(4) What fraction is that which if its numerator be increased by 1 and its denominator be diminished by 1, the value is ; but if its numerator be diminished by 1 and its denominator increased by 1, the value is

?

(5) A farmer wishes to mix 28 bushels of barley at 2s. 4d. a bushel, with rye at 3s. a bushel, and wheat at 4s. a bushel, so that the whole may consist of 100 bushels at 3s. 4d. a bushel: how much rye and wheat must he use for this purpose?

(6) A sum of money was divided equally amongst a certain number of persons: had there been three persons more, each would have received one shilling less; and had there been two persons fewer, each would have received one shilling more: required the number of persons, and what each received.

(7) A and B engage in play; at the first sitting A wins as much as he began with and four shillings more, and finds that he has twice as much as B. At the second sitting B wins half as much as he began with, and finds that he has three times as much as A has: what sum had each at first?

(8) How may a bill of £7 4s. be paid with half guineas and crowns, so that twice the number of crowns may be equal to three times the number of half guineas?

(9) Three masons, A, B, C, engage to build a wall: A and B jointly would build it in 12 days; B and C in 20 days, and A and C in 15 days. In what time will it be finished?

(10) A cistern can be filled by the pipes A and B in 70 minutes, by the pipes A and C in 84 minutes, and by the pipes B and C in 140 minutes. What time will each pipe take to fill it, and in what time will it be filled if all the pipes flow together? [This and 9 are easily solved with one unknown.]

(11) A person rows a distance of 20 miles and back in 10 hours, the stream flowing uniformly in the same direction all the time: he finds that, with the stream, he can row 3 miles in

figures the second be double of the first, then three times the sum of those figures added to the number will always invert the digits: thus

Hence, in the question in the text, three times the sum of the digits is 18, and, consequently, their sum is 6; and as the second is double the first, the first will be a third of 6, or 2, and the second 4. It follows from what is here shown that every number of two figures, of which the first is double the second, is equal to seven times the sum of those figures; so that when the second figure is double the first, seven times their sum will invert the digits.

the same time that it takes him to row 2 miles against it. How long was he going with the stream, and how long against it?

(12) The fore wheel of a carriage makes six revolutions more than the hind wheel in going 120 yards; but if the circumference of each wheel were increased by 3 feet, the fore wheel would make only four revolutions more than the hind wheel in the same distance. What is the circumference of each wheel?

(13) A, B, C, possess certain sums of money, such that if A receive in addition half what B and C have, he will possess £a; if B receive a third of what A and C have, he will possess £b; and if C receive a fourth of what A and B have, ne will possess £c. What sum has each?

(14) All that is known respecting the coefficients a, b, and the quantity c, in the trinomial expression ax2+bx+c, is that when 4 is put for x, the value of the expression is 42, that when 3 is put for x, its value is 22, and that when 2 is put for x, its value is 8. What are the coefficients a, b, and the number c?

(15) Five persons engage in play on the condition that he who loses shall give to each of the others as much as he already has. All lose in turn, and yet at the end of the fifth game they all have the same sum, namely, £32. How much did each begin with?

(16) There are three ingots composed of different metals: a pound of the first contains 7 ounces of silver, 3 ounces of copper, and 6 ounces of pewter; a pound of the second contains 12 ounces of silver, 3 ounces of copper, and 1 ounce of pewter; a pound of the third contains 4 ounces of silver, 7 ounces of copper, and 5 ounces of pewter. It is required to find how much must be taken from each ingot to form a composition which shall contain 8 ounces of silver, 3 ounces of copper, and 4 ounces of pewter in a pound of it.

(17) Divide the number 90 into four parts, such that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the results may all be equal to one another.

(18) A number consists of three digits of which the difference between the first and second is the same as the difference between the second and third. If the number be divided by the sum of the digits the quotient will be 26; but if 198 be added to it, the digits will be inverted. Required the number. (19) A person distributes a shillings among ʼn persons, men

and women: to the men he gives p pence each; to the women pence each. Required the number of men and women.

(20) If from a vessel of wine containing a gallons, b gallons be drawn off, and the vessel then filled up with water, and this operation be repeated n times successively, find the quantity of wine remaining after these n operations are finished.

CHAPTER VI.

ON THE THEORY OF INDICES OR EXPONENTS, AND THE REDUCTION OF SURDS.

THEORY OF EXPONENTS.

61. We have already seen (page 27) how a fractional index placed over the right hand corner of a quantity is made to supply the place of the radical sign prefixed to that quantity, so that if we wish to express the second, third, fourth, &c. roots of any quantity x, instead of employing the radical sign, we may use the notation *, x, x, for the same purpose. The symbol a may, of course, stand for anything: it may, for instance, be a power of some other quantity. Thus, if a represent a3, the above forms become (a), (a), (a)*, &c. which severally stand for the second, third, fourth, &c. root of the fifth power of a; but a better way of expressing a root of a power is to write the index of the root as the denominator, and the index of the power as the numerator, of a single fractional exponent, rather than to exhibit both exponents in a detached state as above. Connecting the exponents in this manner, the expressions just written take the neater forms a3, a*, a3, &c. And it is in this way that a root of a power is generally indicated, the notation a meaning the nth root of the mth power of a. This general convention being fixed, a fractional exponent over any symbol becomes quite as intelligible and quite as significant as an integral exponent. Previously to this article the learner felt no difficulty as to the meaning of a3, a3, a1, &c. and he can now feel as little as to the meaning of a3, a3, a*, &c.; these latter symbols standing each for the seventh root of the corresponding forms or powers above. It remains to be seen whether the rules

hitherto confined to quantities with integral exponents may be extended to those with fractional exponents.

62. When the exponents are positive whole numbers, we know that am xa"a"+", and also that a"÷a"a"-", provided that m>n; that is, that m be greater than n. It is further obvious that xTM×y"=(xy)"; that is, it is plain that to m factors X

XXXX

.........

These identities we have marked (A) and (B) for the purpose of reference in the following inquiry as to whether the rules for multiplication and division are the same when the exponents are positive fractions as when they are positive whole numbers; that is to say, whether the following equations are true, namely,

Put the single symbols x and y for a3 and a1; then (a")"= x2, and(a1)'=y'; but since the qth power of the qth root of a

quantity is that quantity itself, it follows that the first of these equalities is the same as a2=x; and similarly, the second is a=y'; consequently, a"=x, and a"=ya................ (C) and multiplying these together,

a13× a¶=x13×ya; that is, ap+=(x×y)", by (A); and therefore, employing the notation agreed upon above,

a13÷÷aa=x2÷ya; that is, a-=(x÷y)", by (B);

and therefore, by the notation agreed upon,

It follows, therefore, that the rules for multiplication and division are the same, whether the exponents be positive whole numbers or positive fractions.

63. In the rule for division, it has been a stipulation hitherto that the exponent of the divisor be less than that of the divi

dend: when such is the case we have

xm

-"; if, on the con

trary, nm, as for instance n=m+k, then the first member

of this equation would be which, by dividing numerator

and denominator by the common factor x", is the same as

1

also the second member would be xTMm-m- or x. This circumstance suggests a further extension of the theory of exponents, and shows us that quantities with negative exponents may always be used instead of the reciprocals of the same quantities with positive exponents, provided we find that under this change of form the laws of combination in multiplication and division still subsist; that is, provided that in the former rule the exponent of the product is still the sum of the exponents of the factors; and in the latter rule the exponent of the quotient is still the difference between the exponents of dividend and divisor. That such must be the case is pretty obvious; for if, instead of instead of 1

хп

1

we write x-", and

хт

we write "; then instead of

1

‚m+n'

, we must, con

sistently with the same notation, write x-m-n '; and instead of

we must write "-"; and it is plain that this being agreed