ANALYSIS. 76. Analysis in arithmetic consists in determining the solu lion of an example from the relations of the numbers given in that example. The given number which is of the same denomination as the required answer forms the basis of all the reasoning, and should be the first written in performing an example. The value of any number of things may be obtained by first finding the value of a single thing or unit of the same denomination. This unit is sometimes called the unit of computation. ILLUSTRATIVE EXAMPLE. If 25 barrels of flour cost $175, what cost 17 barrels ? $175 is the term of the same denom OPERATION. 175 25 Before × 17 = $119. Ans. ination as the required answer. know the value of 1 barrel.* 1 twenty-fifth of $175, and 17 barrels will cost 17 X 1 twenty-fifth of $175,$119. EXAMPLES. 1. If 13 acres of land produce 780 bushels of bushels will 5 acres produce? 2. If 5 boxes of oranges cost $21.80, what cost 21 boxes? Ans. $91.56. 3. If a car runs 207 miles in 9 hours, how far will it run in 25 hours? 4. If 18 rows of potatoes yield 54 bushels, how many bushels will 405 similar rows yield? 5. If $19.74 were paid for 14 bushels of rye, what must be paid for 25 bushels ? 6. If 19 tons of coal run an engine 266 miles, how far will 14 tons run it? 7. If 5 oxen consume 85 pounds of hay in 1 day, how much will be required for 1 yoke of oxen cf the same size, and for the Jame time? * 1 barrel is the unit of computation. - 8. How many pounds of coffee can be bought for $15, if 40 lbs. cost $8? NOTE. - If $8 pay for 40 pounds, $1 will pay for 1 eighth of 40 pounds, and $15 will pay for 15 × 1 eighth of 40 pounds = 75 pounds. 9. If 150 barrels of apples were bought for $200 and sold for $350, what would be gained by selling 45 barrels at the same rate ? 10. If a quantity of hay lasts 22 oxen 105 days, how many days will it last 5 yoke? NOTE. If it lasts 22 oxen 105 days, it will last 1 yoke 11 × 105, and it will last 5 yoke 1 fifth of 11 × 105 days 231 days. = 11. A field of wheat was reaped by 10 men in 6 days; what length of time would be required for 15 men to reap the same amount? 12. A cistern can be emptied in 35 minutes by 7 pipes; in what time can it be emptied, if 5 only of the pipes are open? 13. If 1423 operatives can do a piece of work in 12 days, in what time will 2400 operatives perform the same work? 14. If a certain piece of work can be performed by 250 men in 14 weeks, how many more must be employed to perform it in a week? 15. A garrison of 10000 men have provision to last them 6 weeks; if 2000 men be killed in a sally, how long will the provisions last the remainder? 77. QUESTIONS FOR REVIEw. 1. FEDERAL MONEY. What are the denominations of federal money? Give the table. How do you write numbers in federal currency ? What is considered the unit? Give the sign for dollars. How do you reduce eagles to dollars? dollars to cents? dollars to mills? cents to mills? mills to dollars? you sub 2. How do you add numbers in this currency? How do tract? When you multiply, of what denomination is the product? When you divide by an abstract number, of what denomination is the quotient? Divide $185 by 7, continue the division to mills, and explain. What is necessary in order to divide mills by dollars? by cents? In dividing cents by dollars, is the quotient abstract or con arete? In dividing dollars by an abstract number, is the quotient abstract or concrete? 3. BILLS. What is a bill? Ans. It is a writing given by the creditor to the debtor, showing the amount of the debt. Who is the creditor? the debtor? What is the receipt of a bill? 4. ANALYSIS. What is analysis? Which number forms the basis of the reasoning. 78. GENERAL REVIEW, No. 2. 1. 2875 million +36 thousand + 59481=? 2. Add 567 to the sum of the following numbers: 121; 232; 343; 154; 565; 676; 787; 898. 3. Take 987 from each of the following numbers, and add the remainders: 9876; 5678; 3644; 7573; 2432; 4001. 4. What number must be added to the difference between 58 and 7003 to equal 938425 ? 5. What number, taken from the quotient of 1833000 ÷ 47 leaves 25? 6. What number equals the product of 1785, 394, and (624 —48)? 7. If 5872 is the multiplicand, and half that number the multiplier, what is the product? 8. If 4832796 is the product, and 1208199 the multiplicand, what is the multiplier ? 9. If 894869 is the minuend, and the sum of all the numbers in the third example is the subtrahend, what is the remainder ? 10. If 700150 is the dividend, and 3685 the quotient, what is the divisor? 11. If 28936 is the divisor, and 86 is the quotient, what is the dividend? 12. Divide 87 million by 15 thousand. 13. $3.75+$9.32+ $.75 + $10. + $2.185+4 cents 14. $19.$.75-$8.25 + $3.54 =? 15. From 18 X $5.873, take $3.684. ? 16. If $183.30 is the dividend, and $3.90 the divisor, what is the quotient? 17. If $98 60 is the dividend, and 17 the divisor, what is the quotient? For changes, see Key. CAVIT PROPERTIES OF NUMBERS. 79. SIGNS. signifies plus, or more. signifies minus, or less. signifies greater than. signifies less than. RECAPITULATION. signifies equal to. X signifies multiplied by. .. signifies therefore. () parenthesis, and, vinculum, signify that the same operation is to be performed upon all the quantities thus connected. DEFINITIONS. 80. Numbers are either integral or fractional. 81. Integral numbers, or Integers, are whole numbers. 82. Fractional numbers are parts of whole numbers. 83. A Factor or Divisor of a number is any number which is contained in it without a remainder; thus, 2 is a factor of 6. 84. A Prime Number is a number which contains no integral factor but itself and 1; as, 1, 2, 3, 11. 85. A Composite Number is a number which contains other integral factors besides itself and 1; as, 4, 6, 8, 25. 86. A Prime Factor is a factor which is a prime number. 87. A composite number equals the product of all its prime. factors; thus, 12 = 2 × 2 × 3. 88. Two numbers are said to be prime to each other when they contain no common factor except 1; thus, 8 and 15 are prime to each other. 89. The Power of a number is the number itself, or the product obtained by taking that number a number of times as a factor. The number itself is the first power; if it is taken twice as a factor, the product is called the second power, or square; if three times, it is called the third power, or cube; if four times, the fourth power, &c. Thus, the second power =9; the third power of 3 is 3 × 3 × 3 power of 3 is 3 × 3 × 3 × 3 × 3 = 243. of 3 is 3 × 3 27; the fifth 90. The Index or Exponent of a power is a figure which shows how many times the number is taken as a factor. It is written at the right of the number, and above the line. Thus, in 53, 72, 24, the exponent 3 shows that 5 is taken three times as a factor, 2 that 7 is taken twice, and 4 that 2 is taken four times as a factor. 91. The Root of a number is one of the equal factors which produce that number. If it is one of the two equal factors, it is the second, or square root; if one of the three, the third, or cube root; if one of the four, the fourth root, &c. Thus the square root of 9 is 3, the cube root of 125 is 5. 92. is the Radical Sign, and, by itself, denotes the square root; with a figure placed above, it denotes the root of that degree indicated by the figure; thus, signifies the third root, the sixth root. DIVISIBILITY OF NUMbers. 93. (1.) Any number whose unit figure is 0, 2, 4, 6, or 8, is even. (2.) Any number whose unit figure is 1, 3, 5, 7, or 9, is odd. (3.) Any even number is divisible by 2. (4.) Any number is divisible by 3 when the sum of its digits is divisible by 3; thus, 2814 is divisible by 3, for 2+8+1+4 = = 15, is divisible by 3. (5.) Any number is divisible by 4, when its tens and units are divisible by 4; for, as 1 hundred, and consequently any number of hundreds, is divisible by 4, the divisibility of the given number by 4 must depend upon the tens and units; thus, 86324 is divisible by 4, while 6831 is not. (6.) Any number is divisible by 5 if the units' figure is either 5 or 0; for, as 1 ten, and consequently any number of tens, is divisible by 5, the divisibility of the given number by 5 must depend upon the units. (7.) Any number is divisible by 6, if divisible by 3 and by 2. (8.) Any number is divisible by 8, if its hundreds, tens, and enits are divisible by 8; for, as 1 thousand, and consequently any |
