(15.)

Let x denote the number of apples which John hal, and y the number Charles had.

Let x denote the number of dollars in A's salary, and y that in B's; then,

Let x denote the number of dollars which A has, and y the number B has; then,

Let x denote the value of the first, and y that of the second.

Then, by the first condition,

x + 7 = 3y'

and, by the second condition, 5x = y + 7; from which we have x and y.

(13.)

Let the numbers be denoted by x and y.

Then, by 1st condition,

and by the second,

from which we have,

6x=5y,

Let the numbers be denoted by x and y.

Let the present ages of the father and son be denoted by

(5.)

Let x denote the greater, and y the lesser part.

Then, by 1st condition,

and by 2d condition,

which give,

x + y = 75;

3x7y+15;

x = 54, and y = 21.

(6.)

Let x denote the number of gallons of wine, and y the

Let a denote the number of guineas, and y the number of moidores used.

Now, it is evident that the number of pieces used, of each kind, multiplied by the number of shillings in the piece, will give the number of shillings paid in that particular kind of money. That is, 21x will be the number of shillings paid in guineas, and 27y the number paid in moidores. Then, observing that the whole bill, £120 = 2400s., we have, x + y = 100; 21x+27y = 2400;

by 1st condition,

by 2d condition,

which give,

x = 50, and y = 50.

(8.)

Let x denote the distance traveled by the first, and y the distance traveled by the second.

But, since the first travels 8 miles, while the second travels but 7, the distance which they respectively travel will be in the proportion of 8 to 7; that is,

or,

x: y:: 8:7;
7x= 8y;

from which we find x = 80, and y =

70; and if the entire

distance traveled by each be divided by the distance trav eled each day, the quotient will be the time, 10 days.

(9.)

Let x denote the number of votes cast for the first, and y the number cast for the second.

which give, x = 233, and y = 142.

(10.)

Let x denote the value of the poorest horse, and y that of the other.

In this example, we must bear in mind that the minute hand goes entirely round the face of the clock, while the hour hand passes from one hour to the other; that is, the minute hand travels twelve times as fast as the hour hand.

If, then, we suppose the face of the clock to be divided into twelve equal parts, corresponding to the hours, and a and y to represent the distances passed over by the hour

and minute hands, from the time of separating until they are again together, we shall have,

since, when the hands come together, the minute hand will have gained the entire twelve spaces or the hour hand. Multiplying the second equation by 12, and adding them together, we have,

12y = y + 144;

or,

y =

144
11

= 13;

that is, the minute hand will have gone once around the face, and 1 of the hour spaces in addition; consequently, the time required will be 1 hour, 5 minutes, of 5 minutes, or of one minute.

If we subtract the second from the first equation of condition, we have,

Denote by x the portion of the beer which the man would drink in a single day.

Then, by the conditions of the question, the man and woman together would drink of the cask in a single day, and the woman of it; hence, what the man would drink must be equal to the difference; that is,