of.

82. If of a ship's crew can load of her cargo in a day, how long would it take the whole crew to load the whole vessel? how long would it take of the crew to load of the cargo?

83. If of a ton may be carried § of a mile for 14 of a dollar how far may of a ton be carried for of a dollar? How far may of a ton be carried for 33 dollars?

X. MISCELLANEOUS QUESTIONS.

84. If a pole is under water, what part is out? if it is under the water, what part must be out? Solution thus: Three-thirds, (the whole pole,) minus, (less,) two-thirds, equals ONE-THIRD, i. e., -2=3. Answer. Hence, if a pole is under the water, then must be out. Again, §=. Answer. Hence, if a pole is under the water, it follows that must be out.

85. If of a school are present, how many must be absent? How many would be present if were absent? if were absent? if were absent?

86. 1 is what part of 5? 2 is what part of 5? 3 is what part of 5 4 is what part of 5? 6 is what part of 5? 7 is what part of 5?

Answer, 1 is part (one-fifth part,) of 5; 2 are part (two-fifths part,) of 5; 3 are part of 5; 4 are part of 5; 6 are part of 5, and 7 is part of 5. For the sake of brevity we more frequently say 1 is 3 of 5; 2 are 2 of 5, &c.

87. If a pole has of its length in the mud, and in the water, how much is out of the water? Answer, one half equals three-sixths, and one-third equals two-sixths; and three-sixths, PLUS (more,) two-sixths, are five-sixths; or, thus, 2, and 3=2; and + are .

88. If of the length of a pole is in the mud and water, and of the covered part is in the mud, how much of the covered part is in the water?

Answer, two-thirds of three-fourths are equal to six-twelfths, equal to one half, i. e., 3 of 3 to in the mud, and three-fourths, less one half, equal to THREE-fourths, less Two

fourths, equal to ONE-FOURTH is in the water, i. e., 3-3=30 -, in the water.

89. A. B. and C. owned a vessel of which A. owned and B., what was C.'s share? Thus: if from the whole vessel we take one half, we shall have a remainder of ONE HALF, or THREE-SIXTHS, from which if we take one-third, or two-sixths, we shall have remaining ONE-SIXTH, which is C.'s share, or the ANSWER. The same result will be found if we find the sum of and to +2=3, and subtract it (§), from the whole, (§), thus, &—5=}. Answer.

he

90. A man had a certain quantity of money, of which gave to A., and to B., after which he had 6 dollars left, how many had he at first? How many dollars did he give to each person?

5

3

5

2

Thus: ++ to, and the whole, or 18-% +10 10. ; therefore, T is given to 6 dollars, and, and 1, or the whole, are respectively required; i. e., 1, 10, and 18 are required, which may be found as before, viz. If 3=f dollars, then will 10 off dollars if =20 dollars, then will

=

=

of 20

10 dollars, and if=20 dollars, then

to 20 dollars, and

to 20 of a dolwill = 3 of 2o=

to 20 of a dollar = 4 dollars. Hence, he had 20 dollars at first, and gave 10 dollars to A., and 4 dollars to B.

91. There is a pole standing so that of it, is under the water, and 12 feet out, how long is the pole? Take from §, (whole,) and we have left, to 12 feet. Therefore, given to 12 feet, and (the whole,) required.

is

= to

92. There is a pole of which is in the mud, and as much in the water, and the part which is out of the water measures 29 feet, how long is the pole? Thus: of 3, in the mud, and to, is in the water; therefore, = to, in the mud and water, and the remainder, to), is out of the water, which (9), by the conditions is = equal to 29 feet, here 3 of the pole is given = to 29 feet, and the whole (1) required.

93. A boy seeing a liberty pole wished to know how high it was. A gentleman standing by, said that and of the pole were equal to 24 feet, how high was the pole? Here

5

+to+3 to 1 of the height is given = to 24 feet and (the whole height,) is required.

94. A. owned of a vessel, B. owned as much as A., and C. owned as much as B., how much did each own? What part of the whole vessel would the sum of their shares make? Thus of 1=1, B.'s share; of =2, C.'s share. Now,+} +=22+2+24=17·

Hence, A. owned 1, B. †, and C. 24, and they all owned 11 of the vessel. ANSWER.

95. If and of a yard of cloth cost 15 dollars, what would a yard cost at that rate?

96. If of of a yard cost 1 dollar, what would a yard cost? what 2 yards cost? what would 5 yards cost?

97. Two men, A. and B., talking of their money, A. said to B., and of my money is just equal to of of yours, and B. said to A., and of my money will make exactly 10 dollars, how much money had each of them? Thus: +=+= to, and of, i. e., of A.'s money equals of B.'s; therefore, the whole (1), of A.'s money equals of B.'s = to of to of B.'s. Now, and of B.'s money is exactly equal to 10 dollars, i. e., 5 of B.'s money = 10 dollars. Hence, of B.'s money is given to 10 dollars, and the whole (†), and respectively required.

=

98. Two men, A. and B., found a purse of money, and disputed who should have it. A. said,, and of the money were equal to 47 dollars, and if B. could tell how much there was he should have of it; how much was there?

99. There is a pole erected, so that 5 feet is in the ground, which is of the length of the pole, how long is it?

100. There is a school where of the pupils study Arithmetic, study Grammar, and study Geography, and (the remainder,) 5 are learning Geometry, how many are there in the school?

101. If of 6 be 3, what will of 20 be?

102. A gentleman being asked how old he was, answered, if 20 years be increased by, this sum will be g of my age. How old was he?

103. What is

of of, increased by of % of ?

104. If 21 pounds

a pound cost?

of indigo cost 5 dollars, what will of

105. Charles had of a dollar, Stephen of a dollar, and John of a dollar, what part of a dollar had they all? How much more had Stephen than John?

and ? What is

and

? between

and ? be

or

or, and

and

greater or

106. What is the difference between the difference between 4 and? between and What is the difference between tween and, (i. e., which is the larger how much?) Is the difference between less than the difference between and ?? 107. What is the ratio of 1 to 2? i. e., 2? It is plain that 1 is of 2, and hence, the ratio of 1 to

2 is .

1 is what part of

108. What is the ratio of 2 to 3? i. e., 2 is what part of 3 Thus: 1 is of 3; hence, 2 is

ratio of 2 to 3 is 3.

of 3; therefore, the

of 5 to 6

ratio of 7 to 9? of 4 to 5 of 3 to 4? of 7 to 8 of 8 to 9

of 9 to 10? of

of 6 to 7 10 to 11 of 11 to 12? 111. What is the ratio of 3 to 2? Thus to 2 is; hence, the ratio of 3 to 2 is 31. principle, what is the ratio of 4 to 3? of 5 to 4 of 7 to 6 of 8 to 7 of 9 to 8 of of 12 to 11 of 9 to 5 of 7 to 3? 112. What is the ratio of 13 to of 15 to 14? of 15 to 16 of 17 to 16 of 17 to 18? of 18 to 17?

to 15

17

10 to 9

of 11 to 10?

14? of 14 to 13 of 14

of 16 to 15 of 16 to

113. What is the inverse ratio of 2 to 3? i. e., 3 is what part of 2? We know that 1 is of 2, and that 3, (which is 3 times 1,) must, therefore, be 3 times of 2, equal to 3 of 2, and hence, the inverse ratio of 2 to 3 is 1. In this case the consequent (3), is compared with the antecedent, (2),

which must always be the case when the ratio is inverse. What is inverse ratio? See Def. 5, page 70.

114. What is the inverse ratio of 3 to 4? i. e., what part same principle, what is the inverse ratio

of 3 is 4? On the of 4 to 5 of 5 to 7 to 8 of 7 to 6

6

of 5 to of 8 to 7

4

of 6 to 5

of 8 to 9

of 6 to 7? of of 9 to 10 of 9

of 10 to 11

of 11 to

12?

to 8 of 10 to 9 of 12 to 11

19 to 20?

of 11 to 10? of 13 to 14 of 14 to 13 of 17 to 19? of

From these examples we learn that, to find the ratio of one number to another, we must, in all cases, begin with the UNIT. The ANALYSIS OF NUMBERS is based upon Four elementary principles which are in themselves self-evident, and may, therefore, be styled ARITHMETICAL AXIOMs* as follows:

I. ONE (1), is one (1), time one (1), i. e., is time }, and since 2 is Ì repeated twice, (Article VI., page 35,) it follows, that are times, or, (by leaving off the unit at the bottom, or the denominator, which shows that the number at the top, or numerator, is an INTEGER,) 2 are 2 times 1; 3 are 3 times 1; 4 are 4 times 1; 5 are 5 times 1; 6 are 6 times 1, &c., ad infinitum. We next notice the reverse of this, viz. that,

II. ONE (1), is the whole of one, (†); hence, is (one half,) of, or, (leaving off the denominator, (1), it follows that,) I is (part,) of 2, (by increasing the denominator of the multiplier instead of the numerator, as before, (that also) 1 is of 3, of 4, of 5, of 6 of 7, 1 of 8, of 9, to of 10, &c., ad infinitum.

In the third place we will notice that which grows out of AXIOMS I. and II., combined, viz. that,

III. () of 1 is

of 1, i. e., one half of one is ONE half is twice 1 it, follows that,

and on the same principle

of one, and since 2
(two halves,) of 1;
(three halves,) of 1; of 4 is of 1; of 5 is
ad infinitum.

of 2 is

of 3 is

of 1, &c.,

IV. One (1), is equal to (), two halves, equal (3), threethirds, i. e., (a unit, or a whole one,) ====== ===10, &c., ad infinitum.

* From (Lat.) Axioma, a general principle, or self-evident proposition.