PUPILS’ OUTLINES FOR 
 
 HOME STUDY 
 
 IN CONNECTION WITH SCHOOL WORK 
 
 ARITHMETIC, PART I 
 
 Integers, Decimals, Fractions, Denominate Numbers 
 
 By D. E. AXELSTROM 
 
 Price , Fifteen Cents 
 
 Jennings Publishing Company 
 P. O. Box 17 , Brooklyn, N. Y. 
 
 Copyrighted 1911 by Jennings Publishing Co. 
 
2 
 
 ARITHMETIC-PART I 
 
 ARITHMETIC 
 
 This is the science of numbers showing their various properties and the art of 
 computing with figures, that is applying the properties of numbers to practical 
 examples arising in business or otherwise. 
 
 Arithmetic cultivates observation, imagination, reasoning, accuracy, and clear- 
 ness of expression. 
 
 Number is a unit or collection of units or a part of a unit. 
 
 Integers are whole numbers. 
 
 NOTATION AND NUMERATION 
 
 I. Notation is representing numbers by characters which may be letters, figures, or 
 words ; as, 
 
 Roman Notation — by Letters — V 
 Arabic Notation — by Figures — 5 
 Language — by Words — five 
 
 (a) Roman System of Notation — commonly used before the Arabic system, con- 
 
 sists of 7 letters — 
 
 1 = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000 
 
 Rules: 
 
 1. If the same letter or one of less value follows another, add the values of both, 
 
 as XX=10+ 10 or 20. XV=10+5 or 15. 
 
 2. If a letter of greater value follows another subtract the value of the less from 
 that of the greater; as, XC=10 subtracted from 100 or 90. 
 
 3. A bar over a letter multiplies it by a thousand; as, V = 5 x 1000 or 5000. 
 The same letter should not be repeated more than 3 times in succession; as, 4 use 
 
 IV not IIII ; for 900 use CM not DCCCC . 
 
 (b) Arabic System of Notation. — By the 16th century, this had come to be gen- 
 erally used among most civilized nations. It originated with the Arabs of 
 India some 2000 years ago. It consisted of the nine characters called 
 digits — 1, 2, 3, 4, 5, 6, 7, 8, 9, but later the 0 was added making ten 
 characters. 
 
 It is also known as the decimal system meaning that the increase and decrease is 
 tenfold. 
 
 The place value of the figures, makes it valuable, for instance in 5 51 500 
 
 the same figure 5 has different values according to the place it occupies: in the first, 
 it is 5 units; in the second number the 5 means 5 tens or 50+1=51, etc. ; whereas in 
 the Roman system to show 5 and 51 we must use entirely different characters to 
 indicate each 5, for the numerals have no place value; as, V=5 and L 1=51. 
 
 The Arabic System is a great improvement on the Roman system as is shown by 
 multiplying in each system. 
 
3 ns 
 
 u. ^ 
 
 ARITHMETIC-PART I 
 
 3 
 
 II. Numeration is reading numbers. 
 
 TABLE OF NUMERATION 
 
 Beginning at the right a figure in the 
 
 1st place 
 
 is 
 
 called 
 
 units 
 
 1 
 
 2nd 
 
 i 4 
 
 i i 
 
 
 tens 
 
 units period. 
 
 3rd 
 
 i 4 
 
 i < 
 
 4 ( 
 
 hundreds 
 
 J 
 
 4th 
 
 4 4 
 
 < t 
 
 4 4 
 
 thousands 
 
 
 5th 
 
 i 4 
 
 < * 
 
 4 4 
 
 ten-thousands 
 
 y thousands period. 
 
 6th 
 
 < ( 
 
 * < 
 
 4 4 
 
 hundred-thousands 
 
 J 
 
 7th 
 
 ( < 
 
 4 4 
 
 4 4 
 
 millions 
 
 - i 
 
 8th 
 
 4 4 
 
 4 4 
 
 4 4 
 
 ten-millions 
 
 [ millions period. 
 
 9th 
 
 4 < 
 
 4 4 
 
 4 4 
 
 hundred-millions 
 
 J 
 
 10th 
 
 1 4 
 
 .4 4 
 
 “ 
 
 billions 
 
 i 
 
 Uth 
 
 i 4 
 
 4 4 
 
 4 4 
 
 ten-billions 
 
 > billions period. 
 
 12th 
 
 < * 
 
 4 4 
 
 4 4 
 
 hundred-billions 
 
 1 
 
 J 
 
 13th 
 
 
 4 4 
 
 4 4 
 
 trillions 
 
 
 14 th 
 
 4 4 
 
 4 4 
 
 4 4 
 
 ten-trillions 
 
 y trillions period. 
 
 15th 
 
 4 4 
 
 4 4 
 
 4 4 
 
 hundred-trillions 
 
 1 
 
 J 
 
 For ease in reading and writing numbers, the different places have been grouped 
 in threes, called periods, as shown above, and separated from each other by commas ; 
 as, 67,894,325. 
 
 Rule.— Numeration: 
 
 A. Separate into the periods. Begin at right, count off 3 places, and place a 
 comma after each set of 3’s except the last. 
 
 B . Begin at the left and read the figures in each period as though they stood alone ; 
 and then add the name of that period ; as, 67 million, 894 thousand, 325 units ; 
 but the name units is omitted after the last period. 
 
 Rule,— Notation: 
 
 A. Begin at the left and write each period in order, placing commas after each, 
 filling in the 3 figures required for each period ; if any are omitted supply 
 ciphers ; as, in 37 million 506, write 37 with a comma after it for millions — the 
 next period in order towards the right is thousands; but no figures are given 
 for thousands, so supply 3 ciphers for that period and place a comma after 
 it— the next period in order is units and we have 506 giving us 37,000,506. 
 
 DECIMALS 
 
 The period is used to indicate a decimal and to separate this part from the whole 
 number. 
 
 The places right of the decimal point are the decimal places, the first being 
 tenths, the others following in the same order as for the whole numbers, but each 
 word ends in ths. 
 
 Numeration — Read the decimal as though it were a whole number and then give 
 it the name of the farthest right hand place; as, .0409 would be 409 ten-thousandths — 
 0 being in tenths place, 4 in hundredths, 0 in thousandths, and 9 in ten-thousandths 
 place. 
 
4 
 
 ARITHMETIC-PART I 
 
 1. If we start with unit 1, it will take 10 of those to make 1 for the next place 
 or 1 ten ; as, 10. 
 
 10. The point or period is a mark of separation between the whole and its 
 parts. 
 
 1. Starting with the unit one again, if we divide it by 10 we will get ^ or 1 
 tenth giving us 
 
 . 1 the first decimal place or part. Proof — it takes 10 of the . 1 to make one of 
 the next higher order or 1 unit. 
 
 Rules: 
 
 A. Ten of any order, whether whole or decimal, makes 1 of the next higher 
 order. 
 
 B. Moving the decimal point 1 place to the right multiplies the number by 10; 2 
 places, by 100; 3, by 1000, and etc. 
 
 C. Moving the decimal point 1 place to the left divides the number by 10, etc. 
 
 D. If we annex ciphers to the decimal places we do not change the value as each 
 figure remains in the same place. 
 
 E. If we prefix a cipher in the decimal before the first figure, we divide it by 10 
 for that moves each figure one place to the right and is therefore only Jg as 
 great as before; as .1 — .01 — one now is in hundredths place and is only ^ as 
 great as it was before. 
 
 ADDITION 
 
 It is the process of finding one number which is equivalent to two or more num- 
 bers taken together, and the answer obtained is called the sum or amount. 
 
 Addends — the numbers combined to give the sum. 
 
 Sign of addition is read plus. 
 
 Rules: 
 
 A. Only numbers of the same kind can be added. 
 
 B. It makes no difference in what order the numbers are combined. 
 
 C. If a column comes to 9 or less put it down and there is nothing to add to the 
 next column but if it comes to 10 or more put down the units figure only and 
 add the tens to the next column, 
 
 Proof — Add each column in the reverse order. 
 
 Method — Write the numbers so that all figures of the same order will be in the same 
 column, for instance all units must come in the same column, all tenths in 
 the same column, etc. 
 
 When numbers have all been written in columns one under the other begin 
 at the right to add. In the example given, 5 and 5 are 10, put down the 0 
 9 . 25 and add the 1 to the next column, for the 10 of any order makes 1 of the next, so 
 
 2.75 we have none left for units order but one for the next order. Add the next 
 
 .80 column and we get 18, which is the same as 10-f-8, the 10 will equal 1 of the 
 
 next order so we add 1 to the next column and put down the 8 which we had 
 
 12.80 left for the second column. As soon as the decimal point is reached put it 
 
 down in the answer. Add the third column and we have 12; following the 
 directions above till finished we have 12.80 for the answer or sum. 
 
 If a $ occurs at the top of the column all the numbers in that column are considered 
 dollars. 
 
 In adding it is easier to group the digits in convenient combinations rather than to 
 add one number after the other in the order they occur. 
 
Arithmetic— part i 
 
 5 
 
 SUBTRACTION 
 
 The process of finding how much one number exceeds another or the difference 
 between two numbers is subtraction. 
 
 The sign of subtraction is — , read minus, and shows that the number following 
 the sign is to be taken away from the number preceding it. 
 
 Minuend — is the larger number from which we subtract. 
 
 Subtrahend is the smaller number which is to be subtracted. 
 
 The difference or remainder is the answer. 
 
 Rule — Numbers subtracted must be of the same kind, and the answer will be the 
 same kind. 
 
 Proof — The difference added to the smaller number, or subtrahend, should equal 
 the larger number, or minueud. 
 
 Principle — If we add or subtract the same number from both minuend and sub- 
 trahend the answer or difference is not changed. 
 
 Method — Write numbers under each other keeping the figures of one order in the 
 same column, and decimal points, if any, in a straight vertical line, the same as in 
 addition. 
 
 52.4 Begin at the right. In the example given, 3 tenths from 4 tenths gives 1 
 
 17.3 tenth— put down the decimal point. We cannot take 7 units from 2 units 
 
 so we take 1 ten from the five leaving that 4 and add the 1 ten making 10 
 
 35.1 units to the 2, giving 12 units, then 7 from 12 is 5' units and 1 ten from 4 
 
 tens is 3 tens or 35.1, the Difference. 
 
 AUSTRIAN METHOD 
 
 This is sometimes known as the Additive Method of Subtraction for we find what 
 number added to the subtrahend will produce the minuend. This depends on the 
 principle given above. 
 
 52.4 Beginning at the right we find that .1 added to .3 will give the minuend .4, 
 
 17.3 so we write .1 in tenth’s column, 5 added to 7 gives 12 so write the 5 in 
 
 units column and carry the one ten to the next column — for we added 10 
 
 35.1 to the minuend to make 12, therefore according to the above principle we 
 
 must add one ten to the subtrahend also. 3 added to one, plus the one 
 ten we had to carry gives 5, so we write 3 in tens place. 
 
 MULTIPLICATION 
 
 Is the process of taking a number as many times as there are units in another. 
 The result obtained is the product. 
 
 Multiplicand is the number to be taken or multiplied. 
 
 Multiplier is the number by which we multiply and shows how many times the 
 other number is to be taken. 
 
 The sign is X. and is read times when it follows the multiplier, or multiplied by 
 when it precedes the multiplier. 
 
 Factors are the numbers which multiplied together produce a given number. 
 Concrete Number is one with a name; that is, the number refers to some particular 
 object; as, 7 books. 
 
 Abstract Number is one without a name ; that is, the number does not refer to 
 any particular object; as 7. 
 
6 
 
 ARITHMETIC-PART I 
 
 Principles: 
 
 1. Multiplicand and the Product will be of the same kind or have the same name, 
 if there is any name. 
 
 2. Multiplier should always be considered abstract, without a name or without 
 being of a certain kind. 
 
 3. The figures in the product will always be the same no matter which number is 
 taken as the multiplier. 
 
 Method— Write the multiplier under the multiplicand. Begin at the right to 
 537 multiply. 7 X 7=49. Put down the 9 units and add the 4 tens to the 
 
 X 27 next product. 7X3 tens=21 tens plus the 4 tens gives 25 tens or 5 tens 
 
 and 2 hundreds. Put down the 5 in tens’ place and add the 2 hundreds 
 
 3759 to the next product. 7X5 hundreds plus 2 hundreds=37 hundreds or 7 
 
 1074 hundreds and 3 thousands and, being the last multiplication, write the 7 
 
 in hundreds’ place and the 3 in thousands’ place. This gives 3769 as 
 
 14,499 our partial product. 
 
 But we had 27 as a multiplier giving 7 units and 2 tens. Having gotten the par- 
 tial product by 7, we multiply same as before by 2 and obtain 1074, but the 2 stands 
 for tens, so our partial product must be 1074 tens. 
 
 Adding our partial products, we have the entire product, 14,499. 
 
 Rule — The first right-hand figure in any partial product should be placed in the 
 same column directly under the figure being used as the multiplier, z. e . , multiplying 
 by a figure in thousands’ place, the first figure in that partial product should be placed 
 in the third column or under the thousands figure. 
 
 Ciphers at the End of the Factors 
 
 If either multiplier or multiplicand, or both, have ciphers at the right of the 
 number, multiply the numbers as though there were no ciphers and in the answer 
 annex at the right as many ciphers as there were ciphers in both factors. 
 
 Adding 1 cipher multiplies by 10 ; adding 2 ciphers multiplies by 100, etc. 
 
 1500 X 210 = ? 
 
 15100 
 
 2ll0 
 
 15 
 
 30 
 
 315,000 Ans. 
 
 Multiplication of Decimals 
 
 Multiply as though whole numbers were given, then in the answer begin at the 
 right and count off as many decimal places as there are decimal places in both fac- 
 tors and there place a decimal point. If there are not sufficient figures, supply 
 ciphers. 
 
 1.5 15 by 21 gives 315 and one decimal place in the multiplicand plus three 
 
 •021 i n the multiplier gives 4 decimal places for the answer. Not having suf- 
 
 15 ficient figures beginning at the right to count, supply one cipher and put 
 
 — — . down the decimal point. 
 
 .0315 
 
 15 .2 is the same as 2 X -1 ; hence multiplying by 2 we get 30, and then by .1 
 
 . 2 we move the decimal point 1 place to the left. 
 
 iTo 
 
 Rule — Multiplying by .1 moves the decimal point one place to the left; multiply- 
 ing by . 01 moves the decimal point 2 places to the left. (Taking . 1 of the number is 
 really dividing by 10 — see Rule C, p. 5). 
 
ARITHMETIC-PART 1 
 
 1 
 
 DIVISION 
 
 Is the process of finding how often one number is contained in another. 
 
 Divisor — is the number by which we divide. 
 
 Dividend — is the number divided. 
 
 Quotient — is the answer obtained after dividing. 
 
 Remainder — is the number left over from the dividend if the divisor is not con- 
 tained in the dividend an exact number of times. The remainder is a part of the 
 dividend and should always be less than the divisor. 
 
 The Sign of division is -f-, read divided by and means that the number before 
 the sign is to be divided by the one after it ; as, 6 3 = 2. 
 
 Division is also shown by expressing it in a fractional form the dividend with 
 
 the divisor under it and a straight line between ; 6 dividend ^ 
 
 — — ... =2 quotient . 
 
 3 divisor 2 
 
 The divisor sometimes is put in front of the dividend with a curved line between 
 and a straight line is placed over the dividend to separate it from the answer ; as, 
 2 quotient 
 
 divisor 3)6 dividend. 
 
 When the dividend and the divisor are like numbers, that is of the same kind or 
 name, the quotient is abstract; as 62 rds -r- 32 rds. = 2. 
 
 Short Division- If the work of dividing may be done mentally, the divisor, as a 
 rule, not exceeding 12, only the answer and not the work is put down, as, 
 
 1061 
 
 6)637 or 6)687 
 
 Beginning at the left, 6 is contained in 600 one hundred times, so write the 1 in 
 hundreds’ place or directly over the 6; 6 is contained in 3 tens, no tens times, so place 
 a cipher over tens’ place. We then have still remaining the 3 tens, which'are equal to 
 30 units — these, plus the 7 units, give 37 units — into which 6 is contained 6 times and 
 1 over, so place the figure 6 over units place and the remainder over the divisor and 
 add it to the whole number ; as, 106£ Ans. 
 
 In actual work the names of the places are omitted and wc simply say 6 into 6 
 once ; 6 into 3, no times and 3 over ; 6 into 37, six times and 1 over, or giving alto- 
 gether 106£. 
 
 Proof of Division — Multiply the divisor and the quotient, and, if there is a re- 
 mainder, add it to the product, and the result will be the dividend. 
 
 106 quotient 
 
 X6 divisor 
 
 636 
 
 -(-1 remainder 
 
 637 dividend 
 
 Short Methods of Division 
 
 1. To divide a whole number by 10, take off one place at the right; by 100, two 
 places, and the numbers not cut off become the quotient, and the figures cut off, the 
 remainder. 
 
 100)63 | 72 gives 63 with remainder 72 — expressed 63j 7 o 2 0 , or decimally as 63.72. 
 
 ^ 2. To divide a decimal by 10, move the decimal point one place to the left . by 
 100, two places to the left, etc. If there are not sufficient figures, prefix ciphers, 
 6.37 — r- 100 = .0637. Answer— .0637. 
 
8 
 
 ARITHMETIC-PART I 
 
 Division 
 
 21 into 6 no times, into 63 — three times — place in the quotient, and 
 multiply — 3 times 21 gives 63, and, subtracting, we have no remainder. 
 Bring down the next figure, 7. — 21 into 7, no times, placing 0 in quo- 
 tient; bringdown the next figure one, 1 — 21 into 71, 3 times, place the 3 
 in the quotient — 3 X 21 = 63, 63 from 71 = 8, the remainder or 2 8 T . 
 Ans. 303 ft . 
 
 Division of Decimals 
 
 2.54 + 
 
 Always make the divisor a whole number, in this case by multiplying 
 by 10. In order not to change the quotient, the dividend must be 
 multiplied by the same number as the divisor was. Multiplying by 
 10, move the decimal point 1 place to the right. 
 
 Above the line place the decimal point directly over the decimal point in the dividend 
 and proceed to divide the same as in whole numbers. If there is a remainder, and 
 especially if there are several decimal places, a + is put at the end instead of the 
 remainder. 
 
 We find the decimal places in the answer are equal to the difference in the number 
 of decimal places in the dividend and the divisor; as, in the dividend 6.371 there are 
 3 decimal places and in the divisor 2.5 there is one decimal place, making a difference 
 of 2 decimal places - the same number found in the answer. 
 
 The figure in the quotient or answer each time must be placed directly over the 
 last figure brought down in the partial dividend. 
 
 Principles of Division: 
 
 Multiplying or dividing both dividend and divisor by the same number does not 
 change the quotient, as 
 
 50 -4- 10 = 5 
 
 (50 X?)h- (10 X 2)= 100 -- 20 = 5 
 (50 2) -s- (10 H- 2)= 25 -f- 5 = 5 
 
 Multiplying the dividend multiplies the quotient by the same amount, as 
 50 -r- 10 = 5 
 
 (50 X 2) -s- 10 = 100 10 = 10 or 5 X 2 
 
 Multiplying the divisor divides the quotient by the same amount. 
 
 50 - 5 - 10 = 5 
 
 50 (10 X 2) = 50 20 = 2| or 5 2 
 
 Dividing the dividend divides the quotient by the same amount. 
 
 50 -s- 10 = 5 
 
 (50 -r- 2) -r- 10 = 25 -r- 10 = 2| or 5 2 
 
 Dividing the divisor multiplies the quotient by the same amount. 
 
 50 — 10 = 5 
 
 50 -v- (10 2) = 10 or 5 X 2 
 
 2.5.) 6.8.71 
 W _50_ 
 137 
 125 
 121 
 100 
 21 
 25 
 
 303 ft 
 21)6371 
 63 
 
 71 
 
 63 
 
 _8 
 
 21 
 
ARITHMETIC- PART I 
 
 9 
 
 USE OF SIGNS IN COMBINATION 
 
 Rules: 
 
 1. Use - 4 - and X in the order they occur from left to right. 
 
 2. Use + and — in the order they occur from left to right. 
 
 3. If all 4 signs occur -f- X H in' an Y combination apply 1st Rule until all 
 
 and X are exhausted, then apply Rule 2 until all -f- and — signs are exhausted. 
 
 8 + 10— 6-3X7=? 
 8 + 10 — 2 X? 
 
 8 + 10 — 14 
 
 18 — 14 
 
 4 
 
 4. If “ of” occurs in an expression it indicates a closer relation than a sign, so 
 perform the multiplication indicated by the “ of ” first, then apply Rule 1 and then 
 Rule 2 if necessary. 
 
 Parenthesis (4 + 2) X 7- This means that the operation indicated within the 
 parenthesis is to be performed first and regarded as one whole ; so 6 X 7 = 42. 
 
 Divisibility of Numbers: 
 
 Figures, or any of the ten Arabic characters making up a number are called 
 digits ; as, in 23 two and three are each a digit. 
 
 A number is divisible by 
 
 2 — if the units digit can be divided by 2 or 0. 
 
 3 — if the sum of the digits can be divided by 3. 
 
 4— if the two right hand digits are 0, or if they can be divided by 4. 
 
 5 — if units digit is 0 or 5. 
 
 9— if the sum of the digits can be divided by 9. 
 
 FACTORING 
 
 If the same number is contained an exact number of times in two or more numbers, 
 that number is a common divisor of all the numbers. 
 
 If it is the greatest number that will be contained exactly in all the numbers, it is 
 the Greatest Common Divisor (G. C. D.) 
 
 Prime Number - is one whose factors are 1 and itself. 
 
 Composite Number— is one that has other factors than one and itself. 
 
 Even Number— is one divisible by 2. 
 
 Odd Number— is one not divisible by 2. 
 
 Factoring —Separating numbers into factors. 
 
 Method 
 
 2 |_144 
 3 |_72 
 3 |_24 
 2 | 8 
 2J 4 
 2 
 
 Ans. 2 4 X 3 2 
 
 Divide the number by one prime factor and the 
 quotient will be the other — divide this in turn by a 
 prime factor and so on till the division is finished. 
 All the divisors and the last quotient are the prime 
 factors, for if they are multiplied together they will 
 produce the number. 
 
 Exponent — The little figure 4 and 2 written above 
 and to the right of the factors, indicate how many 
 times those factors have been used. 
 
10 
 
 ARITHMETIC-PART I 
 
 Greatest Common Divisor: 
 
 Find the G. C. D. of 96 and 120. 
 
 Method — Factoring 96 we have 2 5 X 3. 
 
 Factoring 120 we have 2 3 X 3 X 5. 
 
 So the Common Divisors of both are 3 and 2 3 , or (3 X 2 X 2 X 2) = 24, the G. C. D. 
 
 Rule — Find the factors common to both numbers and multiply them together for 
 the G. C. D. 
 
 Old Method by Long Division is often used if the numbers are large. 
 
 1 
 
 96)120 
 
 96 4 
 
 G. C. D. 24) 96 
 96 
 
 Rule: 
 
 Divide the greater number by the less, then the last divisor by the last remainder 
 until none remains. If there are three numbers take the last divisor and divide it into 
 the third number, continuing as before — the last divisor being the Greatest Common 
 Divisor (G. C. D.). 
 
 The principle involved being that a factor of two numbers is also a factor of their 
 sum and difference. 
 
 Least Common Multiple: 
 
 Multiple— A number containing another is a multiple of the number. 
 
 Common Multiple — Is a number containing each of two or more numbers. 
 
 Least Common Multiple (L. C. M.) of two or more numbers is the least number 
 that contains each of the two or more numbers. 
 
 Method — Least Common Multiple: 
 
 I. Find the prime factors of each number given. The product of the highest 
 power of each factor (or the greatest number-of times it is used as a factor in any one 
 number) will be the L. C. M. 
 
 Example— What is the L. C. M. of 28, 21 and 15. 
 
 28 = 2X2X7 
 
 21 = 7X3 (L. C. M. contains 2 twice, 7 once, 3 once, 
 
 15 = 8X5 and 5 once.) 
 
 2X2X7X3X 5 = 420 L. C. M. 
 
 II. Write the numbers in a horizontal line and divide by any prime factor which 
 will exactly divide two or more of the given numbers. Directly under the numbers 
 copy any undivided numbers and the quotients of the others. 
 
 Use these as a new set of numbers and divide as before until nothing but the 
 prime factors remain ; the product of all the divisors, and of the numbers on the last 
 line, will give the L. C. M. 
 
 Example — Find the L. C. M. of 28, 21 and 15. 
 
 3)28 21 15 
 
 7)28 7 5 3 X 7 X 4 X 1 X 5 = 420 L. C. M. Answer. 
 
 4 1 5 
 
 If the different numbers have no common factor the Least Common Multiple will 
 be the product of the different numbers. 
 
 CANCELLATION 
 
 Dividing both dividend and divisor by the same number does not alter the quotient, 
 
ARITHMETIC— PART I 
 
 11 
 
 so if the equal factors in both dividend and divisor be removed it shortens the work, 
 and this is known as Cancellation. 
 
 4 
 
 3 3 10 
 
 W X U X W 36 
 
 MX 0 ~ i ~ 
 
 i 9 
 
 1 l 
 
 The factor 5 is canceled or removed from 80, leaving 16, and from 45 leaving 9 ; 
 9 is canceled from 27 leaving 3, and from 9 leaving 1 ; 8 is canceled from 24 leaving 
 3 , and from 32 leaving 4; 4 is canceled from 16 leaving 4, and from 4 leaving 1. The 
 numbers remaining in the dividend are prime to those remaining in the divisor, so we 
 can cancel no further. Then perform the operation indicated by the signs and we have 
 
 86 
 
 3 X 3 X 4 = 36 for the dividend, and the divisor IX l = l,° r — — 86 . Answer. 
 
 FRACTIONS 
 
 Fractions are the equal parts into which a unit is divided. 
 
 There are two parts to each fraction, called its terms. They are: 
 
 1. Denominator which is the number written below the line and shows into how 
 many parts the unit has been divided, by naming them, as in f, four shows that the 
 unit has been divided into four equal parts. 
 
 2; Numerator which is the number written above the line and shows the number of 
 those parts that have been used; as in three gives the number of fourths taken. 
 
 Reading a Fraction — Give the numerator or number of parts first, then the 
 denominator or name of the parts; as f, three-fourths. 
 
 Proper Fraction — One whose numerator is less than its denominator, indicating 
 that it is a portion of the unit ; as |. 
 
 Improper Fraction— One whose denominator is less than the numerator, making 
 its value greater than a unit, hence is not properly spoken of, as part of a unit; f = 1 *. 
 
 Mixed Number— Is a whole number and a fraction; as 2f, read 2 and three-fourths. 
 
 Decimal Fraction is one whose denominator is not put down, but is 10 or a 
 power of 10. It begins with a decimal point. 
 
 Principles of Fractions: 
 
 I. Multiplying the numerator or dividing the denominator increases the value of 
 the fraction. 
 
 4^2 g The denominator 12 indicates that the unit has been divided into 
 
 a - ^lS - 12 equal parts. The numerator 4 indicates that 4 of the 12 parts 
 
 have been taken. Multiplying the numerator by 2 gives 8 ; the 
 parts remaining the same size, we have increased the value of the fraction by multi- 
 plying the numerator. 
 
 4 4 The denominator 12 indicates that the unit has been divided into 
 
 b- 12 - 4 - 2 “ 6 12 equal parts, and the numerator 4 indicates that 4 of the ]2 
 
 equal parts have been taken. Dividing 12 by 2 gives 6 , but if 
 the unit has only been divided into 6 parts instead of 12 , each part must be twice as 
 large, and taking the same number of parts, that in this case are larger, the value of 
 the fraction must have increased. 
 
 II. Dividing the numerator or multiplying the denominator of a fraction decreases 
 the value of the fraction. 
 
 4 2 2 We have the unit divided into 12 equal parts and 4 of these taken ; 
 
 a * 12 but dividing the 4 by 2 we have only 2 parts taken instead of 4, 
 
 and as the size of the parts are the same, the value has been 
 decreased. 
 
12 
 
 ARITHMETIC-PART I 
 
 4 4 Twelve indicates the number ol' equal parts into which the unit 
 
 k* lax 2 = 24 _ has b een divided; multiplying by 2 gives 24. If the unit has 
 been divided into 24 parts instead of 12, each part can only be 
 half the size, and taking 4, the same number of them, the value of the fraction has been 
 decreased. 
 
 III. Multiplying or dividing both terms of a fraction by the same number does not 
 change the value. 
 
 4^2 8 Denominator 12 indicates that the unit has been divided into 12 
 
 a " 12 X % ~2+ equal parts, and the numerator that 4 of those parts have been 
 
 taken. Multiplying each term by 2 gives 2 8 ? . The denominator 
 indicates that the unit has been divided into 24 equal parts instead of 12, so each one 
 must be just half the size. If 8 parts are taken instead of 4, we have twice as many 
 parts, but each part is half the size, so we have not changed the value. 
 
 4 g 2 Instead of 12 equal parts the unit has been divided into 6 equal 
 
 12 -i- 2 ~ 6 parts, so each part must be twice as large, but we have taken 
 
 only half as many, 2 instead of 4, so the value has not been 
 
 changed. 
 
 IV. Adding the same number to both terms of a fraction increases its value. 
 
 412 6 Instead of 12 equal parts the unit has been divided into 14 equal 
 
 parts, so the size of the parts must be smaller, but instead of 4 
 parts being taken, G of the smaller parts have been taken. Sub- 
 
 b. 
 
 12 + 2 
 
 14 
 
 tracting we find that is 
 
 2 ^ 4 
 
 “ 2 i greater than 
 
 84 
 
 6 
 
 14 
 
 4 
 
 12 
 
 36 
 
 28 
 
 8 2 
 
 84 ~~ 21 
 
 V. Subtracting the same number from both terms of a fraction decreases its vatue. 
 
 4 2 2 The un ^ h as been divided into 10 equal parts instead of 12, so 
 
 42 2 the parts must be larger ; but instead of 4 only 2 of the 
 
 larger parts have been taken. Subtracting we find that 
 2 2 , , 4 
 
 IT is IT less than 12 
 
 60 
 
 4 
 
 20 
 
 12 
 
 60 
 
 _ 2 _ 
 
 15 
 
 REDUCTION OF FRACTIONS 
 
 This is changing the form, but not the value of the fraction. 
 
 I. To change whole or mixed numbers to fractions ; as 2| to 4 ths. In one unit 
 there are 4 fourths, so in 2 there will be 2 X 4 or 8 fourths ; this plus the 3 fourths 
 given make y. 
 
ARITHMETIC— PART I 
 
 13 
 
 Rule — Multiply the whole number by the denominator, add the numerator and 
 write the whole result over the denominator. 
 
 II. To change fractions to whole or mixed numbers; as In one there are 4 
 fourths (|), so in 15 there will be as many as 4 is contained times in 15, or 3 and 3 
 fourths over = 31- 
 
 Rule— Divide the numerator by the denominator and the answer equals the whole 
 or mixed number. 
 
 III. To reduce to higher or lower terms. 
 
 Principle— Multiplying or dividing both numerator and denominator of a fraction 
 by the same number does not change its value. 
 
 a. To reduce to higher terms multiply both terms of the fraction by the 
 number which gives the desired denominator ; as f to 44 ths: 4 is contained in 44 eleven 
 times — multiply 3 by 11 and 4 by 11, thus obtaining af. Answer. 
 
 b. To reduce to lower terms — divide both terms of the fraction by the same 
 
 number. 
 
 75 -s- 3 25 + 5 5 
 
 105 -j- 3 ~ 35 h- 5 “ 7 
 
 We find that both terms are divisible by 3 and 5, and dividing by these we have f . 
 As both of these, 5 and 7, are prime to each other, it is in its lowest terms. 
 
 In reducing to lowest terms one may also divide both terms directly by their 
 
 „ ^ ^ 75 15 5 
 
 G. C. D„ as Jog -35 =~ 7 ~- 
 
 IV. To reduce common and decimal fractions. Fractions are called decimals 
 if they express a number of tenths, hundredths, etc., and are written .61, the denomi- 
 nator being omitted and the decimal point used instead. 
 
 To express decimally: Place a decimal point, then write the number and see 
 that there are as many places as there are ciphers in the denominator. If there are 
 not a sufficient number of figures, prefix ciphers to them; as, i0 5 00 — decimal point — 
 3 ciphers in the denominator indicates 3 decimal places; but there is only one 
 figure given, 5, so prefix 2 ciphers, giving .005, so that five stands in the place called 
 for by the denominator (thousandths). 
 
 a. To reduce decimals to fractions — .025 expressed as a fraction with a denomi- 
 nator, f § 0 0 , and reducing to lowest terms, gives 
 
 b. To reduce fractions to decimals ; as, |. The value of the fraction is found by 
 dividing the numerator by the denominator. 
 
 Divide 5 by 8 , annexing decimal ciphers to 5, gives 
 . decimals. 
 
 .625 
 
 8)5.000 
 
 as in the division of 
 
 i of 5 is the same as saying | of 50 tenths, which is six tenths, and the 2 tenths 
 remaining is equal to 2 hundredths, dividing that by 8 , etc. 
 
 V. To reduce fractions to the Least Common Denominator. 
 
 Fractions with the same denominators have Common Denominators. 
 
 If the Common Denominator is the L. C. M. of the given denominators, we have 
 the Least Common Denominator. (L. C. D.) 
 
 If this L. C. D. cannot be determined by inspection, factor the denominators, 
 multiply all prime factors the greatest number of times they occur in one factor, and 
 the result is the L. C. D. 
 
 Reduce to L. C. D. f, |, J. 
 
 The L. C M. of the denominators 4, 8 , 6 , is 24- Making 24 the denominator of 
 
14 
 
 ARITHMETIC— PART I 
 
 each fraction, we find 4 into 24 goes 6 times ; multiplying both terms by 6, we have 
 3 X 6 = 18 
 4X6 24 
 
 5 X 3 = 15 
 
 8X3 24 
 
 5 X 4 = 20 
 6X4 24 
 
 and 8 into 24 goes 3 times ; multiplying both terms by 3, we have 
 
 and 6 into 24 goes 4 times; multiplying both terms by 4, we have 
 4 8 15 2 0 Ans 
 
 2 4 ’ 24 * 24 
 
 Addition and Subtraction of Fractions 
 
 Rule — To add or subtract fractions, they must be of the same kind or similar 
 fractions ; that is, having the same or a common denominator. 
 
 12 
 
 g Fractions not being similar must be reduced to the same 
 
 3 denominator. 
 
 5§ 
 
 2 % 
 
 9i i n = ii 9 2 
 
 L. C. D. 12 is written, as indicated, above the column. 3 into 12, 4 times ; multi, 
 plying both terms, gives , but writing the 12 above, will avoid its repetition 
 for each fraction ; 4 into 12, 3 times, multiplying both terms, gives j * 2 3 4 * * 2 ; 6 into 12, 
 twice, multiplying both terms, gives 8 twelfths -f- 3 twelfths -j- 10 twelfths gives 
 ||, which, being reduced, gives 1 whole number and Y 9 2 . The fraction Y 9 2 can be re- 
 duced to the lowest terms by dividing both terms of the fraction by 3, giving f. 
 Add the whole number one to the other integers, giving in all 9|, Answer. 
 
 Reducing to a Common Denominator, we have 8, which 
 8 gives us respectively i and |, but | cannot be taken from 
 
 54 HT+ $ = 9 e< so take 1 f rom the integer 6, leaving that 5; the 1 taken 
 
 being equivalent to § ; this -f- 
 
 4 I 
 
 gives | 
 1 from 
 
 and 
 5 is 4. 
 
 ; from 
 Giving 
 
 and as 
 or sub- 
 
 k - ns * 4 % | gives |. Subtract the integers, 
 
 for the answer, 4|. 
 
 In examples where the fractional parts are expressed both as decimals 
 common fractions, change the common fractions to decimals before adding 
 tracting. 
 
 Multiplication of Fractions 
 
 I. Multiplying a fraction by an integer. 
 
 Multiplying the numerator or dividing the denominator, multiplies the fraction. 
 1X3= = 2| or 2 1 
 
 | 3 = | or 2| This may be simplified by expressing the 
 
 5 1 5 
 
 multiplication and canceling— X">j =~2 : 
 
 2 
 
 2 
 
 i 
 
 2 * 
 
 II. Multiplying a fraction by a fraction. 
 
 X sign is read of. This gives a compound fraction. 
 
 | X I = ? | multiplied by 1 gives §, so § X I equals | of § or 2 \, and § X I will 
 be seven times 2 2 5 or || reduced, giving T 7 2 . 
 
 ? 7 7 
 
 Simplified by cancellation -g X"^~ gives 
 
 4 
 
 R u | e — Multiply the numerators together for the product and the denominators 
 
 together for the product. 
 
ARITHMETIC— PART I 
 
 15 
 
 When multiplying by a fraction, we multiply by the numerator and divide by 
 the denominator, but the order makes no difference, as § of 21. We may say 2 times 
 i of 21, which would be two times 7, or 14. In the example, § of 11, it is easier to say 
 t of 2 X H> f° r 3 is not exactly contained in 11 ; so we have £ of 22 , or 7£. 
 
 III. Multiplying an integer or mixed number by a fraction. 
 
 a. Express mixed numbers as fractions before beginning and whole numbers as 
 fractions by making 1 the denominator. 
 
 1 
 
 11 4 2 22 
 
 2| X 4 X I =~J~X Y X 3 “ = 3 ^ = 7 3 - Answer. 
 
 1 
 
 Cancel where possible, multiplying the numerators for the product and the de- 
 nominators for the product. If the answer is an improper fraction, change it to a 
 mixed number. If the answer is a fraction not in the lowest terms, it shows that all 
 cancellation has not been done. 
 
 b. In multiplying a whole number by a mixed number, it is sometimes simpler 
 to use this method. 
 
 215 multiplied by |, as above, it would be \ of 215 X 3 ; so 
 multiplying 215 by 3 gives 645 and dividing by 4 gives 1611, 
 then multiply the whole numbers as usual, placing each partial 
 product under the figure being used as the multiplier, and 
 uniting this result with the result of multiplying by f , gives 
 79011 for the answer. 
 
 DIVISION OF FRACTIONS 
 
 I. To divide a fraction by an integer. 
 
 Rule— Divide the numerator or multiply the denominator of the fraction by the 
 integer. 
 
 4 4-^-2 2 (used if the numerator contains the divisor an exact 
 
 5 5 5 number of times). 
 
 7 7 7 (used if the numerator does not contain the divisor 
 
 9 ' 9X2 18 exactly). 
 
 Or by cancellation: 
 
 7 
 
 21 n X 1 7 
 
 5 ^ 9 — 5 x ? ” ' 15 
 3 
 
 215 
 
 36| 
 
 4)645 
 
 1611 
 
 1290 
 
 645 
 
 79011 
 
 II. To divide either a fraction or an integer by a fraction. 
 
 Rule — Change the denominators to a Common Denominator and divide the 
 numerators ; or invert the divisor and cancel. 
 
 11 ) 128 
 ll-/ T Ans. 
 
 a 5 5 15 20 15 3 
 
 * 8 6 = n ^ U “ 20 “ 4 
 
 b. _ 5 _ _L_JL V 8 __3_ 
 
 8 * 6 $ S' ft 4 
 
 4 
 
 The (a) method is sometimes used when a mixed number is to be 
 divided, as in example given, changing both to 4 ths, and dividing, 
 
 so that 128 fourths contains 11 fourths, 11 andy-r times. 
 
16 
 
 ARITHMETIC— PART I 
 
 A complex fraction is one whose numerator and denominator are either or both a 
 fraction. 
 
 a. 
 
 b. 
 
 ( 1 ) 
 
 ( 2 ) 
 
 ? Simplify the numerator, then the denominator, and divide 
 the results (numerator by denominator). 
 
 £ 
 
 6 
 
 7_ 
 
 6 
 
 3 
 
 _L _?L V 
 
 6 — 3? A 
 
 16 
 
 3 
 
 0 9 A 
 
 ~Y~ = jg- Answer. 
 
 1 
 
 Fractional Relations and Solutions 
 
 I. To find what part one number is of another. 
 
 a. What part of 3 is 2? Since 1 is £ of 3 ; 2 is 2 times £ of 3 or § of 3, that is 
 2 is § of 3. 
 
 b. What part of f is §? Since f = and £ = T 4 2 , then £ is the same part of 
 | that r «2 is of t 9 2 ; but that is the same part as the numerator 9 is of the numerator 4, 
 or 2\. The same answer is obtained by dividing f by §, which = | X f = |. or 2\. 
 
 II. To find the whole when a pa.rt is given. 
 
 a. If | of a number is 20, what is the number? If 20 is 2 fifths, one will be 
 \ of 20 or 10, and the whole § will be 5 times as much or 50. 
 
 b. § is f of what number? Since § is f of a number, \ would be \ of f , or |. 
 If i is £ the whole f = 7 times £ or ^ or 3£. Answer. 
 
 Analysis 
 
 Reasoning from something given to o?ie , and from one to the required quantity. 
 
 1. If it takes 7 boxes to fill 1 case, how many boxes are needed for 5 cases? 
 Solution — If 1 case requires 7 boxes, 5 cases will require 5 times as many, or 7 X 
 
 5 = 35 boxes. 
 
 2. If peaches are 3 for 10 cents, what will 15 cost? 
 
 Solution — If 3 cost 10 cents, one will cost £ of 10, or 3£ cents. If one costs 3J cents, 
 
 5 
 
 10 W 
 
 15 will cost 15 times that amount, or X "j - = 50 cents. Answer. 
 
 1 
 
 3. If 1 book cost 20 cents, how many books can be bought for $2.00? 
 
 Solution — If one cost $.20 we can buy as many as $.20 is contained m $2.00, or 10. 
 
 4. The box contains 75 blue envelopes and 25 white envelopes, what part is blue 
 envelopes? 
 
 Solution — 75 plus 25 gives the entire number, or 100. Out of the 100 we have 75 
 blue, or which is f of the box. 
 
 5. If 4| lbs. cost 80 cents, what will 7| lbs. cost? 
 
 Solution — If 4| lbs. cost 80 cents, one pound will cost as much as 4| is contained 
 times in 80, or $.80 h- 4£ = $.80 X I = — = $.17|. 
 
 4 
 
ARITHMETIC— PART I 
 
 17 
 
 If one pound cost 17| cents, lbs. will cost times that amount, or 17| X 7 J'= 
 
 $ .80 5 
 
 .160 16 4.00 
 
 — p — X ~ 2 ~ — XT = ^ 1 - 33 3- Answer. 
 
 3 1 
 
 6. If t 5 6 of the men are away and 143 remain in the village, how many went 
 away? 
 
 Solution — If represents the whole number of men, T 5 g went away, which left (If — 
 T y = in the village, so {A of the men = 143 men. If that is 11 sixteenths, one will 
 be of 143 or 13 men ; and if is 13, the whole j-f will be 16 times as much, or 16 X 
 13 .= 208 men. Answer. 
 
 7. A man lost £ of his money and spent | of what remained. He then had $18. 
 What did he have at first? 
 
 Solution — The whole of his money was § ; he lost so must have had § remain- 
 ing. Of this he spent or \ of f , which is He had § and spent £, so he must have 
 had left (f — £), or £. But £ was equal to $18, so the whole f must have been 3 times 
 as much or $54. 
 
 8. A can do a piece of work in 4 days and B in 6 days. How long will it take 
 them working together? 
 
 Solution — If it takes A 4 days, in one day he will do \ of the work. 
 
 If it takes B 6 days, in one day he will do £ of the^work. 
 
 Both working together, in one day they will do £ + £, or ^ • 
 
 To do the whole work if, it will take as many days as the amount done in 
 one day, is contained in J | ; or, 
 
 1 
 
 12 5 12 12 
 
 “i2 -*■ i2" = w y ' t = x = 25 days - Answer - 
 
 i 
 
 9. There are two pipes in a cistern ; one fills it in 40 minutes and the other 
 empties it in 60 minutes. If the cistern is empty, and both pipes started at once, how 
 long will it take to fill the cistern? 
 
 Solution — Find the least number which will hold each of them, or the L. C. M. of 
 60 min. and 40 min., which is 120 min. So the first pipe, if it fills it once in 40 minutes, 
 will fill it three times in 120 minutes; and if the second empties it once in 60 minutes, 
 will empty it twice in 120 minutes; hencS the cistern filled 3 times, minus 2 times 
 emptied in 120 minutes, will be filled once in 120 minutes, or 2 hours, both pipes work- 
 ing together. Once in 2 hours. Answer. 
 
 10. There is food enough to last 2,000 men for 30 days, but 1,000 leave. How 
 long did the food last the others? 
 
 Solution — If it lasted 2,000 men 30 days, it will last one man 2,000 X 30, or 60,000 
 days. 
 
 Removing 1,000 men leaves 1,000. If it lastsone man 60,000days it will last 1,000 
 men as many days as 1,000 is contained in 60,000, or 60 days. 
 
 11. A boat will go down the river 36 miles in 3 hours, but it takes 4 hours to come 
 back. What is the rate of the current of the river? 
 
 Solution — If it goes 36 miles in 3 hours, in 1 hour it will go 12 miles. As it is going 
 down stream it is aided by the current, so 
 
 Rate in still water -f- rate of the current = 12 miles in 1 hour, but returning it 
 takes 4 hours to go 36 miles, or 9 miles in 1 hour. This time the current lessens the 
 speed, so 
 
18 
 
 ARITHMETIC— PART I 
 
 Rate in still water — rate of current = 9 miles per hour. So the current must 
 have made a difference of 3 miles in 1 hour, but this current was met both ways — 
 
 2 X rate °f th e current = 3 miles in 1 hour, therefore the rate of the current must 
 have been one-half of 3, or 1| miles an hour. Answer. 
 
 12. To 3 times a number add 56, divide the sum by 7, subtract 6 from the quotient, 
 and the remainder will be 20. What is the number? 
 
 Solution — Beginning at the end we have the remainder 20 and the subtrahend, 
 the number subtracted is 6, hence we have 20 + 6 = 26, the minuend, which was also 
 the quotient of the part before, so we have divisor 7, quotient 26, to find the dividend 
 or 26 X 7 = 182 dividend ; but this was also the sum of the previous part, 56 had been 
 added to three times the number and gave 182; so 182 — 56 = 126, which is 3 times 
 the number . •. the number must be r of 126, or 42. Answer. 
 
 13. The sum of two numbers is 56, their difference is 28, what are the numbers? 
 
 Solution A. 
 
 Greater -f less = 56 
 Greater — less = 28 
 
 2 X greater = 84 
 greater = ** or 42 ) Answer 
 56 — 42 14 f Answer - 
 
 Sum -)- Difference 
 
 — „ — = greater nnmber. 
 
 14. 
 
 Solution B. 
 
 Sum — Difference 
 
 2 
 
 56 + 28 
 2 
 
 less number. 
 
 14 | 
 
 J 
 
 Answer. 
 
 1. At what time between 1 and 2 will the hands of a clock be together? 
 
 Solution — Two hands are together at noon 12 o’clock. In the next hour the minute 
 
 hand goes through 60 minute spaces while the hour hand goes through 5 minute spaces, 
 the minute hand having gained 55 spaces in the hour. 
 
 The minute hand is now at 12, and the hour hand at one, so to be together the 
 minute hand would have to go 5 more spaces, or 60 altogether. 
 
 If 55 is the number of spaces covered in 1 hour, it will take as many hours to go 
 60 spaces as 55 is contained in 60, or §§ = 1^ T hour = 
 
 1 hour, 5 minutes, 27 T \ seconds p. m. 
 
 2. At what time will they be together between 4 and 5 o’clock? 
 
 Solution — From noon the minute hand would have to go 60 minutes each hour to 
 be together, so in 4 hours it would have to gain 240 minutes in order to be together 
 with the hour hand. If the gain in 1 hour is 55 minutes, to gain 240 it will take as 
 long as 55 is contained in 240, or = 4 hours, 21 minutes, 491 1 2 * seconds p. m. 
 
 RATIO 
 
 Ratio is the relation that one number bears to another of the same denomination. 
 The colon (:) is the sign of ratio. 
 
 Ratio may be expressed 9 : 3, or as a fraction § ; or by division 9 -5- 3. 
 
 A ratio is always abstract ; as, the ratio $9 : $3 is 3, or $3 : $9 is r. 
 
 Antecedent is the name of the first number of the ratio. 
 
ARITHMETIC— PART I 
 
 19 
 
 Consequent is the name of the second number of the ratio. 
 Rule — Antecedent divided by Consequent = Ratio. 
 
 PROPORTION 
 
 This is an equality between two ratios. 
 
 The sign of proportion is the double colon ( : : ) or equality (=) sign between the 
 ratios ; as, 4 : 8 : : 7 : 14, or 4 : 8 = 7 : 14. 
 
 It is read 4 is to 8 as 7 is to 14. 
 
 (I or £) = (i 7 4 or £) 
 
 Extremes are the first and fourth terms. 
 
 Means are the second and third terms. 
 
 Principle : 
 
 a. The product of the means equals the product of the extremes ; as, 
 
 4 : 8 : : 7 : 14 
 4 X 14 = 8 X 7 
 56 = 56 
 
 b. 
 
 c. 
 
 Product of the means divided by one extreme will give the other extreme, 
 Product of the extremes divided by one mean will give the other mean. 
 
 4 : 8 : : 7 : ? 
 (means) 8X7 
 Extreme 4 
 
 = 14 other extreme. 
 
 (Extremes) 4 
 
 14 
 
 X 14 
 
 8 other mean. 
 
 one mean 7 
 
 Rule for Solving Problems : 
 
 1. Make the answer the fourth term. 
 
 2. Take for the third term the number that is of the same kind as the answer. 
 
 3. Decide whether the answer is to be greater or less than the third term. 
 
 4. Arrange the remaining numbers for the first and second terms so that they 
 will bear the same relation to each other, as the third and fourth do to each other. 
 (Greater to less or less to greater). 
 
 5. Apply principle b. 
 
 Example— What will 20 tons of hay cost if 7 tons cost $42? 
 
 Less : greater : : less : greater 
 
 20 
 
 $42 
 
 6 
 
 20 X 4? 
 1 
 
 1 
 
 $120. 
 
 The answer, the number of $ the cost 
 of the hay, is made the fourth term. 
 The third term is $42, also the cost of the 
 hay; 20 tons cost more than 7 tons, so the 
 ratios are arranged less to greater. 
 
 The remaining terms are 7 and 20 tons to be similarly arranged for first and second 
 terms, or 7 : 20 — less to greater, giving the proportion 7 : 20 : : $42 : ? $ 
 
 MONEY 
 
 It is a measure of value, and is either in coin or paper. 
 
 UNITED STATES MONEY 
 
 Is the currency used by the United States, adopted by Congress in 1786. It is the 
 decimal system — dimes, cents, and mills being written as decimals. 
 
 The coins are gold, silver, nickel and bronze. 
 
 Gold coins now coined are the double- eagle, $20; eagle, $10; half-eagle, $5. Gold 
 can legally be given for any amount in settlement of a debt. 
 
20 
 
 ARITHMETIC- PART I 
 
 Silver coins are the dollar, one-half dollar, one-quarter dollar, and dime. Silver 
 can legally be given for amounts not more than $5-00 at any one time. 
 
 Gold and silver coins are T 9 0 pure metal, ^ alloy, in order to give hardness to the 
 coin and so prevent its wearing off so quickly. 
 
 Nickel coins are the five cent pieces. They are made of nickel and copper. 
 
 Bronze coins are the one cent pieces. They are 95 parts copper and 5 parts tin 
 and zinc. 
 
 Bronze and nickel coins can legally be given only for amounts not more than 
 twenty-five cents in one payment. 
 
 The mill is never coined. 
 
 Paper Money = $1, $2, $5, $10, $20. $50, $100, $500, $1,000, $5,000 and $10,000 
 dollars are issued m bills and are used instead of coin. These bills are gold and silver 
 certificates, bank bills, and Treasury notes of United States. 
 
 Names : 
 
 Mill— from the Latin mille, meaning thousand, is the one-thousandth part of a 
 dollar. 
 
 Cent — from the Latin centum, meaning a hundred, is the one-hundredth part of a 
 dollar. (It was proposed by Robert Morris.) 
 
 Dime — from the French disme, meaning tenth, is the one-tenth part of a dollar. 
 
 Dollar — from Dale-town, where it was first coined. 
 
 Eagle— from the national bird. 
 
 Table 
 
 10 mills = 1 cent 10 dimes = 1 dollar 
 
 10 cents = 1 dime 10 dollars = 1 eagle 
 
 The dollar is the unit of measure. When cents are less than 10 a cipher must be 
 put in tenths place, as $5.01. 
 
 The mill is placed in thousandths place, being To l 0Q of a dollar. In answers for 
 business purposes 5 or more mills are counted as another cent, less than five mills are 
 not taken account of. 
 
 In documents such as checks, etc., write cents not as decimals, but as common 
 fractions. 
 
 Aliquot Parts of a Dollar 
 
 $.10 = A 
 
 $.60 = § 
 
 $.62£ = | 
 
 $.50 = | 
 
 $.66| = I 
 
 • 20 = 1 
 
 12|= h 
 
 ,87| = | 
 
 .75 = | 
 
 •161 = h 
 
 •40 = | 
 
 II 
 
 §5 
 
 .25 = * 
 
 *h!W 
 
 II 
 
 th'M 
 
 CO 
 
 CO 
 
 ■m = i 
 
 Canadian Money 
 
 In 1858 the currency of Canada was changed from that of England to * a decimal 
 one with coins of practically the same value as those of the United States. 
 
 The gold coin, however, is the English Sovereign worth $4.8665. 
 
 Silver coins are the fifty cent piece, the twenty-five cent piece, shilling or twenty 
 cent piece, dime, half dime. 
 
 The copper coin is the cent. 
 
 The Table is the same as for the United States money. 
 
 English Money 
 
 The early Germans who came to trade with the English were called Easterlings , 
 or those coming from the East. Their money was called by the same name, but has 
 since been changed to Sterling. 
 
 The unit is the pound or sovereign = $4.8665, but in estimating values $5.00 is 
 
 used. 
 
ARITHMETIC-PART I 
 
 21 
 
 Table 
 
 4 farthings = 1 penny (d) (denarius Latin. ) 
 
 12 pence = 1 shilling (s) (solidus “ ) 
 
 20 shillings = 1 pound (:) (libra “ ) 
 
 The pound value was secured by weight — 240 pence making one pound. The sign 
 £ always precedes the number of pounds ; as, £2 6s. 
 
 Gold coins are the sovereign or pound; half sovereign (10s.); guinea (21s.), this is 
 not coined now. It was so named because the first one was made from gold obtained 
 in Guinea, Africa. Half guinea (10s. 6d.) not coined now. 
 
 Silver coins are the crown = 5s. (not coined now) ; half crown = 2s. 6d. (not coined 
 now); florin = 2s., shilling; six pence; four pence; three pence. 
 
 Copper coins are the penny and half penny. 
 
 Farthing means four things — at first the English penny was cut by a cross so 
 deeply that it could be easily broken into four parts. They are not coined now and 
 are given usually as the fraction of a penny. 
 
 French Money 
 
 The unit of measure is the franc. The money is the decimal system. 
 
 100 centimes (c.) = 1 franc (fr.) 
 
 The same names and values are applied to the money of Switzerland and 
 Belgium. 
 
 The units of Spanish, Italian and Grecian money have the same value as the 
 franc. 
 
 1 franc = $.193 — but in quick estimates 20 cents is used. 
 
 German Money 
 
 The unit of measure is the mark , value $.2385, but in quick estimates taken as 
 $.25. This is the decimal system. 
 
 100 pfennings (pf.) = mark (M.) 
 
 MEASURES 
 
 Capacity 
 
 Dry Measure — used for dry substances. The unit of measure is the bushel. 
 
 2 pints (pt.) =1 quart. . . .qt. 
 
 8 quarts =1 peck. . . .pk. 
 
 4 pecks = 1 bushel, .bu. 
 
 2| bushels = 1 barrel, .bbl. 
 
 1 bushel = 2150.42 cubic inches. 
 
 1 bushel = 11^ cubic feet. 
 
 Liquid measure used for liquids. The unit of measure is the gallon. 
 
 4 gills (gi) = 1 pint pt. 
 
 2 pints =1 quart qt. 
 
 4 quarts = 1 gallon gal. 
 
 31 1 gallons =1 barrel bbl. (These are not fixed 
 
 63 gallons =1 hogshead . . hhds. f rates in business. 
 
 1 gallon = 231 cubic inches. 
 
 7 \ gallons = 1 cubic foot. 
 
 Time Measure 
 
 This is determined by the rotation and revolution of the earth. The unit of 
 measure is the day. 
 
22 
 
 ARITHMETIC-PART I 
 
 Rotation on its axis = 1 day. 
 
 Revolution around the sun. . = 1 year. 
 
 60 seconds (sec.) 
 
 . = 1 minute 
 
 
 60 minutes . . 
 
 . — 1 hour 
 
 . hr. 
 
 24 hours 
 
 . — 1 day 
 
 ..da. 
 
 365 days 
 
 . — 1 common year. 
 
 .yr. 
 
 366 days. 
 
 . — 1 leap year 
 
 .yr. 
 
 100 years 
 
 . — 1 century 
 
 ..cent. 
 
 7 days 
 
 . — 1 week 
 
 
 4 weeks 
 
 13 lunar months, 1 da. 6 hr. ) 
 
 . — 1 lunar month. . 
 
 I 
 
 .mo. 
 
 or 
 
 12 calendar months j 
 
 - — 1 year 
 
 .yr. 
 
 Calendar 
 
 Julius Caesar changed the calendar in 46 B. C., making the year 365^ days, thus 
 giving a little too much, so that by 1682 this made the time 10 days ahead of what it 
 should be. Pope Gregory dropped 10 days so that the Spring Equinox would fall on 
 March 21st, and decided that only centennial years, divisible by 400, should be leap 
 years, that is a year having the extra day making it 366 days long. Most Catholic 
 countries made the change ; but England especially bitter against the Catholics at that 
 time did not make the change till 1752, the mistake then being 11 days. Russia and 
 Turkey never made the change, and so are now about 12 days behind the United 
 States. The beginning of the year was also changed to January 1st instead of March 
 25th. 
 
 The true year is the time reckoned according to the sun, and consists of 365 days 
 5 hours 48 minutes 49.7 seconds. 
 
 To have a whole number of days, instead of using fractions, the ordinary year is 
 taken as 365 days and the leap year as 366 days. 
 
 Leap Years 
 
 a. Those years that are divisible by 4, except the years ending in 00. 
 
 b. Those years ending in 00 that are divisible by 400. 
 
 The year now has 12 months — 4 seasons, each 3 months long. 
 
 January — Janus, god of the year. 
 
 February — Febru, Roman festival celebrated on the 15th. 
 
 (The above months were added later, as at first there were only 10 months in the year.) 
 March — Mars, god of war (first month of the Roman Calendar). 
 
 April — Latin Operire — meaning to open — shown in the growth of the vegetable 
 
 kingdom. 
 
 May — Maia — Mother of Mercury — sacred to her. 
 
 June — Jrmo — wife of Jupiter — sacred to her. 
 
 July — Birth of Julius Csesar — the month was first called Quintilis 
 August — Augustus Csesar became consul in this month, it was first called 
 Sextilis, or the 6th month. 
 
 September — 7th month "| 
 
 October — 8th month I According to the old reckoning of ten months to the 
 
 November — 9th month 
 
 Y 
 
 December — 10th month j 
 
 year, March being the first month, 
 
ARITHMETIC-PART I 
 
 23 
 
 The number of days in the months can easily bo learned by following : 
 “ Thirty days hath September, 
 
 April, June, and November; 
 
 All the rest have thirty-one, 
 
 Excepting February alone; 
 
 Which hath* but twenty-eight; 
 
 Till leap year gives it twenty-nine.” 
 
 Paper Measure 
 
 24 sheets = 1 quire. 
 
 20 quires ) 
 
 or r = 1 ream. 
 
 480 sheets ) 
 
 Weight 
 
 Measures estimated by a scale or a balance. 
 
 Counting 
 
 12 things = 1 dozen. 
 
 12 dozen = 1 gross. 
 
 12 gross = 1 great-gross. 
 20 things = 1 score. 
 
 Troy Weight is used for weighing gold, silver, etc. 
 
 24 grains (gr.) = 1 pennyweight (pwt.) 
 
 20 pennyweights = 1 ounce (oz.) 
 
 12 ounces = 1 pound (lb.) 
 
 5760 grains =1 pound Troy. 
 
 3.168 troy grains = 1 carat (used in measuring jewels). 
 
 14_ 
 
 24 
 
 A carat used for measuring gold means of the whole, so 14 carats would be 
 
 gold, the rest alloy. 
 
 Avoirdupois Weight is used for weighing articles other than gold, silver, etc. 
 
 16 ounces (oz. ) = 1 pound (lb. ) 
 
 100 pounds. =1 hundred-weight (cwt.) 
 
 2000 pounds or 20 cwt =1 ton (short) 
 
 2240 pounds =1 ton (long). Not used generally except in 
 
 Custom Houses. 
 
 7000 grains = 1 pound (avoirdupois). 
 
 1 cubic foot = 62 1 pounds. 
 
 1 barrel = 196 pounds. 
 
 It is used to measure distance. 
 
 12 inches (in. ) 
 
 3 feet 
 
 16 * feet ) 
 
 5 1 yards ) 
 320 rods j 
 
 5280 feet ) 
 4 inches . . 
 1 inch 
 
 Length 
 
 = 1 foot (ft.) 
 
 = 1 yard (yd.) 
 
 = 1 rod (rd.) 
 
 = 1 mile (mi.) 
 
 = 1 hand ( used in measuring horses ). 
 = 3 sizes (used in measuring shoes'). 
 
 Surface — Square Measure 
 
 Square measure is used to measure surfaces of land, streets, rooms, etc. 
 144 square inches (sq. in.) = 1 square foot (sq. ft.) 
 
 9 square feet (sq. ft.). . . = 1 square yard (sq. yd.) 
 
 30^ square yards ) 
 
 or >- = 1 square rod (sq. rd. ) 
 
 272^ square feet ) 
 
 160 square rods = 1 acre (A) 
 
 640 acres = 1 square mile (sq. mi.) 
 
 100 square feet =1 square (roof, floors, etc.) 
 
24 
 
 ARITHMETIC -PART I 
 
 Volume — Cubic Measure 
 
 Cubic measure is used for measuring articles having three dimensions, length , 
 breadth and thickness. 
 
 1728 cubic inches (cu. in.) . =1 cubic foot (cu. ft.) 
 
 27 cubic feet = 1 cubic yard (cu. yd.) 
 
 2150.42 cubic inches =1 bushel. * 
 
 1 | cubic foot . = 1 bushel. 
 
 231 cubic inches =1 gallon. 
 
 1 cubic foot = 62 1 pounds. 
 
 Wood 
 
 Cord of Wood is so called because the pile was first measured by a cord, the 
 
 long high wide 
 dimensions of the pile being g-^r X 4 “ ft X 4 “ ft" 
 
 Cord Foot was one foot in length of the above pile. 
 
 16 cubic feet. =1 cord foot (cd. ft.) 
 
 128 cubic feet ) 
 
 or >- =1 cord (cd.) 
 
 8 cord feet ) 
 
 Circular Measure 
 
 This is used in measuring angles — latitude and longitude. 
 
 60 seconds ( B ) =1 minute (') 
 
 60 minutes = 1 degree ( p ) 
 
 360 degrees =1 circumference. 
 
 The circumference of a circle contains 360° whether the circle is large or small. 
 
 Longitude and Time 
 
 15° of longitude =1 hour of time. 
 
 15' of longitude = 1 minute of time. 
 
 15" of longitude =1 second of time. 
 
 DENOMINATE NUMBERS 
 Reduction 
 
 Reduction Descending — changing to lower denominations without changing the 
 values. 
 
 Examples: 
 
 1. Change 5 gallons, 2 quarts, 1 pint— to pints. 
 
 5 gal. 
 
 4 
 
 — Method — There are 4 quarts in 1 gallon, hence in 5 gallons 
 
 ^2 ^ S there are 5X4 or 20 quarts, this added to the quarts 
 
 — given in the example makes 22 quarts. 
 
 ^2 ^ In 1 quart there are 2 pints, therefore in 22 quarts there are 
 
 — 22 X 2 or 44 pints, this, plus the 1 pint of the example, gives 
 
 44 \ 
 
 l 45 pints. 
 
 45 pts. Answer. 
 
 2. Change | miles to inches. 
 
 7 40 140 11 J 
 
 a. p mi. X rds. = 280 rds. b. 280 rds. X 5J = X 0 ~ 1^40 yds. 
 
 1 X 
 
 yds. ft. 
 
 c. 1540 X 3 = 4620 ft. d. 4620 X 12 = 55440 inches. Answer. 
 
ARITHMETIC— PART I 
 
 25 
 
 Reduction Ascending — Changing to higher denominations without changing the 
 value. 
 
 Example : 
 
 1. How many rods in 214 feet? 
 
 3 ) 214 ft. 
 
 .A 71 yds. 1 ft. 
 
 2 / X 2 Answer 12 rods, 5 yards, 1 foot. 
 
 11) 142 half yds. 
 
 12 rds. 10 half yds. or 5 yds. 
 
 2. Reduce 3 cwt. 70 lbs. to tons. 
 
 70 lbs. -v- 100 = t 7 o% or ^ cwt. This plus 3 cwt. = 3/ 0 cwt. 
 
 3/o cwt * -s- 20 = i o X — 2 Y 0 to ns - Answer. 
 
 Since 100 lbs. make 1 cwt., 1 lb. is cw£., so 70 lbs. would be 70 times 1 or / Q ° 6 
 cwt. or /o cwt. This, added to the 3 given, makes 3/g cwt. 
 
 20 cwt. = 1 ton, so 1 cwt. would be ^ of a ton, and 3 T 7 0 would be 3 T 7 0 X 20 — 2 Y 0 
 tons. Answer. 
 
 Addition, Subtraction, Multiplication, Division 
 
 is the same as for other numbers, except that care must be taken to use the correct 
 number of units of one denomination in changing to the next, as they vary so much. 
 
 Add. 
 
 15 ft. 8 in. 
 2 ft. 10 in. 
 
 17 W 
 1 6 
 
 Ans. 18 ft. 6 in. 
 
 Subtract. 
 
 14 20 
 
 X^ft. $ in. 
 2 ft. 10 in. 
 Ans. 12 ft. 10 in. 
 
 10 in. plus 8 in. = 18 in. 
 
 15 ft. plus 2 ft. = 17 ft. 
 
 but 12 inches makes a foot, so in 18 inches there are 1 
 foot 6 inches, which gives a total sum of 18 feet 6 inches. 
 
 10 inches cannot be taken from 8 inches, so change 1 foot 
 to 12 inches, giving 14 feet, 20 inches, then subtracting 2 
 feet 10 inches we have 12 feet 10 inches. 
 
 Difference of Dates 
 
 Represent years, months and days by figures and subtract as in denominate num- 
 bers. Always count 30 days to a month in subtracting unless the exact number of 
 days is called for. 
 
 Example — What is the difference between July 1, 1907, and December 12, 1905? 
 
 18 
 
 6 6+12 31 
 
 1907 — 7 X 
 
 1905 12 12 
 
 1 6 19 
 
 Difference in Hours 
 
 There are 24 hours to a day, divided in groups of 12, one set before noon A. M. 
 and one set after, P. M.; but in subtracting do not separate into these groups — for the 
 time after noon, or p. m., count up to the 24th hour. 
 
 Example — What is the difference between 5 p. m. and 11 a. m.? 
 
 Hrs. 
 
 17 
 
 11 5 p. m. is the 17th hour of the day. 
 
 6 hours. Answer. 
 
 Writing Time— the numbers representing the hours, minutes and seconds are 
 written one after the other with a colon between, as, 10:20 meaning 20 minutes after 
 
26 
 
 ARITHMETIC-PART i 
 
 10. The minutes arc always added to the hour they follow, as 20 minutes of eleven 
 would be written 10.40 and read “ten forty,’' as the 11th hour has not been reached 
 to use as a starting point from which to reckon. 
 
 Multiply 
 
 12 ft. 8 in. Multiplying by 5 gives 60 feet 40 inches, but 40 inches 
 
 contains 3 feet 4 inches, making in all 63 feet, but 63 feet 
 
 fifd ft. 0 in. 
 
 3 ft. 4 in. is equal to 21 yards, 0 feet, so the answer is 21 yards, 
 
 Ans. 21yds. 0 ft. 4 in. 0 feet, 4 inches. 
 
 Division 
 
 Divide 11 ft. 8 in. by 4 4 into 11 feet is contained twice and 3 feet over ; 3 feet = . 
 
 4)11 ft. 8 in. 36 inches, this plus 8 inches = 44 inches; this divided by 
 
 2 ft. 11 in. Ans. 4 = 11 inches. 
 
 If both numbers are compound, reduce both to the same denomination (the lowest 
 denomination called for) and then divide. 
 
 A vessel contains 8 gallons 3 quarts, how many of the smaller vessels, each con- 
 taining 3 quarts 1 pint, will it take to fill the larger? 
 a. 8 gal. 3 qts. 0 pts. = 70 pts. b. 
 
 gal. qts. qts. qts. 3 qts 1 pt. = 7 pts. 
 
 8 X 4 = 32 + 3 = 35 qts. pt. pt. 
 
 35 qts. X 2 = 70 pts. 3X2 = 6+ l= =r 7 pts. 
 
 c. 70 pts. — j— 7 pts. = 10. Answer. 
 
 Note — In denominate numbers, if a fraction occurs, reduce it to lower denomina- 
 tions and add it to the proper column. 
 
 LONGITUDE AND TIME 
 
 Longitude is the distance east or west of a given meridian. It is measured in 
 degrees, minutes and seconds. It can never be greater than 180°. 
 
 A Meridian is a half of a great circle passing from pole to pole. It is sometimes 
 called the mid-day line, for all places on that line have mid-day or noon at the same time. 
 
 The Prime Meridian is the meridian selected to measure from. 
 
 The Meridian going through Greenwich, London, is generally taken as the 
 Prime Meridian. A place on this meridian has no longitude. 
 
 Difference in Longitude — If both places have the same name, i. e., are East or 
 both West subtract to find the difference, but if they have different names, one being 
 East and the other West, add to find the difference. 
 
 A.. . 
 
 . . 15° E . 
 
 C 
 
 . . 10° E. 
 
 West 
 
 
 East 
 
 B 
 
 5° E. 
 
 D . . . . 
 
 . . 15° W. 
 
 
 5 
 
 5° 15° 
 
 
 
 
 
 
 s 
 
 ..... B . .A 
 
 
 10° Dif. 
 
 
 25° Dif. 
 
 
 03 
 
 j=3 
 
 V ; 
 
 
 
 
 
 15° 
 
 
 10° 
 
 
 
 
 
 D 
 
 
 .c 
 
 The earth revolves on its axis from west to east once in 24 hours, the sun appear- 
 ing to move round the earth from east to west, so it is afternoon for places east of the 
 meridian and forenoon for places west of it. 
 
 Hence places East receive the sun first, and moving on will have later time than 
 places West, just receiving the sun. 
 
 Therefore, of two places, no matter whether both are West Longitude or East 
 Longitude, the easternmost one has later time and clocks will be faster , while the 
 other place will have earlier time and the clocks will be slower. 
 
 In 24 hours 360° pass under the vertical rays of the sun. 
 
 Rotation— 360 9 in 24 hrs. = 15° in 1 hr 
 
 Rotation— 15° or 900 1 in 60 min. (1 hr.) = 15' in 1 min. 
 
 Rotation — 15' or 900" in 60 sec. (1 mm.) = 15“ m 1 sec. 
 
 Tables 
 
 Difference in Longitude = Difference in Time 
 
 15° (degree) =1 hour of time 
 
 15' (minute) = 1 minute of time 
 
 15" (second). . =1 second of time 
 
 Circular Measure 
 60 (") seconds = 1 minute (') 
 
 60 (') minutes = 1 degree (°) 
 360° degrees = 1 circumference 
 
ARITHMETIC— PART 1 
 
 27 
 
 Rules : 
 
 1. Difference in Time = Difference in Longitude 15. 
 
 2. Difference in Longitude = Difference in Time X 15. 
 
 Examples : 
 
 When it is 10 a. m. at San Francisco 122° 24' 32" W., what is the time at Washing- 
 ton 77- 03' 06" W.? 
 
 Before we can find the time at Washington we must know the difference in time 
 between the two places, so use 
 
 Rule 1 . Difference in Time = Difference in Longitude -s- 15. 
 
 Before applying the rule we must know the difference in longitude. 
 
 As both places are West Longitude, find the difference by subtracting the less 
 
 from the greater. 
 
 122 r 
 
 77 c 
 
 24' 32" W. 
 03' 06" W. 
 
 45° 21' 26" Difference in Longitude. 
 
 Applying rule — Difference in Longitude -t- 15 gives difference in Time, of the 
 degrees, minutes, and seconds, will equal the “Difference in Time ” in hours, minutes 
 and seconds. 
 
 15 1 45° 21' 26" 
 
 3 hrs. 1 min. 25J| sec. Dif. in Time 
 
 West 
 Wash. 77°. . . 
 
 ? 
 
 S. Frisco 122° 
 
 10 A.M. 
 
 East 
 
 Places East have later time. 
 
 We see that though Washington is West Longitude it is further East than San 
 Francisco, so must have later time. If the time at San Francisco is 10 a. m. and the 
 difference in time is 3 hrs. 1 min. sec. — at Washington, which has later time, it 
 
 10 — 0 — 0 
 
 must be the 13th hour, etc., or 1 : 1 : 25 f. m. 3 — 1 — 
 
 13 — 1 — 25H 
 
 2. What is the longitude of Washington, whose time is 1 hr. 1 min. 26 sec. p. m. 
 when it is 10 a. m. at San Francisco, whose longitude is 122° 24' 32" West? 
 
 Before finding longitude of Washington we must know the difference of longitude. 
 Rule — Difference in Time X 15 = Difference in Longitude. 
 
 hr. 
 
 mm. 
 
 sec. 
 
 
 
 13 
 
 1 
 
 26 
 
 
 
 10 
 
 0 
 
 0 
 
 
 
 3 
 
 1 
 
 26 
 
 Difference in Time. 
 
 
 
 
 15 
 
 
 
 45° 
 
 15' 
 
 m* 
 
 Difference in Longitude. 
 
 * 
 
 
 + 
 
 
 
 390" -f- 60 = 6' — 30" over. 
 
 
 6 
 
 30 
 
 
 15' plus 6' = 21". 
 
 
 21 
 
 
 or 
 
 21' 60 = 0° — 21" over. 
 
 45° 
 
 21' 
 
 30" 
 
 Difference in Longitude. 
 
 
 122 ° 
 
 45° 
 
 24' 
 
 21 ' 
 
 32" W. 
 30" W. 
 
 77° 3' 2" W. 
 
 Wash. 1 p. m. 
 
 ? ° 
 
 (Dif. 45°) 
 
 S. Frisco 10 a.m 
 
 122 ° 
 
 Washington having later time must be further east and so nearer the prime 
 meridian, and therefore would have less degrees, so we subtract. 
 
 International Date Line 
 
 Persons traveling 180° west would see the sun twelve hours after they would see 
 it at Greenwich. Persons traveling 180° east would see it twelve hours before they 
 would see it at Greenwich. 
 
 When these persons meet at the 180th meridian from the place of starting, they will 
 find when they compare reckoning of time that it differs in a great degree. The one 
 
2S 
 
 Arithmetic-part i 
 
 who has gone in an easterly direction will find that his time is twelve hours later 
 than when he started. The one who went in a westerly direction will find that his is 
 twelve hours earlier than when he started. Suppose that it was New Year’s Day at 
 noon when they started, the one who went east would find that by his reckoning it 
 was twelve midnight of January first; the one who went west would find by his reckon- 
 ing that it was twelve midnight of December thirty-first — yet they are together at the 
 180th meridian from where they started. 
 
 To overcome this difficulty an International Date Line, at the 180th meridian east 
 and west of Greenwich, has been agreed upon by the nations as the place where time 
 is changed. Those sailing west crossing the line add a day, and those going east 
 crossing the line subtract a day from their time. 
 
 The person traveling west crossing this line on Sunday would by this agreement 
 call it Monday. The persons traveling east crossing the line . on Sunday would call it 
 Saturday, at the same hour of the day. 
 
 This line does not exactly' follow the 180th meridian, but avoids land bodies as 
 much as possible and so is irregular. 
 
 Standard Time 
 
 For convenience of railroads, etc., the United States has been divided into 5 sec- 
 tions, each about 15° (longitude) wide. All places within a section using the time of 
 the central meridian of that section. 
 
 The time of each district is the solar time of meridians 15° apart. 
 
 The Districts are known as : 
 
 Atlantic or Colonial - Solar time of 60 meridian W. L. 4 hours earlier than Greenwich. 
 
 Eastern — Solar time of 75 meridian W. L. 5 hours earlier than Greenwich. 
 
 Central— Solar time of 90 meridian W. L. 6 hours earlier than Greenwich. 
 
 Mountain — Solar time of 105 meridian W. L. 7 hours earlier than Greenwich. 
 
 Pacific - Solar time of 120 meridian W. L. 8 hours earlier than Greenwich. 
 
 As the railroads change the time of running only at important stations or junc- 
 tions, the lines of division of the districts are irregular. 
 
 The boundary between the Colonial and Eastern — From the St. Lawrence River 
 near 65 meridian, west longitude irregularly to the Atlantic Coast of Maine, near the 
 68 meridian west longitude. 
 
 The boundary between the Eastern and Central Districts — From Fort William on 
 the Canadian side of Lake Superior, it follows the Canadian lake shore line to Buffalo; 
 thence in an irregularly southeasterly direction to near Gainesville, Georgia; thence 
 southeasterly to the coast at Savannah, Georgia. 
 
 The boundary between the Central and Mountain Districts— From a point on the 
 northern boundary of the United States near the 103 meridian west longitude, south- 
 easterly to a place in Nebraska near Long Pine ; thence westerly to Alliance, Nebraska ; 
 thence irregularly in a southwesterly and southeasterly direction to Cheyenne; thence 
 easterly to the Arkansas River; thence southwesterly to the boundary of the United 
 States and Mexico near El Paso. 
 
 The boundary between the Mountain and Pacific Districts — From a point on the 
 boundary of the United States and Canada, just east of the 117 meridian west longi- 
 tude, in a generally southwesterly direction to a place just east of the 120 meridian 
 west longitude, south of the 40° parallel north latitude ; thence in a southeasterly 
 direction to a point near El Paso, on the boundary of the United States and Mexico. 
 
 There being 15° longitude difference between each section gives 1 hour’s difference 
 in time in each. 
 
 Difference in Time == Difference in Longitude -4- 15. 
 
 Solar Time — Is the actual time of a place. It differs as places are east or west of 
 a meridian from which the time is reckoned. 
 
 Example 3. When was a message sent from Cairo 30° East to London 0° if 
 received there at 5.15 p. m.? 
 
 30° E. West 
 
 ^ London 0° . 
 
 15)30° Difference in Longitude. 5 pm 
 
 2 hrs. Difference in Time. 
 
 East 
 
 ..Cairo 30° 
 
 7 
 
 Cairo is further East, so must have later time, or 7.15 p. m. Answer. 
 
 Example 4. What is the difference between the standard and the local time, if 
 the longitude of Boston is 71° 3' 30" E.? 
 
ARITHMETIC— PART I 
 
 29 
 
 Boston being 70°, etc., must be in the section using 75° longitude. 
 
 Difference in Time = Difference in Longitude -r- 15. 
 
 59 
 
 74 00 60 
 
 0' 0" E. 
 
 J71 Q _ 3 3 0 E. 
 
 15) 3 V 56' 30" Difference in Longitude. 
 
 0 hrs. 15 min, 46 sec. Difference in Time. Ans. 
 
 METRIC SYSTEM 
 
 It was invented in France in 1800. The unit of measure in this system is the 
 meter. It is nearly one ten-millionth part of the distance between the equator and the 
 poles; or nearly 39.37 inches. 
 
 It is a decimal system and in each table there is joined to the word standing for 
 the unit, Latin prefixes to indicate the decimal parts and Greek prefixes to indicate 
 the multiples. 
 
 The abbreviations are the first letter of the prefix and the first letter of the unit, 
 both written with small letters if it is to indicate a part; but if it is to indicate a 
 multiple the first letter is usually a capital, though some do not use the capital at all. 
 
 Prefixes 
 
 deci ..... ...... = .1 deka . . = 10 
 
 cent.... = .01 hekto . =100 
 
 mill ...... .001 kilo =1000 
 
 Table of Length. (Unit = Meter) 
 
 10 Millimeters (mm.). = 1 centimeter (cm.) 
 
 10 Centimeters =1 decimeter (dm.) 
 
 10 Decimeters . . =1 meter (m) 
 
 10 Meters. . . =1 dekameter (dm.) 
 
 10 Dekameters. . = 1 hektometer (hm.) 
 
 10 Hektometers. =1 kilometer (km.) 
 
 The above measures most used are the Centimeter (scientific work), Meter 
 (cloth), Kilometer (long distance). 
 
 The table is on the scale of 10, so in reduction one decimal place must be allowed 
 for each denomination; as 
 
 525. m. = 5250. dm. = 52500 cm. or 525. m. = 52.5 Dm. =5.25 Hm. = .525 Km. 
 
 Table of Capacity. (Unit = Liter) 
 
 The table of capacity is used for liquid and dry measure. 
 
 10 Milliliters (ml. ) =1 centiliter 
 
 10 Centiliters =1 deciliter 
 
 10 Deciliters =1 liter 
 
 10 Liters .... =1 dekaliter 
 
 10 Dekaliters =1 hektoliter 
 
 The measures most used are the Liter (small quantities) and the 
 quantities). 
 
 Table of Weight. (Unit = Gram) 
 
 A gram is the weight of 1 cubic centimeter of water. 
 
 10 Milligrams (mg.) = \ centigram 
 
 10 Centigrams ... = l decigram 
 
 10 Decigrams =1 gram 
 
 10 Grams = 1 dekagram 
 
 10 Dekagram . = 1 hektogram 
 
 10 Hektogram =1 kilogram 
 
 1000 Kilograms =1 metric ton. 
 
 (cl.) 
 
 (dl.) 
 
 (!•) 
 
 (dl.) 
 
 (HI.) 
 
 Hektoliter (larger 
 
 (eg.) 
 (dg.) 
 (g-) 
 (Dg.) 
 (H g.) 
 (Kg) 
 
 Table of Surface— Square Measure. (Unit = Square Meter) 
 
 100 Square millimeters (sq. mm.) or (mm 2 ) = 1 square centimeter (sq. cm.) or (cm 2 ) 
 
 100 Square centimeters = 1 square decimeter (sq. dm.) or (dm 2 ) 
 
 100 Square decimeters = 1 square meter (sq. m.) or (m 2 ) 
 
 100 Square meters. =1 square dekameter (sq. Dm.) or (Dm 2 ) = are 
 
 100 Square dekameters = 1 square hektometer (sq. Hm.) or (Hm 2 ) = hektare 
 
 100 Square hektometers =1 square kilometer (sq. Km.) or (Km 2 ) 
 
30 
 
 ARITHMETIC-PART I 
 
 The above measures most used are the square meter (area of floors, walls, etc.), 
 square kilometer (area of land, etc.). 
 
 The names are (a) and hektare ( ha ) are used in measuring land. 
 
 This table increases by 10 2 or 100, so in reduction two decimal places must be 
 allowed for each denomination; as, 
 
 52535 sq. mm. = 525 35 sq. cm. = 5.2535 sq. dm. = .052535 sq. m. = .00052535 
 sq. Dm., etc. 
 
 Table of Volume— Cubic Measure (Unit = Cubic Meter) 
 
 1000 cubic millimeters (cu. mm.) or (mm 3 ) = 1 cubic centimeter (cu. cm.) or (cm 3 ) 
 1000 cubic centimeters =1 cubic decimeter (cu. dm. ) or dm 3 ) 
 
 1000 cubic decimeters cubic met “ < cu ' m > or < m3 > 
 
 This table increased by 10 3 or 1000, 
 allowed for each denomination ; as, 
 
 2535 cu. mm. = 52.535 cu. cm. 
 
 1 stere (wood measure) 
 so in reduction three decimal places must be 
 
 052535 cu. M. 
 
 Equivalents 
 
 1 meter ...... ...= 39.37 inches. 1 kilometer = | miles. 
 
 1 cubic decimeter = 1 liter. 1 liter =1 quart. 
 
 1 liter. . = 1 kilogram. 1 kilogram = 2£ lbs. 
 
 1 five cent piece weighs 5 grams. 1 metric ton. ... = 2204-6 lbs. 
 
 1 hektare = nearly 2\ Acres or 2.47 Acres. 
 
 ADDITION, SUBTRACTION, MULTIPLICATION, DIVISION, 
 in the Metric System is the same as for whole numbers and decimals in our system. 
 
 Reduce the different denominations all to the same denomination by moving the 
 decimal point or annexing ciphers, and perform the operation indicated by the signs. 
 
 Reduction— Changing from one table to another, or from the metric system to the 
 common system, use the above equivalents. 
 
 Example — 1, What will 3 cubic meters of water weigh? 
 
 1 cu. dm = 1 iiter. 1 liter = 1 kilogram. 
 
 hence 3 cu. m . 3000 cu. dm., and 
 
 3000 cu. dm. . = 3000 liters, and 3000 liters . . . . = 3000 kilogram. Answer, 
 or if we wish to reduce it to our system we know, 1 kilogram : . . = lbs. 
 
 600 11 
 
 3000 kg. will weigh 2\ times as much, or X = 6600 lbs. Answer. 
 
 2. How many lbs. will a person weigh if he weighs 45 kilos? 
 
 * 1 kilogram . . . = 2\ lbs. 
 
 9 11 
 
 45 kilos. = 45 X = 99 lbs. Answer. 
 
 3. How many 5 cent pieces in 2 kilograms? 
 
 one 5 cent piece = 5 grams. 
 
 2 Kg = 2000 grams. 
 
 2000 -i- 5 s 400 five cent pieces. 
 
 By specific gravity of any material is meant the number of times heavier a sub- 
 stance is in air than an equal volume of water. 
 
 4. If 4 liters of milk weighs 4-12 Kg. what is its Specific Gravity? 
 
 (water), 1 1. = 1 kg. 
 
 (milk). .1 1. = i of 4.12 or 1.03 Kg. 
 
 Weight of milk is 1.03 times that of water, so the Specific Gravity is 1.03. Ans. 
 
 PAPERING 
 
 A single roll of wall paper is 24 feet long by 1| feet wide. 
 
 A double roll is 48 feet long and \\ feet wide. 
 
 A part of a roll cannot be bought, and prices are always quoted by the single roll. 
 Borders are sold by the yard in length. 
 
 Rule : 
 
 1. Find in Jeet the perimeter of the room by adding the length of the 4 walls; 
 divide this by 1^ feet, the width of a roll, to find how many strips are needed. If 
 there is any remainder count it an extra strip. 
 
 2. Divide the length of 1 roll, 24 feet if single, or 48 feet if double, by the height 
 
ARITHMETIC— PART I 
 
 31 
 
 of the room in feet , and this will give the number of strips that can be cut from 1 roll. 
 If there is any remainder, reject it. 
 
 3 . Divide the number of strips required for the room by the number obtained 
 from 1 roll and the result will be the number of rolls required. 
 
 Example — How many single rolls of paper will it take for a room 45 feet by 33^ 
 feet and 11 \ feet high? 
 
 45 X 2 = 90 ft. — 2 long walls 
 (a) 33f X 2 = 67| ft. — 2 short walls 
 157| ft. — Perimeter. 
 
 105 
 
 m ? 24 2 ‘ 48 2 
 
 (b) 157| 1 | = -j X -j- = 105 strips, (c) 24 -f- 11 J = y X = 23 = 2 23 or 
 
 2 strips from each roll. 
 
 105 strips required - 4 - 2 strips from 1 roll = 52| rolls, or 53 rolls must be bought. Ans. 
 
 Note — Dealers use 36 square feet as the area of a single roll, and then find the 
 area of all the walls in square feet, subtracting from this the area of all openings, 
 dividing by 36, the area of the single roll, to obtain the number of rolls required. 
 The area of the ceiling is also divided by 36 to obtain the number of single rolls re- 
 quired for it. If an allowance is made for waste in matching paper, then 30 square 
 feet instead of 36 are used as the single roll 
 
 PAINTING, PLASTERING, KALSOMINING 
 
 Find the area of all the surfaces to be covered, by multiplying the length by the 
 width of each. If any allowance is made for doors, windows, etc., find their area in 
 the same way and deduct this from the whole area. 
 
 Laths are 4 feet long and come usually in bundles of 1000, no less than a bundle 
 can be bought. 
 
 Example — What will it cost to paint the walls and ceiling of a room 24 feet by 15 
 feet by 10 feet high at 10 cents a square foot, allowing for 2 windows 6 ft. X 3 ft.. 
 
 and for one door 7 ft. X 3 ft. ? 
 
 Ceiling— 24 ft. X 15 ft 360 sq. ft. 
 
 2 Long Walls— 24 ft. X 10 ft. X 2. . . . . = 480 sq. ft. 
 
 2 Short Walls— 15 ft. X 10 ft. X 2. = 300 sq. ft. 
 
 1.140 sq. ft. 
 
 Area, 2 Windows (6 ft. X 3 ft.) X 2 = 36 sq. ft. ) _ r 7 .. 
 
 Area Door (7ft. X 3 ft. ) = 21 sq. ft. sq ‘ 
 
 1,083 sq. ft. 
 
 $.10X1083 = $10.83 
 $10.83 Answer. 
 
 CARPETING 
 
 Rule — Find the number of strips required by dividing the width of the room if 
 laid lengthwise, or the length of the room if laid crosswise, by the width of 1 strip of 
 carpet. If the result has a fraction count it an extra strip — for if the exact number of 
 strips is too wide, a part of one breadth or strip is turned under. 
 
 In matching figures the ends of the strips after the first strip, will sometimes have 
 to be- turned under. or cut off, thereby necessitating more carpet than is called for by 
 the actual size of the room. 
 
 Multiply the length of a strip required for the room by the number of strips and 
 the result will be the quantity required. 
 
 Example — How many yards carpet f yard wide laid lengthwise will be required 
 to cover a floor 16 feet by 14 feet? 
 
 Width of room 14 ft. -f- 3 = 4§ yards. 
 
 Width of room 4§ yds. -s- width of carpet | yds. = *§* Xf = 5 § 6 = 6| or 7 strips. 
 
 One strip the length of the room is 16 feet or 5£ yards ; 5£ yds. X 7 the number of 
 strips = J g 6 X 7 = 1 i- .= 37^ yds. Answer. 
 
 ROOFING AND FLOORING 
 
 This is usually done by the square , meaning 100 square feet. 
 
 1000 shingles are usually allowed to a square. Shingles are sold by the bundle — 
 meaning— 1000 shingles measuring 18" by 4", or by the bunch meaning 250 shingles. 
 No less than a bunch can be bought. 
 
 Sometimes roofing and flooring is done by the square foot or square yard. 
 
32 
 
 3 0112 105669409 
 
 ARITHMETIC -PART I 
 
 Example — How much would it cost to shingle both sides of a 40 foot roof measur- 
 ing from eaves to ridge 25 feet at $6.75 a thousand? 
 
 Solution— 2 sides roof = 40 ft. X 25 ft. X 2 = 2,000 square feet (for every 100 
 square feet, 1 bundle or 1000 shingles will be required. ) . \ 2000 sq. ft. -r- 100 sq. ft. 
 
 = 20 bundles. Each bundle of 1000 cost $6-75, therefore 20 will cost 
 
 5 
 
 3 27 W 
 
 $6 
 
 X 20= j- 
 
 X 
 
 = $135. Answer. 
 
 LUMBER 
 
 A board foot is the unit of measure and is a board one foot long , one foot wide 
 and one inch thick. 
 
 When lumber is 1 inch or less thick, to find the number board feet, multiply the 
 length of the board by the width. If it is more than an inch then multiply the length 
 in feet by the width in feet and that result by the number of inches thick. Therefore 
 a board 10 feet by 6 inches by 2 inches would contain 10' X I 1 X 2" = 10 board feet. 
 
 CAPACITY OF CISTERNS 
 
 Rule — Divide the contents of the cistern in cubic inches by 231, to find the number 
 of gallons. 
 
 Example — 1. How many, cubic feet in a cistern containing 50 hogsheads? 
 
 Solution — 50 hhds X 63 = 3150 gallons. As 231 cubic inches = 1 gallon, in 3150 
 gallons we will have 231 X 3,150 or 727,650 cubic inches 
 
 There are 1728 cubic inches in each cubic foot, so 727650 - 4 - 1728 = 421 g Vi cubic 
 feet. Answer. 
 
 2. How many gallons in a tank 12 feet by 5 feet by 8 feet? 
 a. 12 ft. X 5 ft. X 8 ft. = 480 cubic feet. b. 480 cu. it. X 1728 = 829440 cu. in. 
 
 829440 cubic inches 231 = 3,590f-f gallons. Answer. 
 
 CAPACITY OF BINS 
 
 Rule — Divide the contents of the bin in cubic inches by 2150.42 to find the number 
 of bushels, for 1 bushel == 2150.42 cubic inches. 
 
 Example 1. How many bushels in a bin 12 feet by 5 feet by 8 feet? 
 
 12 ft. X 5 ft. X 8 ft. = 480 cubic feet. 480 X 1728 = 829440 cubic inches. 
 
 76415 
 
 829440 - 4 - 2150. 42 = 385 mb2l ~ bushels. 
 
 The above gives the exact result, but ordinarily the following method gives 
 accurate enough results and is easier. 
 
 Rule — Divide contents of the bin in cubic feet by 1* cubic feet to find the number 
 of bushels. 
 
 2150.42 cubic inches (in 1 bushel) is to 1728 cubic inches (In. 1 cu. ft.) (very nearly) 
 as 5 is to 4 ; so 1 bushel = ® or 1 * cubic feet. 
 
 Example above according to this method would be solved. 
 
 96 4 
 
 12 X 5 X 8 = 480 cubic feet. 480 - 4 - 1 } = X = 384 bushels. Ans. 
 
 THERMOMETERS 
 
 In Centigrade thermometers the freezing point is 0° and the boiling points 100°, 
 giving 100 ° — 0 ° or 100 °. 
 
 In Fahrenheit thermometers the freezing points 32° and the boiling point 212°. 
 
 212 3 — 32° = 180° number degrees between freezing and boiling on the Fahrenheit 
 thermometer. 
 
 100 5 Q 
 
 Therefore 180° Fahrenheit = 100° Centigrade, hence 1° Fahrenheit = ^ or -g- 
 Centigrade.” 
 
 Example — What degree on the Centigrade thermometer corresponds to 68 ° on the 
 Fahrenheit? 
 
 4 5 
 
 68 ° — 32° = 36° Fahrenheit. $$ X — = 20° Centigrade. Answer. 
 
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