are successively subtracted from the corresponding parts of the dividend, until the whole is exhausted. Now we have just shown by the operation of multiplication, that the sum of these products, taken in the order in which they stand, is equal to the dividend. Therefore the above rule for long division must be correct. PROOF From what has been said, we also infer that this method of long division proves itself as we proceed with the work, since we have only to add the successive products, and the remainder, if any, to obtain the dividend. What is long division? How do you place the numbers? Repeat the rule. If, after having brought down a new figure, the result is less than the divisor, how do you proceed? When the partial product is greater than the number which was supposed to contain the divisor, how do you do? When the remainder is greater than the divisor, how do you proceed? Explain the method of proof. 2. Divide 175678 by 223. OPERATION. 223)175678(787 1561 first product. 1957 1784 second product. 1738 1561 third product. 177 remainder. If we take the sum of the successive products and the remainder, adding them as they now stand in the above work, we shall obtain 175678; which, agreeing with the dividend, proves the accuracy of the division. This method of proving division, is perhaps, as simple and brief as any method which can be devised. The common method of proving division, and one which is applicable to short division as well as long division, is to multiply the divisor and quotient together, and to add in the remainder, if any. 3. Divide 7892343 by 139. Ans. 5677925· Ans. 235678. Ans. 539 379 64534 Ans. 156472381 789 678509 Ans. 8335375. Ans. 70056 with 7856 remainder. 10. Divide 166168212890625 by 12890625.* 11. Divide 11963109376 by 109376. CASE III. Ans. 12890625. Ans. 109376. 28. When the divisor is a composite number. In this case we evidently have the following RULE. Divide successively by each of the factors of the divisor. It makes no difference which factor we first use. To obtain the true remainder, we must observe the following RULE. Multiply the last remainder by the preceding divisor, and add in the preceding remainder; multiply this sum by the next preceding divisor, and add in the next preceding remainder; so continue this reverse process until you have multiplied by all the divisors. How do you proceed when the divisor is a composite number? Does it make any difference which factor we first divide by? When there are several remainders, explain how the true remainder is obtained. EXAMPLES. 1. Divide 839 by 120. In this case we will resolve 120 into the three factors 4x5×6=120. Now proceeding agreeably to the rule, we have this This question and the next are worthy of notice, since the terminal figures of the dividend, divisor, and quotient, are the same. OPERATION. 4)839 5)209 3 first remainder. 6 5 third remainder. Now, to obtain the true remainder, we have this 5×5+4=29 again, 29×4+3=119 Had there been more than three factors, the method of operation would have been equally as simple, but a little more lengthy. 2. Divide 8217 by 35. The factors of 35 are 5 and 7. Hence, we have this 4. Divide 9591 by 72=8×9. Ans. 133 with 15 remainder. 5. Divide 10859 by 49=7×7. Ans. 221 with 30 remainder. CASE IV. 29. When the divisor ends with one or more ciphers. We have seen under ART. 3, that a number is multiplied by 10 by annexing a cipher; it is multiplied by 100 by annexing two ciphers; by 1000 by annexing three ciphers, &c. Conversely, a number is divided by 10 by cutting off one figure from the right; it is divided by 100 by cutting off two figures from the right, &c. Hence we have this RULE. Cut off from the right-hand of the dividend, as many figures as there are ciphers in the divisor, then divide what remains by the significant figures of the divisor after the ciphers are omitted. To the remainder bring down the fig ures cut off from the dividend. This will give the true remainder. How do you proceed when there are ciphers at the right of the divisor? EXAMPLES. 1. Divide 4567894 by 3700. OPERATION. 3700)4567894(1234 quotient. 37 86 74 127 111 168 148 2094 remainder. If we regard the divisor 3700 as a composite number whose factors are 100 x 37=3700, this example will properly fall under the last Case. According to which the 94 cut off from the right of the dividend is to be considered the first remainder, and the 20 is the last remainder. Hence the true remainder is 20×100+94=2094, from which we discover the correctness of the above rule. 2. Divide 7123545 by 421000. Ans. 16 with 387545 remainder. 3. Divide 1212121212 by 42000. Ans. 28860 and 1212 remainder. 4. Divide 123456789 by 12300. Ans. 10037 and 1689 remainder there are 7 days. EXERCISES IN DIVISION. 1. In one year there are 365 days, and in one week How many weeks in one year? Ans. 52 weeks and 1 day. 2. Nine square feet make one square yard. How many square yards are there in 495 square feet? Ans. 55 square yards. 3. Three men are to share equally in the sum of 1236 dollars. How many dollars will each have? Ans. 412 dollars. 4. Divide 1245 acres of land equally between five brothers. Ans. Each has 249 acres. 5. It is about 95000000 miles from here to the sun. Now admitting that it requires 8 minutes for light to pass from the sun to the earth, how many miles does it pass in one minute? Ans. 11875000 miles. 6. Allowing 22 brick to. be sufficient to make one cubic foot of masonry, how many cubic feet are there in a work which requires 100000 brick? Ans. 4545 cubic feet and 10 brick remaining. 7. The circumference of the `earth is about 24900 miles. How long would it require for a person to travel around it, if he could pass uninterruptedly at the rate of 200 miles per day? Ans. 1244 days. |