GENERAL PRINCIPLES IN DIVISION. 140. From the nature of division, it is evident, that the value of the quotient depends both on the divisor and the dividend. 141. If a given divisor is contained in a given dividend a certain number of times, the same divisor will obviously be contained, In double that dividend, twice as many times. In three times that dividend, thrice as many times, &c. Hence, If the divisor remains the same, multiplying the dividend by any number, is in effect multiplying the quotient by that number. Thus, 6 is contained in 12, 2 times; in 2 times 12 or 24, 6 is contained 4 times; (i. e. twice 2 times ;) in 3 times 12 or 36, 6 is contained 6 times; (i. e. thrice 2 times ;) &c. 142. Again, if a given divisor is contained in a given dividend a certain number of times, the same divisor is contained, In half that dividend, half as many times; In a third of that dividend, a third as many times, &c. Hence, If the divisor remains the same, dividing the dividend by any number, is in effect dividing the quotient by that number. Thus, 8 is contained in 48, 6 times; in 48÷2 or 24, (half of 48,) 8 is contained 3 times; (i. e. half of 6 times;) in 48÷3 or 16, (a third of 48,) 8 is contained 2 times; (i. e. a third of 6 times;) &c. 143. If a given divisor is contained in a given dividend a certain number of times, then, in the same dividend, Hence, Twice that divisor is contained only half as many times; Three times that divisor, a third as many times, &c. If the dividend remains the same, multiplying the divisor by any number, is in effect dividing the quotient by that number. Thus, 4 is contained in 24, 6 times; 2 times 4 or 8 is con QUEST.-140. Upon what does the value of the quotient depend? 141. If the divisor remains the same, what effect has it on the quotient to multiply the dividend? 142. What is the effect of dividing the dividend by any given number? 143. If the dividend remains the same, what is the effect of multiplying the divisor by any given number? 7 tained in 24, 3 times; (i. e. half of 6 times ;) 3 times 4 or 12 is contained in 24, 2 times; (i. e. a third of 6 times ;) &c. 144. If a given divisor is contained in a given dividend a certain number of times, then, in the same dividend, Hence, Half that divisor is contained twice as many times; A third of that divisor, three times as many times, &c. If the dividend re.nains the same, dividing the divisor by any number, is in effect multiplying the quotient by that number. Thus, 6 is contained in 36, 6 times; 6÷2 or 3, (half of 6,) is contained in 36, 12 times; (i. e. twice 6 times;) 6÷3 or 2, (a third of 6,) is contained in 36, 18 times; (i. e. thrice 6 times ;) &c. 145. From the preceding articles, it is evident that any given divisor is contained in any given dividend, just as many times as twice that divisor is contained in twice that dividend; three times that divisor in three times that dividend, &c. Conversely, any given divisor is contained in any given dividend just as many times, as half that divisor is contained in half that dividend; a third of that divisor, in a third of that dividend, &c. Hence, 146. If the divisor and dividend are both multiplied, or both divided by the same number, the quotient will not be altered. Thus, 6 is contained in 12, 2 times; 2 times 6 is contained in 2 times 12, 2 times; 12 2 is contained in 48÷2, 4 times; 12÷3 is contained in 48÷3, 4 times, &c. 147. If the sum of two or more numbers is divided by any number, the quotient will be equal to the sum of the quotients which will arise from dividing the given numbers separately. Thus, the sum of 12+18=30; and 30÷6=5. Now, 12÷6=2; and 18÷÷6=3; but the sum of 2+3=5. 84; 24÷8=3; and 40÷÷8=5; but 4+3+5=12 QUEST. 144. What of dividing the divisor? 146. What is the effect upon the quotien if the divisor and dividend are both multiplied, or both divided by the same number? CANCELATION.* 148. We have seen that division is finding a quotient, which, multiplied into the divisor, will produce the dividend. (Art. 112.) If, therefore, the dividend is resolved into two such factors that one of them is the divisor, the other factor will, of course, be the quotient. Suppose, for example, 42 is to be divided by 6. Now the factors of 42 are 6 and 7, the first of which being the divisor, the other must be the quotient. Therefore, Canceling a factor of any number, divides the number by that factor. Hence, 149. When the dividend is the product of two factors, one of which is the same as the divisor. (Ax. 9.) Cancel the factor common to the dividend and divisor; the other factor of the dividend will be the answer. Note. The term cancel, signifies to erase or reject. 1. Divide the product of 34 into 28 by 34. Common Method. 34 28 272 68 34)952(28 Ans. 68 272 By Cancelation. $4)34×28 28 Ans. Canceling the factor 34, which is common both to the divisor and dividend, we have 28 for the quotient, the same as before. 272 150. The method of contracting arithmetical operations, by rejecting equal factors, is called CANCELATION. OBS. It applies with great advantage to that class of examples and problems, which involve both multiplication and division; that is, which require the product of two or more numbers to be divided by another number, or by the product of two or more numbers. 2. Divide 76 X 45 by 76. 4. Divide 65X82 by 82. 3. Divide 63X81 by 81. 5. Divide 9573 by 95. 6. Divide the product of 45 times 84 by 9. *Birk's Arithmetical Collections: London, 1764 Analysis.-The factor 45=5X9; hence the dividend is com posed of the factors 84×5X9. We may therefore cancel 9, which is common both to the divisor and dividend, and 84×5, the other factors of the dividend, will be the answer required. Operation. 9)84X5X9 420 Ans. Proof. 84X5X9=3780 And 3780 9 420. 7. Divide the product of 45X6X3 by 18X5. Operation. 18×5)45×6×3 9 Ans. Proof. 45X6X3=810; and 18×5=90 Now, 810-90=9 Note.-We cancel the factors 6 and 3 in the dividend and 18 in the divisor; for 6X3=18. Canceling the same or equal factors in the divisor and dividend, is dividing them both by the same number, and therefore does not affect the quotient. (Arts. 146, 148.) Hence, 151. When the divisor and dividend have common factors. Cancel the factors common to both; then divide the product of ihose remaining in the dividend by the product of those remaining in the divisor. 8. Divide 15X7×12 by 5×3×7×2. 9. Divide 27X3X4X7 by 9X12X6. 10. Divide 75X15X24 by 25×3×6×4×5. Note.--The further development and application of the principles of Cancelation, may be seen in reduction of compound fractions to simple ones; in multiplication and division of fractions; in simple and compound proportion, &c. 151. a. The four preceding rules, viz: Addition, Subtraction, Multiplication, and Division, are usually called the FUNDAMENTAL RULES of Arithmetic, because they are the foundation or basis of all arithmetical calculations. OBS. Every change that can be made upon the value of a number, must necessarily either increase or diminish it. Hence, the fundamental operations in arithmetic are, strictly speaking, but two, addition and subtraction; that is, increase and decrease. Multiplication, we have seen, is an abbreviated form of addition; division of subtraction. (Arts. 80, 114.) QUEST.-151. a. Name the fundamental rules of Arithmetic. Why are these rules called fundamental? APPLICATIONS OF THE FUNDAMENTAL RULES. 152. When the sum of two numbers and one of the numbers are given, to find the other number. From the given sum, subtract the given number, and the remainder will be the other number. Ex. 1. The sum of two numbers is 87, one of which is 25: what is the other number? Solution.-87-25-62, the other number. (Art. 72.) 2. A and B together own 350 acres of land, 95 of which belong to A: how many does B own? 3. Two merchants bought 1785 bushels of barley together, one of them took 860 bushels: how many bushels did the other have? 153. When the difference and the greater of two numbers are given, to find the less. Subtract the difference from the greater, and the remainder will be the less number. 4. The greater of two numbers is 72, and the difference between them is 28: what is the less number? Solution.-72-28-44, the less number. (Art. 72.) 5. A man bought a horse and chaise; for the chaise he gave 265 dollars, which was 75 dollars more than he paid for the horse how much did he give for the horse? 6. A traveler met two droves of sheep; the first contained 1250, which was 125 more than the second had: how many sheep were there in the second drove ? 154. When the difference and the less of two numbers are given, to find the greater. QUEST.-152. When the sum of two numbers and one of them are given, how is the other found? 153. When the difference and the greater of two numbers are given, how is the less found? 154. When the difference and the less of two numbers are given, how is the greater found? |