SHORT METHOD OF DIVISION. 79. The following method saves about half the figures required by the method just given, and we think that there will be fewer mistakes made when using it. EXAMPLE.-Divide 39,913,910 by 5,494. dividend divisor SOLUTION. 39913910) 5494 14559 7265 quotient. Ans. 35711 27470 EXPLANATION.-The addition method of subtraction (see Art. 43) is used in this case; the divisor is written on the right of the dividend, and the quotient underneath the divisor. The different figures of the quotient are obtained in exactly the same manner as by the preceding method. Thus, the divisor is contained in the first five figures of the dividend 7 times, and 7 is written for the first figure of the quotient. Now, instead of multiplying the divisor by 7, writing the prod uct under the first five figures of the dividend, and then subtracting, we multiply each figure of the divisor by 7, and by the addition method, subtract from the dividend, writing only the remainder. Thus, 7 times 4 is 28, and 5 is 33; write the 5 under the 3 in the dividend, and carry 3. Then, 7 times 9 is 63 and 3 is 66, and 66 and 5 is 71; write the 5 and carry 7. 7 times 4 is 28 and 7 is 35, 35 and 4 is 39; write the 4 and carry 3. 7 times 5 is 35 and 3 is 38, and 38 and 1 is 39; write the 1. Now bring down the next figure of the dividend, 9, and annex it to the remainder. 14,559 ÷ 5,494 = 2; write 2 as the second figure of the quotient. Then, as above, 2 × 4 8, and 8+ 1 9; write the 1 under the 9, as shown. 2x9 carry the 2. 2 X 4 = = = the 5 and carry the 1. = 14; write the 3. the dividend, 35,711 = 18, and 18+ 7 = = 25; write the 7 and 10, and 10 + 5 10, 10 + 1 = 2 x 5 = 5,494 = = 15; write 11, and 11 + 3 next figure of 6, the third figure of the quotient. Proceed as above with the remaining figures. A fast computer would work as follows: In multiplying by 6, he would repeat to himself 6, 24, and 7 is 31 (writing the 7 and carrying the 3). 6, 54, 57, and 4 is 61. 6, 24, 30, and 7 is 37. 6, 30, 33, and 2 is 35. The object of writing the divisor on the right is to make it easier to multiply by the figures of the quotient; it also saves space, as may readily be seen. The student is strongly advised to learn this method thoroughly, and always to use it. The best way to attain facility in division, is first to practice dividing by small numbers, from 2 to 12, and using the method of short division. After he has become proficient in this, he should practice long division by the method just described. Some special methods which may be used when dividing by certain numbers will be mentioned farther on. 80. The principle given in Art. 62 may be used to test the work of division, when the principle has been slightly modified. Add the digits of the divisor, the dividend, the quotient, and the remainder, if any, as described in Art. 62, obtaining a single figure for the sum of each. Multiply the number thus obtained for the divisor by the number obtained for the quotient, and add to the product the number obtained for the remainder, if any. If the work has been done correctly, the result must equal the number obtained for the dividend. Thus, in the last example, the sum of the digits in the divisor (reduced to a single figure) is 4, of those in the quotient, 2, and in the remainder, 0. Hence, 4 X 2 = 8; 808. Adding the digits in the dividend, the result (reduced to a single figure) is also 8; hence, the work is very probably correct. Applying this method to the example in Art. 78, we have, for the divisor 9, for the quotient 4, for the remainder 5. Hence, 9 x 4+5= 41, and 4+1 5. For the dividend, 4+2+3+5 = 14, and 1 + 4 5, also. The student will find this principle very useful. = Addition, subtraction, multiplication, and divison are the four corner stones of arithmetic; everything else in arithmetic depends upon them. 82. Cancelation is the process of shortening operations in division by casting out equal factors from both dividend and divisor. 83. The factors of a number are those numbers which, when multiplied together, will equal that number. Thus, 5 and 3 are the factors of 15, since 5 X 3 8 and 7 are the factors of 56, since 8 X 7 = 15. Likewise, = 56. 84. A prime number is a number that cannot be divided by any number except itself and 1; 1 is not considered a factor. Thus, 2, 3, 11, 29, etc. are prime numbers. 85. A prime factor of a number is any factor that is a prime number. Any number that is not a prime is called a composite number, and may be produced by multiplying together its prime factors. Thus, 60 is a composite number, and is equal to the product of its prime factors, 2 x 2 x 3 x 5. Two numbers are said to be prime to each other when they have no common factor, as, for example, 15 and 28; there is no number, except 1, that will divide both 15 and 28 without a remainder. 86. Canceling equal factors from both dividend and divisor does not change the quotient. The canceling of a factor in both dividend and divisor is. the same as dividing them both by the same number, which, by a principle of division, does not change the quotient. Write the numbers forming the dividend above the line, and those forming the divisor below it. EXAMPLE.-Divide 4 × 45 × 60 by 9 × 24. SOLUTION.-Placing the dividend over the divisor, and canceling, EXPLANATION.-The 4 in the dividend and 24 in the divisor are both divisible by 4, and 24 divided by 4 equals 6. write the 6 under 24. Thus, since 4 divided by 4 equals 1, Cancel the 4, and the 24, and 4 × 45 × 60 9 × 24 60 in the dividend and 6 in the divisor are divisible by 6, since 60 divided by 6 equals 10, and 6 divided by 6 equals 1. Cancel the 60 and write 10 over it; also, cancel the 6. Thus, 10 4 × 45 × 60 Again, 45 in the dividend and 9 in the divisor are each divisible by 9, since 45 divided by 9 equals 5, and 9 divided by 9 equals 1. Cancel the 45 and write the 5 over it; also, cancel the 9. Thus, 5 10 4 × 43 × 60 9 × 24 Since there are no two remaining numbers (one in the dividend and one in the divisor) divisible by any number greater than 1 without a remainder, it is impossible to cancel further. Multiply together all the uncanceled numbers in the dividend and divide their product by the product of all the uncanceled numbers in the divisor. The result will be the quotient. The product of all the uncanceled numbers in = 50, and there are no uncanceled the dividend is 5 × 10 numbers in the divisor. II. Then divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor, and the result will be the quotient. 88. Divide: EXAMPLES FOR PRACTICE (a) 32. (6) 250. 1. (c) (d) 48. Ans. (a) 14 X 18 X 16 X 40 by 7X8X6X5 X 3. |