25. The Illinois Central Railroad is 700 miles long, and cost 31647000 dollars; what did it cost per mile?

Ans. 45210 dollars.

26. What number is that, which being divided by 7, the quotient multiplied by 3, the product divided by 5, and this quotient increased by 40, the sum will be 100? Ans. 700.

27. How many cows at 27 dollars apiece, must be given for 54 tons of hay at 17 dollars a ton?

28. A mechanic receives 56 dollars for 26 days' work, and spends 2 dollars a day for the whole time; how many dollars has he left? Ans. 4 dollars.

29. If 7 men can build a house in 98 days, how long would it take one man to build it? Ans. 686 days.

30. The number of school houses in the State of New York, in 1855, was 11,137; suppose their cash value to have been 5,301,212 dollars, what would be the average value? Ans. 476 dollars.

31. A cistern whose capacity is 840 gallons has two pipes; through one pipe 60 gallons run into it in an hour, and through the other 39 gallons run out in the same time; in how many hours will the cistern be filled? Ans. 40 hours.

32. The average beat of the pulse of a man at middle age is about 4500 times in an hour; how many times does it beat in 24 hours? Ans. 108000 times. 33. How many years from the discovery of America, in 1492, to the year 1900?

34. According to the census, Maine has 31766 square miles; New Hampshire, 9280; Vermont, 10212; Massachusetts, 7800; Rhode Island, 1306; Connecticut, 4674; and New York, 47000; how many more square miles has all New England than New York?

35. What is the remainder after dividing 62530000 by 87900? Ans. 33100.

36. A pound of cotton has been spun into a thread 8 miles in length; allowing 235 pounds for waste, how many pounds will it take to spin a thread to reach round the earth, supposing the distance to be 25000 miles? Ans. 3360 pounds.

37. John has 8546 dollars, which is 342 dollars less than 4 times as much as Charles has; how many dollars has Charles? Ans. 2222 dollars. 38. The quotient of one number divided by another is 37, the divisor 245, and the remainder 230; what is the dividend? Ans. 9295. 39. What number multiplied by 72084 will produce 5190048 ? Ans. 72.

40. There are two numbers, the greater of which is 73 times 109, and their difference is 17 times 28; what is the less number? Ans. 7481.

41. The sum of two numbers is 360, and the less is 114; what is the product of the two numbers? Ans. 28044. 42. What number added to 2473248 makes 2568754?

Ans. 95506.

43. A farmer sold 35 bushels of wheat at 2 dollars a bushel, and 18 cords of wood at 3 dollars a cord; he received 9 yards of cloth at 4 dollars a yard, and the balance in money; how many dollars did he receive? Ans. 88 dollars.

44. A farmer receives 684 dollars a year for produce from his farm, and his expenses are 375 dollars a year; how many dollars will he save in five years?

45. The salt manufacturer at Syracuse pays 58 cents for wood to boil one barrel of salt, 10 cents for boiling, 5 cents to the state for the brine, 28 cents for the packing barrel, and 3 cents for packing and weighing, and receives 125 cents from the purchaser; how many cents does he make on a barrel? Ans. 21 cents.

46. A company of 15 persons purchase a township of western land for 286000 dollars, of which sum one man pays 6000 dollars, and the others the remainder, in equal amounts; how much does each of the others pay? Ans. 20000 dollars. 47. If 256 be multiplied by 25, the product diminished by 625, and the remainder divided by 35, what will be the quotient? Ans. 165.

48. Two men start from different places, distant 189 miles, and travel toward each other; one goes 4 miles, and the other 5 miles an hour; in how many hours will they meet?

GENERAL PRINCIPLES OF DIVISION.

86. The quotient in Division depends upon the relative values of the dividend and divisor. Hence any change in the value of either dividend or divisor must produce a change in the value of the quotient. But some changes may be produced upon both dividend and divisor, at the same time, that will not affect the quotient. The laws which govern these changes are called General Principles of Division, which we will now examine.

If we multiply the dividend by 3, we have

54 X 39162918,

and 18 equals the quotient, 6, multiplied by 3. Hence, Multiplying the dividend by any number, multiplies the quotient by the same number.

II. Using the same example, 5496.
If we divide the dividend by 3 we have

54918÷9= = 2,

and 2 = the quotient, 6, divided by 3. Hence, Dividing the dividend by any number, divides the quotient by the same number.

III. If we multiply the divisor by 3, we have

and 2

549X85427 = 2,

the quotient, 6, divided by 3. Hence, Multiplying the divisor by any number, divides the quotient by the same number.

IV. If we divide the divisor by 3, we have

5454318,

Upon what does the value of the quotient depend? What is the first general principle of division? Second? Third? Fourth?

GENERAL PRINCIPLES OF DIVISION.

65

and 18 the quotient, 6, multiplied by 3. Hence, Dividing the divisor by any number, multiplies the quotient by the same number.

V. If we multiply both dividend and divisor by 3, we have

54 × 39 × 3 = 162 ÷ 27 — 6.

Hence, Multiplying both dividend and divisor by the same number, does not alter the value of the quotient.

VI. If we divide both dividend and divisor by 3, we have

Hence, Dividing both dividend and divisor by the same number, does not alter the value of the quotient.

87. These six examples illustrate all the different changes we ever have occasion to make upon the dividend and divisor in practical arithmetic. The principles upon which these changes are based may be stated as follows:

PRIN. I. Multiplying the dividend multiplies the quotient; and dividing the dividend divides the quotient. (86. I and II.) PRIN. II. Multiplying the divisor divides the quotient; and dividing the divisor multiplies the quotient. (86. III and IV.)

PRIN. III. Multiplying or dividing both dividend and divisor by the same number, does not alter the quotient. (86. V and VI.)

88. These three principles may be embraced in one

GENERAL LAW.

A change in the dividend produces a LIKE change in the quotient; but a change in the divisor produces an OPPOSITE change in the quotient.

NOTE. If a number be multiplied and the product divided by the same number, the quotient will be equal to the number multiplied. Thus, 15 X 4 = 60, and 60 ÷ 4 =15.

Fifth? Sixth? Into how many general principles can these be condensed? What is the first? Second? Third? In what general law are these embraced?

EXACT DIVISORS.

89. An Exact Divisor of a number is one that gives a whole number for a quotient.

As it is frequently desirable to know if a number has an exact divisor, we will present a few directions that will be of assistance, particularly in finding exact divisors of large numbers.

NOTE. A number whose unit figure is 0, 2, 4, 6, or 8 is called an Even Number. And a number whose unit figure is 1, 3, 5, 7, or 9, is called an Odd Number.

2 is an exact divisor of all even numbers.

4 is an exact divisor when it will exactly divide the tens and units of a number. Thus, 4 is an exact divisor of 268, 756, 1284.

5 is an exact divisor of every number whose unit figure is 0 or 5. Thus, 5 is an exact divisor of 20, 955, and 2840.

8 is an exact divisor when it will exactly divide the hundreds, tens, and units of a number. Thus, 8 is an exact divisor of 1728, 5280, and 213560.

9 is an exact divisor when it will exactly divide the sum of the digits of a number. Thus, in 2486790, the sum of the digits 2+4+8+6+7+9+0=36, and 3694.

10 is an exact divisor when 0 occupies units' place. 100 when 00 occupy the places of units and tens.

1000 when 000 occupy the places of units, tens, and hundreds, &c.

A composite number is an exact divisor of any number, when all its factors are exact divisors of the same number. Thus, 2, 2, and 3 are exact divisors of 12; and so also are 4 (= 2 × 2) and 6 (= 2 × 3).

An even number is not an exact divisor of an odd number. If an odd number is an exact divisor of an even number,

An odd num

What is an exact divisor? What is an even number? ber? When is 2 an exact divisor? 4 5 9? 10? 100 ? 1000? When is a composite number an exact divisor? An even number is not an exact divisor of what? An odd number is an exact divisor of what?