EXAMPLES. Divisor. Dividend. Quotient. 1. Divide 11772 by 327. Having set down the divisor on the OPERATION, left of the dividend, it is seen that 327 is not contained in 117; but by observing that 3 is contained in 11, 3 times and something over, we conclude that the 327)11772(36 divisor is contained at least 3 times in 981 the first four figures of the dividend. 1962 Set down the 3 in the quotient, and 1962 multiply the divisor by it; we thus get 981 which being less than 1177, the quo 0000 tient figure is not too great: we subtract 981 from the first four figures of the dividend, and find a remainder 196, which being less than the divisor, the quotient figure is not too small. Annex to this remainder the next figure 2, of the dividend. As 3 is contained in 19, 6 times, we conclude that the divisor is contained in 1962 as many as 6 times. Setting down 6 in the quotient and multiplying the divisor by it, we find the product to be 1962. Therefore the entire quotient is 36, or the divisor is contained 36 times in the dividend. Note 1. After multiplying by the quotient figure, if any one of the products is greater than the number supposed to contain the divisor, the quotient figure is too large, and must be diminished. Note 2. When any one of the remainders is greater than the divisor, the quotient figure is too small, and must be increased by at least 1. Q. If any one of the products is too large, what do you do? If any one of the remainders is greater than the divisor, what do you do? 2. Divide 2756 by 26. We first say, 26 in 27 once, and place 1 in the quotient. Multiplying by 26)2756(106 i, subtracting, and bringing down the 5, 26 we say 26 in 15,0 times, and place the 156 0 in the quotient. Bringing down the 156 6, we find that the divisor is contained in 156, 6 times. OPERATION. Note. If after having annexed the figure from the dividend to any one of the remainders, the number is less than the divisor, the quotient, figure is 0, which being written in the quotient, annex the next figure of the dividend and divide as before. DEMONSTRATION OF THE RULE OF DIVISION. § 33. If 6 simple units be divided by 3, the quotient will be 2. If 6 units of the 2d order, or 60, be divided by 3, the quotient will be 2 tens, or 2 units of the 2d order. If 9 hundreds, or 9 units of the 3d order be divided by 3, the quotient will be 3 hundreds, or 3 units of the 3d order. So, in general, If units of any order be divided by simple units, the units of the quotient will be of the same order as those of the dividend. OPERATION. Let us suppose, as an example, that it were required to divide 11772 by 327. We first consider, as we have a right to do, that 11772 is made up of 1177 327)11772(36 tens and 2 units. We then divide the 981 tens by the divisor 327, and find 3 tens 1962 for the quotient, by which we multiply 1962 the divisor and subtract the product from 1177, leaving a remainder of 196 tens. To this number we bring down the 2 units, making 1962 units. This number contains the divisor 6 times: that is, 6 unit's times. When the unit of the first number which contains the divisor is of the 3d order, or 100, there will be 3 figures in the quotient; when it is of the 4th order there will be 4, &c. Hence, the quotient found according to the rule, expresses the number of times which the dividend contains the divisor, and consequently is the true quotient. Q. When the divisor is contained in simple units, what units will the quotient figure express? When the divisor is contained in tens, what units will the quotient figure express? When it is contained in hundreds ? In thousands ? PROOF OF DIVISION $34. Multiply the divisor by the quotient and add in the remainder, when there is one: the sum should be equal to the dividend. EXAMPLES 1. Divide 67289 by 261. In this example, we find a quotient OPERATION. of 257 and a remainder of 212, which 261)67289(257 being less than the divisor will not con 522 tain it. 1508 1305 PROOF. 2039 261 Divisor. 1827 257 Quotient. 212 Rem. 1827 1305 522 212 Remainder. 67289=the dividend: Hence, the work is right. 2. Divide 119836687 by 39407. PROOF. 39407)119836687(3041 39407 Divisor. 118221 3041 Quotient. 161568 39407 157628 157628 39407 118221 39407 119836687 Dividend. OPERATION. Q. How do you prove division? PROOF OF MULTIPLICATION. § 35. When two numbers are multiplied together the multiplicand and multiplier are both factors of the product; and if the product be divided by one of the factors, the quotient will be the other factor. Hence, if the product of two numbers be divided by the multiplicand, the quotient will be the multiplier ; or, if it be divided by the multiplier, the quotient will be the multiplicand. Q. If two numbers are multiplied together, what are the factors of the product? If the product be divided by one of the factors, what will the quotient be? How do you prove multiplication ? EXAMPLES. 3679 Multiplicand 327 Multiplier 7358 3679)1203033(327 11037 9933 7358 25753 25753 00000 24 rem. 2. The multiplicand is 61835720, the product 8162315040: what is the multiplier? Ans. 132. 3. The multiplier is 270000, the product 1315170000000: what is the multiplicand ? Ans. 4. The product is 68959488, the multiplier 96: what is the multiplicand ? Ans. 718328. 5. The multiplier is 1440, the product 10264849920 : what is the multiplicand ? Ans. 6. The product is 6242102428164, the multiplicand 6795634 : what is the multiplier ? Ans. 7. Divide 7210473 by 37. Ans. 8. Divide 147735 by 45. Ans. 3283. 9. Divide 937387 by 54. Ans. 1 rem. 10. Divide 145260 by 108. Ans. 1345. 11. Divide 79165238 by 238. Ans. 12. Divide 62015735 by 7803. Ans. 79475305 13. Divide 74855092410 by 949998. Ans. 14. Divide 47254149 by 4674. Ans. 15. Divide 119184669 by 38473. Ans. .-33788 rem. 16. Divide 280208122081 by 912314. Ans.--121 rem. 17. Divide 293839455936 by 8405. Ans.--346 rem. 18. Divide 4637064283 by 57606. Ans.---11707 rem. -9 rem. 19. Divide 352107193214 by 210472. Ans. --165534 rem. 20. Divide 558001172606176724 by 2708630425. Ans. .-24 rem. CONTRACTIONS IN DIVISION. CASE I. $36. When the divisor is a composite number. RULE. Divide the dividend by one of the factors of the divisor, and then divide the quotient thus arising by the other factor: the last quotient will be the one sought. EXAMPLES. Let it be required to divide 1407 dollars equally among 21 men. Here the factors of the divisor are 7 and 3. 3)201 1st quotient. Let each one of the 7 men divide 67 quotient sought. his share into 3 equal parts, each one of the three equal parts will be 67 dollars, and the whole number of parts will be 21; there the true quotient is found by dividing continually by the factors. 2. Divide 18576 by 48=4x12. Ans. 387. 3. Divide 9576 by 72=9x8. Ans. 133. 4. Divide 19296 by 96=12x8. Ans. 201. § 37. It sometimes happens that there are remainders after division, for which we have the following RULE. The first remainder, if there be one, forms a part of the true remainder. The product of the second remainder, if there be one, by the first divisor, forms a second part. Either of these parts, when the other does not exist, forms the true remainder, and their sum is the true remainder when they both exist together. |