DIVISION. 65. DIVISION is the operation of finding how many times one number is contained in another; or, of dividing a number into equal parts. 66. THE DIVIDEND is the number to be divided. 67. THE DIVISOR is the number by which we divide. It is the standard which measures the dividend; or, it shows into how many equal parts the dividend is to be divided. 68. THE QUOTIENT is the result of division. It shows how many times the divisor is contained in the dividend, or the value of one of the equal parts of the dividend. 69. THE REMAINDER is what is left after the operation. When it is 0, the quotient is a whole number, and the division is exact. Numbers in Division. 70. There are always three numbers in every division, and sometimes four: First, the dividend; second, the divisor; third, the quotient; fourth, the remainder. There are three methods of denoting division; they are the following: 12 123 expresses that 12 is to be divided by 3; 3) 12 When the last method is used, if the divisor does not exceed 12, we draw a line beneath the dividend, and set the quotient under it. If the divisor exceeds 12, we draw a curved line on the right of the dividend, and set the quotient at the right. Kinds of Division. 71. SHORT DIVISION is the operation of dividing when the work is performed mentally, and the results only written down. t is limited to the cases in which the divisors do not exceed 12. 72. LONG DIVISION is the operation of dividing when all the work is expressed. It is used when the divisor exceeds 12. 73. Operations and Rule. 1. Divide 456 by 4. ANALYSIS.-The number 456 is made up of 4 hundreds, 5 tens, and 6 units, each of which is to be divided by 4. Dividing 4 hundreds by 4, we have the quotient, 1 hundred: 5 tens divided by 4, gives 1 ten, and 1 ten over: reducing this to units, and adding in the 6, we have 16 units, which contains 4, 4 times: hence, the quotient is 114: that is, the dividend contains the divisor 114 times. 2. Divide £11 8s. 7d. 3far. by 5. OPERATION. 4) 456 114 OPERATION. £ 8. d. far. ANALYSIS.-Dividing £11 by 5, the quotient is £2, and £1 remaining. Reducing this to shillings, and adding in the 8, we have 28s., which, divided by 5, gives 5s., and 3s. over. This being reduced to pence, and 7d. added, gives 43d. Dividing by 5, we have 8d., and 3d. remainder. Reducing 3d. to farthings, adding 3 farthings, and again dividing by 5, gives the last quotient figure, 3far. Hence, £2 5s. 8d. 3far., is one of the five equal parts of the dividend. 3. Divide 11772 by 327. 5) 11 8 7 3 2 5 8 3 OPERATION. 327) 11772 (36 981 1962 1962 ANALYSIS.—Having set down the divisor on the left of the dividend, it is seen that 327 is not contained in the first three figures on the left, which are 117 hundreds. But by observing that 3 is contained in 11, 3 times, and something over, we conclude that the divisor is contained at least 3 times in the first four figures, 1177 tens, which is a partial dividend. Set down the quotient figure 3, and multiply the divisor by it: we thus get 981 tens, which being less than 1177, the quotient figure is not too great: we subtract the 981 tens from the first four figures of the dividend, and find a remainder 196 tens, which being less than the divisor, the quotient figure is not too small. Reduce this remainder to units, and add in the 2, and we have 1962. 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. Hence, the entire quotient is 36, or the divisor is contained 36 times in the dividend, and 36 is also one of the 327 equal parts of the dividend. Rule. I. Beginning with the highest order of units, take for a partial dividend the fewest figures that will contain the divisor: divide these figures by it, for the first figure of the quotient: the unit of this figure will be the same as that o he partial dividend: II. Multiply the divisor by the quotient figure so found, and subtract the product from the partial dividend: III. Reduce the remainder to units of the next lower order, and add in the units of that order found in the dividend: this gives a new partial dividend. Proceed in a similar manner until units of every order shall have been divided. 74. Directions for the Operations. 1. There are five steps in the operation of Division: 1st, To write down the numbers; 2d, To divide, or find how many times; 3d, To multiply; 4th, To subtract; 5th, To bring down, to form the partial dividends. 2. The product of a quotient figure by the divisor must never be larger than the corresponding partial dividend: if it is, the quotient figure is too large, and must be diminished. 3. If any one of the remainders is greater than the divisor, the quotient figure is too small, and must be increased. 4. The unit of any quotient figure is the same as that of the partial dividend from which it is obtained. The pupil should always name the unit of every quotient figure. 65. What is Division ?-66. What is the dividend?-67. What is the divisor? What does it show?-68. What is the quotient? What does i show?-69. What is the remainder? 70. How many numbers are there in every division? What are they? How many signs of Division are there? Make and name them. 71. What is Short Division? When is it used?-72. What is Long Division? When is it used?-73. Explain each of the three examples. Give the rule for the division of numbers. 5. If the dividend and divisor are both compound numbers, reduce them to the same unit before commencing the division. 6. If any partial dividend is less than the divisor, the corresponding quotient figure is 0. 7. When there is a remainder, after division, write it at the right of the quotient, and place the divisor under it. 75. Principles resulting from Division. 1. When the divisor is equal to the dividend, the quotient will be 1. 2. When the divisor is 1, the quotient will be equal to the dividend. 3. When the divisor is less than the dividend, the quotient will be greater than 1. The quotient will be as many times greater than 1, as the dividend is times greater than the divisor. 4. When the divisor is greater than the dividend, the quotient will be less than 1. The quotient will be such a part of 1, as the dividend is of the divisor. 74.-1. How many steps are there in division? Name them. 2. If a partial product is greater than the partial dividend, what does it indicate? What then do you do? 3. What do you do when any one of the remainders is greater than the divisor? 4. What is the unit of any figure of the quotient? When the divisor is contained in simple units, what will be the unit of the quotient figure? When it is contained in tens, what will be the unit of the quotient figure? When it is contained in hundreds? In thousands? 5. If the dividend and divisor are both compound numbers, what do you do? 6. If any partial dividend is less than the divisor, what is the corresponding figure of the quotient? 7. When there is a remainder after division, what do you do with it? 75.-1. When the divisor is equal to the dividend, what will the uotient be? 2. When the divisor is 1, what will the quotient be? 3. When the divisor is less than the dividend, how will the quotient compare with 1? How many times will it be greater than 1? 4. When the divisor is greater than the dividend, how will the quotient compare with 1? What part will the quotient be of 1? Proofs of Division. 76. There are three methods of proving division : I. Multiply the divisor by the quotient, and add in the remainder, if any: the result should be the dividend. II. Divide the dividend, diminished by the remainder, if any, by the quotient: the result should be the divisor. III. Find the excess of 9's in the divisor and in the quotient; multiply them together, and note the excess of 9's in the product: this should be equal to the excess of 9's in the dividend, after being diminished by the remainder, if any. 1st Method. In the last example, we had 11772 ÷ 327 = 36. |