many gallons of water were delivered in one hour? Answer, 33,600 gallons. 27. In coaling ship the port boat crane made 54 hoists in one hour. At each hoist it brought aboard 7 bags of coal averaging 560 pounds per bag. How many pounds of coal did it bring aboard per hour? Answer, 211,680 pounds. 28. Two ships are 7,483 miles apart and are steaming toward each other, one at the rate of 240 miles per day and the other at the rate of 180 miles per day. How far apart will they be at the end of 9 days? Answer, 3,703 miles. 29. The earth moves in its orbit at the rate of 19 miles per second. How far does it move in one minute? How far in one hour? Answer, (a) 1,140 miles. (b) 68,400 miles. DIVISION. 1. If you divide 8 tons of coal equally between two firerooms, how much will each have? Answer 4 tons. Why? 2. How many times is 2 contained in 10? Answer 5. Why? The process by which the preceding examples are solved is called division. DEFINITIONS. 1. Division is the process of finding how many times one number is contained in another. 2. The Divisor is the number by which to divide. The Dividend is the number to be divided. The Quotient is the number denoting how many times the divisor is contained in the dividend. Thus 3 is contained in 12, 4 times. Here 3 is the divisor, 12 is the dividend, and 4 is the quotient. 3. Since 3 is contained in 12, 4 times, 4 times 3 are 12; that is, the divisor and the quotient multiplied together produce the dividend. 4. Since 3 and 4 are factors of the product 12, the divisor and quotient correspond to the factors in multiplication, the dividend to the product. Therefore: Division is the process of finding one of the factors of a product when the other factor is known. Example: A man has 80 cents. How many pairs of socks can he buy at 10 cents per pair? Analysis. He can buy 8 pair, because 8 pair at 10 cents per pair will cost 80 cents. The man would give 10 cents for 1 pair and have 70 cents left. After giving 10 cents for the second pair he would have 60 cents left. After giving 10 cents for the third pair he would have 50 cents left. After giving 10 cents for the fourth pair he would have 40 cents left. After giving 10 cents for the fifth pair he would have 30 cents left. After giving 10 cents for the sixth pair he would have 20 cents left. After giving 10 cents for the seventh pair he would have 10 cents left. After giving 10 cents for the eighth pair he would have nothing left. The natural method of performing this operation is by subtraction; but when it is known how many times 10 can be subtracted from 80, instead of subtracting 10 eight times, say: 10 in 80 eight times, and 8 times 10 are 80. Therefore, division may be termed a short method of making many subtractions of the same number. The divisor is the number subtracted; the dividend the number from which the subtraction has been made; the quotient shows how many subtractions have been made. Division is indicated in three ways: 1st. 3)12, which means that 12 is to be divided by 3. 2d. 12, which means that 12 is to be divided by 3. 꿍, 3d. 123, which means that 12 is to be divided by 3. In using the first sign, when the divisor does not exceed 12, draw a line under the dividend and write the quotient underneath. If the divisor exceeds 12, draw a curved line on the right and place the quotient to the right of this. If 7 dollars be equally divided between 3 men, each man would receive 2 dollars and there would be left or remaining 1 dollar. The number left after dividing is called the remainder. Since the remainder is a part of the dividend, it must be of the same denomination. If the dividend be dollars, the remainder will be dollars; if pounds, the remainder will be pounds. The remainder is always less than the divisor; if the remainder were equal to or greater than the divisor, the divisor would be contained at least once more in the dividend. If the dividend and divisor are simple numbers, the operation is called division of simple numbers. SHORT DIVISION. When the division is performed mentally and only the quotient written, it is termed short division. Short division is used when the divisor does not exceed 12. 1. How many times is 2 contained in 468? Here the dividend is composed of three numbers, 4 hundreds, 6 tens, and 8 units-that is 400, 60, and 8. The same result can be obtained without actually separating the dividend into parts. Thus: 2 in 4 (hundreds), 2 times, which write in hundreds place; then 2 in 6 (tens), 3 times, which write in tens place; then 2 in 8 (units), 4 times, which write in units place. Divisor 2)468 dividend 2. How many times 3 in 693? Answer, 231. 1. How many times is 3 contained in 129? 3)129 Solution. Here, 3 is not contained in 1; but 3 is contained in 12 (tens) 4 times, which write under tens place; 3 in 9 (units), 3 times, which write under units place. 2. How many times is 3 contained in 735? 3)735 Solution. Here 3 is contained in 7 (hundreds) 2 times, and 1 (hundred) over. The 1 (hundred) united with the 3 (tens) makes 13 (tens) in which 3 is contained 4 times and 1 (ten) left over. The 1 (ten) is then united to the 5 (units) making 15 (units) and the quotient. figure 5 is placed in the units place. 3. How many times is 3 contained in 618? 3)618 Solution.--Here 3 is contained in 6 (hundreds) 2 times. As the 1 in tens place will not contain 3, a cipher is placed in tens place and the 1 (ten) is then added to the 8 (units) making 18 and the quotient figure 6 is placed in the units place. 4. How many times is 3 contained in 609? 3) 609 203 Solution. Here the solution is indicated as in the above example; there being no tens their order is indicated by a 0. 5. How many times is 3 contained in 743? 3)743 Solution. After dividing, there is 2 left, the division of which is merely indicated by placing the divisor under the remainder; thus . The quotient is written thus 2473; read 247 and 2 divided by 3 or 247 with a remainder of 2. 6. How many times is 3 contained in 462? 7. How many times is 5 contained in 1170? 8. How many times is 4 contained in 948? Answer, 154. Answer, 234. RULE: Write the divisor at the left of the dividend with a curved line between them and draw a line beneath the dividend. Begin at the left hand, divide successively each figure of the dividend by the divisor, and write the result in the same order in the quotient. If there is a remainder after dividing any figure, prefix it to the figure in the next lower order and divide as before. If the number in any order does not contain the divisor place a cipher in the same order in the quotient, prefix the number to the figure in the next lower order and divide as before. If there is a remainder after dividing the last figure place the divisor under it and annex it to the quotient. PROOF: Multiply the quotient by the divisor and add the remainder, if any, to the product; if the work is correct the sum will be equal to the dividend. This method of proof depends on the principle that a dividend is a product of which the divisor and quotient are factors. 9. Divide 653 cents by 3. When any number is divided into two equal parts, one of the parts is called one-half of that number. If divided into three equal parts, one of the parts is called one-third; if into four equal parts, one-fourth; if into five equal parts, one-fifth; and so on. Hence, to find one-half of a number divide by 2; to find one-third, divide by 3; to find one-fourth, divide by four, etc. 10. Divide 8,652 by 2. Answer, 4,326. 15. If coal costs $3 per ton, how many tons can be bought for $894? Answer, 298 tons. 16. If a certain sized brass globe valve costs $4, how many can be bought for $140? Answer, 35 valves. 17. If a bunker contains 64 tons of coal, how long will it supply 2 boilers whose daily consumption is 16 tons? Answer, 4 days. 18. A chief machinist's mate receives $924 for a year's work. What is his salary per month? Answer, $77. LONG DIVISION. When the entire work of division is written down it is called Long Division. Long division is commonly used when the divisor exceeds 12. 1. Divide 3,465 dollars among 15 men. 15)3465(231 46 tens 15 units Solution. Fifteen is not contained in 3 (thousands), therefore there. will be no thousands in the quotient. Take 34 (hundreds) as a partial dividend; 15 is contained in 34, 2 times; that is, 15 men have 200 dollars each, which requires in all 15×2, or 30 hundreds of dollars. Subtract 30 (hundreds) from 34 (hundreds) and 4 (hundreds) remain. Bring this down and add to the 6 (tens) and you have 46 (tens) for a second partial dividend. Forty six (tens) contains 15, 3 times; that is, each man has 30 dol-. lars more, and all require 15×3, or 45 tens of dollars. Subtract 45 and bring down the 5 (units), which gives 15 (units) for a third partial dividend. In this the divisor is contained once, giving each man 1 dollar more. Hence each man receives 2 hundred dollars, 3 ten dollars, and 1 dollar, a total of 231 dollars. By this process the dividend is separated into parts, each part containing the divisor a certain number of times. The first part, 30 hundreds, contains the divisor 2 times; the second part, 45 tens, contains it 3 times; the third part, 15 units, contains it 1 time. The several parts together equal the dividend, and the several partial quotients make up the entire quotient. RULES: Place the divisor on the left of the dividend and place a curved line between them, and another on the right of the dividend. |