13. Divide 205896 into 46 equal parts.

Ans. 4476.

14. A man divides 318 dollars among 6 boys; how much

will each one receive?

SOLUTION 1ST.-To give each boy 1 dollar will require 6 dollars, and if in giving 6 dollars each boy receives 1 dollar, in giving 318 dollars each boy will receive as many times 1 dollar as 6 dollars are contained times in 318 dollars, which are 53. Therefore, etc.

SOLUTION 2D.-If 6 boys receive 318 dollars, one boy will receive 1 sixth of 318 dollars, which, by division, we find is 53 dollars. Therefore, etc.

15. If 12 men earn 384 dollars in a week, how much does one man earn? Ans. 32 dollars. 16. A boat goes 1584 miles in 24 hours; how far will it go in 1 hour? Ans. 66 miles. 17. There are 1575 gallons in 25 hogsheads; how many gallons are there in one hogshead? Ans. 63 gallons.

18. There are 8316 cubic inches in 36 gallons of wine; how many cubic inches are there in one gallon?

Ans. 231. 19. There are 7614 cubic inches in 27 gallons of milk; how many cubic inches in one gallon?

20. There are 221760 feet in 42 miles; how there in one mile?

Ans. 282.

many feet are

Ans. 5280.

21. Sound moves 61545 feet in 55 seconds; how far does it move in one second?

Ans. 1119 feet.

22. If a road 132 miles long cost 339240 dollars, how much did it cost per mile? Ans. 2570 dollars. 23. The salary of the President of the United States is 25000 dollars a year; how much is that a day? 24. A man wills 8000 dollars to his

Ans. 68+ dollars. wife, 2000 dollars to 9 children, in equal

the church, and the remainder to his shares; what did each child receive, the fortune being 75000 dollars? Ans. 7222 dollars. 25. A man having 20000 dollars buys 150 acres of land, at 75 dollars an acre; how much land can he buy with what remains, at 125 dollars an acre? Ans. 70 acres. 26. The earth is 95 millions of miles from the sun; how long

does it take light to pass from the sun to the earth, its velocity being 192500 miles per second? Ans. 493 sec.+.

27. Mercury is 37 millions miles from the sun, Venus 68 millions, Mars 145 millions, Jupiter 495 millions, Saturn 900 millions, Uranus 1800 millions, and Neptune 2800 millions; in how many seconds will light pass from the sun to each planet; in how many minutes, there being 60 seconds in one minute? Ans. To Mer. 192 sec.+; or 3 min.+.

52. CASE III.-To divide by using the factors of the divisor. 1. Divide 2952 by 24, using the factors 4 and 6.

SOLUTION 1ST.—To multiply by 24 we may multiply by 6, and then multiply that product by 4; hence, to divide by 24 we may divide by 4, and then divide that quotient by 6. Dividing by 4 we have 738, and dividing 738 by 6 we have 123; hence, etc.

OPERATION.

4)2952

6)738

123

SOLUTION 2D.-Since 24 times a number equals 6 times 4 times the number, of the number equals of of the number; of 2952 is

738, and of 738 is 123; hence, etc.

RULE.-Divide the dividend by one factor of the divisor, the quotient by the other factor, and so on for all the factors used. The last quotient will be the quotient required.

NOTE.-When remainders occur it is evident that the true remainder is not the last remainder, nor the sum of the remainders; it is therefore necessary to ascertain how to find the true remainder.

14. Divide 791 by 24, using the factors 2, 3, and 4.

SOLUTION.-In dividing by 2 we see that 791 equals 395 twos and 1 remaining. Dividing 395 twos by 3 we find that 395 twos equals 131 sixes and 2 twos remaining. Dividing by 4 we find that 131 sixes consists of 32 twenty-fours and 3 sizes remaining. Hence the entire remainder is 3 sixes or 18, plus 2 twos or 4, plus 1, which is 23. Hence we have the following

OPERATION.

2)791

3)395,

1

4)181, 2 twos = 4

32, 3 sixes=18

True remainder, 23

RULE.-Multiply each remainder by all the divisors preceding the one which obtained it, and take the sum of the products.

Divide the following and find the true remainder :

NOTE. The above article is generally called "Contractions in Division." It will be seen, however, that the so-called contracted method is often longer than the ordinary method.

53. CASE IV. To divide when there are ciphers at the right of the divisor.

From the preceding solution we derive the following

RULE.-I. Cut off the ciphers at the right of the divisor, as many places at the right of the dividend.

and

II. Divide the remaining part of the dividend by the remaining part of the divisor, and prefix the remainder to the figures cut off for the true remainder.

NOTE.-The above subject is usually presented in two cases: 1st, where the divisor is a unit with ciphers; 2d, where the divisor is any number with ciphers on the right. The division and the derivation of the remainder are explained by the case of dividing by the factors of a The explanation here given, it is believed, is not only shorter, but simpler and much more easily understood.

number.

54. EXERCISE UPON THE USE OF THE PARENTHESIS.

The Parenthesis (), denotes that the quantities included are to be subjected to the same operation: Thus, (8+6—4)×3 denotes that the value of 8+6-4, which is 10, is to be multiplied by 3.

1. What is the value of (12+9-7) X5?

SOLUTION.-12+9 equals 21, and 21 minus 7 equals 14, and 14 multiplied by 5 equals 70. Therefore, etc.

55. Arithmetical Analysis is the process of solving problems by a comparison of their elements.

All arithmetical reasoning consists of comparison, and in this comparison the unit is made a standard of reference. Thus, in two numbers whose relation is not seen, we compare them by their relation to the unit. The unit thus becomes a standard of comparison.

It will be noticed that when we pass from a collection to the unit the process is analytic, but when we pass from the unit to a collection the process is synthetic.

1. If 25 oxen cost 1125 dollars, what will 30 oxen cost at the same rate?

2. If 7 hogsheads contain 441 gallons, how many gallons do

9 hogsheads contain?

Ans. 567 gallons. 3. If there are 2080 perches in 13 acres, how many perches are there in 15 acres? Ans. 2400 perches. 4. In 17 miles there are 29920 yards; how many yards are there in 21 miles? Ans. 36960 yards. 5. Twenty-five barrels of flour weigh 4900 pounds; what is the weight of 36 barrels ? Ans. 7056 pounds.