RULE: Begin at the left and write each period as a number composed of hundreds, tens, and units, filling the vacant orders with ciphers. Write the following numbers: 2. Four thousand; thirty-five thousand; eight hundred thousand. 3. Forty-eight million; seventy-nine million; sixty-three million. 4. Five billion; seventy billion; nine hundred billion. 5. One thousand five hundred; three thousand two hundred. 6. Four thousand two hundred and fifty; seven thousand nine hundred and forty-seven. 7. Eighteen thousand four hundred and fifty-six. 8. Five hundred and eighty-six thousand six hundred and fortythree. 9. Nine hundred and seventy-six million eighty-seven thousand six hundred and twenty-one. 10. Nine hundred and ninety-six million eight hundred and sixty thousand seven hundred and two. Numeration is a method of reading numbers expressed by characters. To read numbers in Arabic system, read: 785,431,416,523,144. RULE 1: Begin at the right and point off the number into periods of three figures each. 2. Begin at the left and read each period as a number composed of hundreds, tens, and units, giving the name of the period. (1) The left-hand period will sometimes contain but one or two figures. (2) It is customary to omit the name of the unit period. Read the following numbers: 2. 41582, 796543, 987546. 6. 1000570201, 8606021, 40001. THE FUNDAMENTAL RULES. DEFINITIONS. An integer is a whole number. Numbers are either concrete or abstract. A concrete number is one that has reference to a particular thing, as five ships, 9 men, 24 guns, etc. An abstract number is one used without reference to any particular thing, as 1, twenty-five, eight, etc. The name of the object of a concrete number is its denomination. Thus in 9 men, the denomination is men. Numbers are either simple or compound. A simple number is a single number, either abstract or concrete, as 3, 7 men, etc. A compound number is one made up of two or more numbers of different denominations, as 3 tons, 4 hundredweight, 36 pounds. There are four primary operations of arithmetic, namely, addition, subtraction, multiplication, and division. They are called the Fundamental Rules. ADDITION OF SIMPLE NUMBERS. The sign of addition (+) is called plus, and means more; when placed between two numbers it shows that they are to be added; thus 4+2 means that 4 and 2 are to be added together. The sign of equality (=) denotes that the quantities between which it stands are equal; thus the expression 4+2=6 means that the sum of 4 and 2 is 6. It is read 4 plus 2 equals 6. Addition is the process of finding the sum of two or more numbers. The number obtained by addition is called the sum or amount. Like numbers only can be added. Unlike numbers can not be added. 8 men can be added to 5 men; but 8 men can not be added to 5 ships. For addition, place the numbers to be added directly under each other, taking care to place units under units, tens under tens, hundreds under hundreds, and so on. When the numbers are thus written, the right-hand figure of one number is placed directly under the right-hand figure of the one above it, making the right-hand side of the column to be added in a line. Proceed as in the following example: Example: What is the sum of 242, 121, 32, 1, and 303? Solution. 242 121 32 1 303 Sum- 699 Ans. Explanation: After placing the numbers in the proper order begin at the bottom of the right hand column and add; thus, 3 1 2129, 3+1+2+1+2=9, the sum of the numbers in the units column. Place the 9 directly beneath as the first, or units figure in the sum. The sum of the numbers in the next or tens column equals 9 tens, which is the second or tens figure in the sum. The sum of the numbers in the next, or hundreds column, equals 6 hundreds, which is the third or hundreds figure in the sum. The sum or the answer is 699, or six hundred and ninety-nine. Example: What is the sum of 526, 98, 7,654, 3, 452, and 51? Solution. 526 98 7654 3 452 51 24 260 1500 7000 Sum.. 8,784 Ans. Explanation: The sum of the numbers in the first or units column is 24 units, i. e., two tens and four units. Write 24 as shown. The sum of the numbers in the second or tens column is twenty-six tens, or 260. Write 260 underneath the 24, as shown. The sum of the numbers in the third or hundreds column is 15 hundreds or 1,500. Write 1,500 under the two preceding results, as shown. only 1 number in the fourth or thousands column, 7, which represents 7,000. Write 7,000 under the three preceding results. Adding these four results the sum is 8,784, which is the sum of 526, 98, 7,654, 3, 452, and 51. The addition may also be performed as follows: There is 526 98 7654 3 452 51 Sum.. 8,784 Ans. Explanation: The sum of the numbers in the units column 24, or two tens and 4 units. Write the 4 units as the first or right hand figure in the sum. Reserve the two tens and add them to the figures, in the tens column, making 28 in the tens column; write the 8 as the second figure in the sum. Reserve the 2 and add it to the figures in the hundreds column, making 17 in the hundreds column; write the 7 as the third figure in the sum. Reserve the 1 and add it to the figures in the thousands column, making 8 in the thousands column; write the 8 as the fourth or last figure in the sum. Then the sum of 526, 98, 7,654, 3, 452, and 51 is 8,784. RULE: Begin at the right, add each column separately, and_write the sum, if it be only one figure, under the column added. If the sum of any column consists of two or more figures, put the right hand figure of the sum under that column and add the remaining figures or figure to the next column. PROOF: To prove addition add each column from top to bottom, and if you obtain the same result by adding from bottom to top the work is probably correct. 1. Find the sum of: EXAMPLES FOR PRACTICE. (a) 14,204 +8,173+1,065 +10,042. Answer, 33,484. (b) 6,354+2,145 +20,042 +1,111+5,000. Answer, 34.652. ADDITION OF SIMPLE NUMBERS. 1. A week's record of coal used in a steam launch is as follows: Monday, 1,800 pounds; Tuesday, 1,665 pounds; Wednesday, 1,725 pounds; Thursday, 1,690 pounds; Friday, 1,648 pounds; Saturday, 1,020 pounds. How much coal was burned during the week? Answer, 9,548 pounds. 2. A steam pump pumps out of a cistern in one hour 4,200 gallons, in the next hour 5,420 gallons, and in 45 minutes more an additional 3,600 gallons, when the cistern becomes empty. How many gallons were in the cistern at first? Answer, 13,220 gallons. 3. What is the total cost of a steam plant, the several items of expense being as follows: Steam engines, $900; boiler, $775; fittings and connections, $225; erecting the plant, $125; engine house, $.50? Answer, $2,675. 4. A man bought an automobile for $3,500, and sold it at a gain of $173. What was the selling price? Answer, $3,673. 5. The following stores were received aboard a ship: Two Mason reducing valves, at $75 each; 1 bale of waste, at $4; 50 gallons machine oil, valued at $17; 2 hundredweight of charcoal, at $1 a hundredweight; 1 injector, at $65; 1 steam hose, at $18. What was the total value of the stores? Answer, $256. 6. In 1905 the United States Navy consisted of 265 vessels fit for sea service, 47 under construction, 15 out of commission undergoing repairs. What was the total number of vessels? Answer, 327. 7. A fleet of ships in making a cruise steamed 235 miles the first day, 260 miles the second day, 245 miles the third day, 186 the fourth day, 270 the fifth day, and 13 miles the last day. What was the total number of miles steamed? Answer, 1,209 miles. 8. A battleship sailed from New York to Jamaica, a distance of 1,457 miles, in 5 days; from Jamaica to Buenos Aires, a distance of 5,166 miles, in 19 days; from Buenos Aires to Cape Town, 3,778 miles, in 14 days; from Cape Town to Gibraltar, 5,124 miles, in 19 days; from Gibraltar to New York, 3,206 miles, in 15 days. What was the total number of miles steamed and how many days did it take? Answer, 18,731 miles; 72 days. 9. The evaporators on a battleship made the following amounts of water during four days' running: First day, 2,350 gallons; second day, 2,180 gallons; third day, 1,945 gallons, and fourth day, 2,575 gallons. What was the total number of gallons? Answer, 9,050 gallons. 10. In weighing coal in a box containing 16 cubic feet of coal during a recent coaling on a man-of-war, the weights of the boxes ranged as follows: 1,150, 1,045, 1,215, 963, 1,189, 1,201, 1,071, 899, 1,104, 972. How much did the 10 boxes weigh? Answer, 10,809 pounds. SUBTRACTION Subtraction is the process of finding how much greater one number is than another. The greater of the two numbers is called the minuend. The smaller of the two numbers is called the subtrahend. The number left after subtracting the subtrahend from the minuend is called the difference or the remainder. The sign of subtraction is . It is read minus, and means less. Thus 12-7 is read 12 minus 7, and means that 7 is to be taken from 12. Example: From 7,568 take 3,425. Solution. minuend 7, 568 subtrahend 3, 425 remainder 4, 143 Ans. Explanation: Begin at the right hand, or units column, and subtract in succession each figure in the subtrahend from the one directly above it in the minuend, and write the remainders below the line. The result is the entire remainder. When there are more figures in the minuend than in the subtrahend and when some figures in the minuend are less than the figures directly under them in the subtrahend, proceed as in the following example: Example: From 8,435 take 844. Solution. minuend 8, 453 subtrahend 844 remainder 7,609 Ans. = Explanation: Begin at the right hand, or units column, to subtract. We can not take 4 from 3, and must therefore borrow 1 from 5 in the tens column and annex it to the 3 in units column. The 1 ten, or 10 units, when added to the 3 in units column 13 units. 4 from 13=9, the first, or units, figure in the remainder. Since we borrow 1 from 5, only 4 remains; 4 from 4=0, the second, or tens, figure. We can not take 8 from 4, so borrow 1 thousand, or 10 hundreds, from 8; 10 hundreds +4 hundreds = 14 hundreds. 8 from 14-6, the third, or hundreds, figure in the remainder. Since we borrow 1 from 8, only 7 remains, from which there is nothing to subtract; therefore, 7 is the next figure in the remainder or answer. The operation of borrowing is placing 1 before the figure following the one from which it is borrowed. In the above example the 1 borrowed from 5 is placed before 3, making it 13, from which we subtract 4. The 1 borrowed from 8 is placed before 4, making 14, from which 8 is taken. Example: Find the difference between 10,000 and 8,763. Solution. minuend 10, 000 subtrahend 8,763 remainder 1,237 Ans. Explanation: In the above example we borrow 1 from the second column and place it before 0, making 10; 3 from 10-7. In the same way we borrow 1 and place it before the next cipher, making it 10; but as we ave borrowed 1 from this column and taken it to the units column, only 9 remains from which to subtract 6, 6 from 9 3. For the same reason we subtract 7 from 9 and 8 from 9 for the next two figures, and obtain a total remainder of 1,237. = RULE: Place the subtrahend (or smaller) number under the minuend (or larger) number, in the same way as for addition, and proceed a‹ in the above examples. PROOF: To prove an example in subtraction, add the subtrahend and the remainder. The sum should equal the minuend. If it does not, a mistake has been made, and the work should be done over. 1. From EXAMPLES FOR PRACTICE. (a) 94,278 take 62,574. (b) 310,465 take 102,141.. (c) 2,040+1,213+542 take 3,791 31,704 Ans. 208,324 Ans. 4 Ans. |