Ex. Find the time between 3 and 4 o'clock, when the hour and minute hands of a clock are (1) coincident; (2) exactly opposite to each other; (3) at right angles to each other.

The lines, OA, OH, denote the position of the hands at 3 o'clock, and the lines, OB, OC, the respective positions of the minute hand and the hour hand in the three different cases. Suppose x = number of minute spaces passed over by the hour hand since 3 o'clock; then 12x=number of minute spaces passed over by the minute hand since 3 o'clock. The following statements will now be easily understood : (1) Arc AHB 12 times arc HB=AH+HB; that is, 12x=15+x, whence x=11, or the hands are coincident at 15+1=16 minutes past 3.

(2) Arc AHCB=12 times arc HC=AH+HC+CB; that is, 12x= 15+x+30, whence x=41, or the hands are exactly opposite at 45+4=49 minutes past 3.

(3) Arc AHCB=12 times arc HC=AH+HC+CB; that is, 12x= 15+x+15, whence x=21, or the hands are at right angles at 321 minutes past 3.

Observe, if we had taken x to represent the number of minute spaces passed over by the minute hand since 3 o'clock, then we should have had directly

(a) x=15+ whence x= =16 minutes past 3.

12'

180

11

540

11

360
11

(b) x=15+ +30, whence x= =49 minutes past 3.

12

(c) x=15+ 12+15, whence x= =32

minutes past 3.

These results the student ought to be able to write down at once, as soon as he has mastered the application of the principle that the minute hand moves over twelve minute spaces whilst the hour hand moves over one minute space. Let him practise himself by finding the respective times between 7 and 8 o'clock, when the hour and minute hands of a watch are (1) exactly opposite each other, (2) at right angles to each other, (3) coincident.

12. If a man can do a piece of work in x hours he can do

work in 1 hour; and, on the same principle, if a tap can fill a vessel in x hours the part of the vessl filled by the tap in 1 hour will be 1.

x

Ex. 1. If A can do a piece of work in 8 days, and B the same in 10 days, in what time will they finish it together?

1

х

Let x =

number of days required, then in 1 day they will do of the work; .. also in 1 day A will do of the work, and B in

1 day will doo; and .. in 1 day they will both do (+1%) of the = whence x= - 4 days.

work; we have +

1

Observe, we might have reasoned thus: In 1 day A does part of the work, in x days he does part of it; and, in the same way,

х

Then, taking unity to represent the work, we should have

It is sometimes usual to let W or w represent the work; when the xW xW equation would stand thus: + =W. But this symbol is not 8 10

necessary, for the equation, when divided by W, stands as before, and unity may as well represent the work.

It is well for the student to notice generally that if a person does

fore the whole work in days. For example, if he does in 1 day

p

he will do in of a day, and therefore, or the whole, in or 2 days.

Ex. 2. A can do a piece of work in 10 days; but after he has been at work upon it 4 days B is sent to help him, and they finish it together in 2 days. In what time would B have done the whole?

If x = number of days B would have taken, then in 1 day B would

do of the work, and A would do of it in the same time ;

х

.. in 4 days A does 1%, and in 2 days A and B together do 2 (1 +

Taking unity to denote the work, we have + 10 +

2

=

1, whence

Ex. 3. If 3 pipes fill a vessel in a, b, c minutes, running separately, in what time will the vessel be filled when all three are opened at once?

1

Let x the required time in minutes, then will denote the part

of the vessel filled in 1 minute.

will be the respective parts filled by the three

1 1 1 1

taps in 1 minute, and .. we have + + = whence

x=

α

abc

́ab + ac+bc'

с

Or let x= number of minutes required. In one minute the first tap would fill part of the vessel, .. in x minutes it would

a

Similarly the second and third taps would fill

х

the same time. Then taking unity to denote or the whole, we

х x х

have +

a

+= 1, whence we get the same value of x as before.

с

Ex. 4. A cistern can be filled in half-an-hour by a pipe A, and emptied in one-third of an hour by another pipe B. After A has been opered one-third of an hour B is opened for 12 minutes, when A is closed, and B remains open for 5 minutes more, and now there are 13 gallons in the cistern. How many gallons will it contain when full?

Let x= number of gallons required; in 1 minute A pours in gallons, and B lets out gallons, but A is opened for 20+12 or 32 minutes, and B for 12 5 or 17 minutes,

13. If a rower can pull at the rate of x miles an hour in still water, and if he be rowing on a stream running at the rate of y miles an hour, then x+y will denote his rate down the stream, and x-y his rate up the stream.

Ex. 1. A crew which can pull at the rate of 16 miles an hour down a river finds that it takes twice as long to come up the river as to go down. At what rate does the river flow?

Herex+y=16, and 2 (x − y) = 16; whence y=4. Hence the rate required is 4 miles an hour.

Ex. 2. A crew which can pull at the rate of 9 miles an hour in still water finds that it takes twice as long to come up the river as to go down. At what rate does the river flow?

Let x= required rate, then x+9= distance per hour down the river, and 9-x=distance per hour up the river. But by question x+9=2(9-x), whence x=3.

Ex. 3. A waterman finds that he can row 5 miles in of an hour with the tide, and that it takes him 1 hour to row the same dis ance against the tide when it is only half as strong. Find the rate of the tide per hour.

If x= rate of tide in miles per hour, then

but half as strong. Also,

5

x

2

= rate when it is

20 x, or -x rate of the boat in still 3 5

2

10 x

water, that is, when there is no tide; and + or + also =

3 2

14. If x and y represent the significant digits, the algebraical representation of a number of two digits is 10x+y; similarly 100x + 10y+z will represent a number of 3 digits, x, y, z. If the digits composing the last number be inverted the form will be 100% + 10y+x.

Ex. 1. A number consisting of 3 digits, the absolute value of each being the same, is 37 times the square of any digit. Find the number.

Here 100x+10x+x=37x2, that is 111x=37x2, or 37x=111, whence x=3, and the number is 333.

Ex. 2. A certain number consisting of two digits is equal to six times the sum of the digits, and if 117 be subtracted from three times the number, the digits are reversed. Find the number.

10x+y=6(x+y) 3(10x+y) - 117=10y+x

Here we get the equations x=5 and y=4, and the number is 54.

whence

The above principles are capable of indefinite extension to far more complex examples than those we have presented. We have not attempted to exhaust the subject. The student should regard the cases we have given both as hints for practical work and as suggestions of the way in which the elementary points of a problem should be firmly grasped before we attempt its solution, or get lost in its labyrinthine mazes.

TEXT-BOOKS IN ALGEBRA.

I. Elementary.

Among the many smaller works which include all the requirements of the First B.A. Examination in Algebra, Hamblin Smith's Elementary Algebra (3s. 6d., Key 9s., Rivington) holds a distinguished place, and deservedly so, for it presents a fresh, clear, and somewhat original treatment of the subject. It is remarkably well graduated and classified, and we do not know of any work with which the learner may more profitably commence his studies; whilst many a pupil who is more advanced, but has had the misfortune to use an unsystematic and confusing work at the beginning, may find many suggestive points in this excellent treatise. The author has " carefully abstained from making extracts from books in common use," and hence the freshness of the work. It contains a useful chapter on the Resolution of Expressions into Factors, and gives about 2,300 examples, with which the student may surely have abundant exercise; or, if he wants further tests of his ability, he may find them in a book of Miscellaneous Exercises (2s. 6d., Rivington) by the same author, and specially adapted to the above treatise. Algebra, Part II. (8s. 6d., Rivington), by E. J. Gross, M.A., is designed to be a continuation of Hamblin Smith's more elementary work. There is prefixed to it an Appendix containing several proofs and propositions which properly belong to the portion of Algebra treated of in Part I., but were thought too difficult for the student when first beginning the subject. This Appendix, which is limited to 13 pages, among other things gives the methods of cross multiplication, of summing the squares and the cubes of the first ʼn natural numbers, of finding the values of recurring decimals by geometrical progression, and the number of combinations and of permutations of n things taken r at a time. The other subjects of the book are mainly above the requirements of the Pass candidate; but they are treated in a full and masterly style, which the Honours candidate will know how to appreciate, and to him we most cordially commend it as a first-rate text-book, not open," as a critic has well said, "to the standing reproach of most English mathematical treatises for students-a minimum of teaching and a maximum of problems."

Two other smaller Algebras may be mentioned as containing all the subjects required of the Pass candidate, and a little more of the theory of Algebra than the elementary work mentioned above, though scarcely as much as it is desirable for the candidate to be acquainted with. We allude to Pryde's Algebra, Theoretical and Practical (2s. 6d., Key 2s. 6d., Chambers), and to Atkins's Elements of Algebra for Middle-class Schools and Training Colleges (3s., Collins). The method adopted in the first treatise is "to state the rule concisely and clearly, to illustrate it by appropriate gradational examples, to

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