in 33, and how many remain ? in 38? in 46? 11. How many times 8 in 46, and how many remain ? in 54? in 71? 12. How many times 11 in 37, and in 49? in 93 ? 13. How many times 12 in 52, and in 79? in 101? 14. How many times 9 in 61, and in 86? in 107 ? 15. How many times 7 in 51, and in 83? in 78? 16. How many times 6 in 39, and in 57? in 47 ? how many remain ? how many remain ? how many remain ? how many remain ? how many remain ? 17. How many times 12 in 79, and how many remain ? in 63? in 95? 18. How many times 11 in 81, and how many remain ? in 69? in 98 ? 19. How many times 10 in 64, and how many remain? in 73? in 117? SIMPLE DIVISION.-For the Slate. § 78. From the previous questions and answers we learn the nature of Division; viz., that it is a process of ascertaining how many times one number contains another. Two numbers are given, one of which is to be divided by the other. That to be divided is called the dividend; that by which we divide, is called the divisor; and if, after the operation is performed, a number be left undivided, that is called the remainder. The quotient, or number, obtained by dividing the dividend by the divisor, is one of the fac tors, (§ 64. 1,) or equal parts into which the dividend is separated. The number of these factors equals the units 78. What do we ascertain by division? How many numbers are given? What is that to be divided called? What that by which we divide? What call the number left undivided? What is the quotient? in the divisor; hence if the quotient be written down a number of times equal to the units composing the entire divisor, and their sum obtained, it will exactly correspond with the dividend in all instances, except when there is a remainder; if a remainder exist, that must likewise be added to the amount of factors. ILLUSTRATION. If we divide 20 by 4 we obtain 5 as a quotient; 4 and 5 are therefore the factors of 20; and since the divisor 4 contains four units, the quotient 5 must be repeated 4 times to produce the dividend 20; thus 5+5+ 5+5=20. Again, 23÷4=5 and 3 remainder, and 5+ 5+5+5+3=23; that is, the four factors into which 23 is divided, and the number (3) which remains over and above these four factors, being added, produce the dividend 23. It is obvious that the remainder must in every instance be less than the divisor, for were it larger than the divisor, the dividend would contain that divisor one time more than is indicated. At paragraph 73, we gave the character, as that used to imply division. It is also expressed by writing the dividend over the divisor with a short line between them; thus 27 3 implies that 27 is to be divided by 3. The dividend may be a mere abstract number, or a number representing some particular objects, or valuation. The divisor is always to be regarded as merely a number, showing the number of equal factors or quantities into which the dividend is to be resolved; while the quotient, being one of the equal parts into which the dividend is resolved, must be of the same name or kind as the dividend. The remainder being also an undivided part of the dividend, agrees with it in kind. Division is the reverse of Multiplication, and is proved by it. The divisor showing the number of factors What is a factor? (§ 64. 1.) The number of these factors equals what? How proved? What exception? How then? Give illustrations? How does the remainder compare with the divisor? Why? What is the sign of division? How else expressed? What may the dividend be? The divisor what? Showing what? The quotient must agree in kind with what? The remainder with what? How is division proved? What does the divisor show? obtained from the dividend, and the quotient being one of these factors, the product of these two terms necessarily produces the dividend in all cases except where there is a remainder. If a remainder exist, it must be added to the product of the divisor and quotient to produce the dividend. Division is usually separated into two sections: the first embracing all cases when the divisor is not more than 12; the second, all cases when the divisor is more than 12. CASE I. 879. The following rule is to be used only when the divisor does not exceed 12. RULE. Write down the dividend, and placing a short curve line on the left, write the divisor at the left of that curve, and draw a horizontal line directly beneath the dividend. Take from the left of the dividend the smallest number of figures capable of containing the divisor, and observe how many times it is contained, and whether there be any remainder, writing the figure expressing the times, beneath the line, directly under the figure or figures divided. If there be a remainder, write the following figure of the dividend on the right of that remainder, and divide as before, placing the figure obtained, under the dividend figure last taken. Continue this process, till all the figures of the dividend have been taken; the number obtained will be the quotient. If the figure or figures taken divide without remainder, proceed with the remaining figures as before; and if in assuming any figure of the dividend, the number obtained What is the quotient? Their product, what? What exception? How, if there be a remainder? How many sections? The first, what? The second, what? 79. What is the rule? How write the sum? How many figures are taken, and where? Where write the quotient figure? If there be a remainder, what is done? Where place the figure obtained? How long continue the process? If the figures divide without remainder, how proceed? When place a cipher in the quotient? |