Then from (2), we obtain 2v=20, or v=10; and from (1) we obtain (v + 2)2 + (v — ≈)2 = 202,

or 2v2+2x2=202,

.. 2x2=202-2v2,

=202-200=2,

.. x=10+1, or 10-1=11, or 9,

and y=10-1, or 10+1= 9, or 11.

The above equation might have been solved thus:

(3),

PROBLEMS WHICH PRODUCE SIMULTANEOUS QUADRATIC EQUATIONS, INVOLVING TWO

UNKNOWN QUANTITIES.

Ex. LV.

Examples worked out.

Ex. 1. There is a certain number consisting of two digits. The lefthand digit is equal to 3 times the right-hand digit; and if 12 be subtracted from the number itself, the remainder will be equal to the square of the left-hand digit. What is the number?

Let x= the left-hand digit,

y=the right-hand digit.

Then 10x+y=the number.

which equations solved give x=9 and y=3.

The number=10x+y=90+3=93.

Ex. 2. What two numbers are those whose difference multiplied by

the greater gives 40, and by the less 15?

Let x=the greater number, y = the less number.

Ex. 3. The fore-wheel of a carriage makes 6 revolutions more than the hind-wheel in passing over 120 yards; but if the circumference of each wheel be increased by one yard, it will make only 4 revolutions more than the hind-wheel in the same space. Find the circumference of each wheel.

Let x=number of yards in the circumference of the larger wheel. y= number of yards in the circumference of the smaller wheel. By the question,

which equations solved give x= 5 yards, and y=4 yards.

Ex. LV. Examples for practice.

1. There are two numbers, such, that if the less be taken from three times the greater, the remainder will be 35; and if four times the greater be divided by three times the less plus one, the quotient will be equal to the less number. Find the numbers.

2. A number consists of two digits; if 9 be added to it, the digits will be inverted; and if 10 be subtracted from it, the remainder will be equal to the sum of the squares of the digits; find the number.

3. The product of two numbers is 128, and the difference of their squares is 192; find the numbers.

4. There is a number consisting of 2 digits: the number is equal to three times the sum of its digits, and if it be multiplied by 3, the result shall be equal to the square of the sum of its digits: find the number.

5. What number is that the sum of whose digits is 15, and if 31 be added to their product the digits will be inverted?

6. The product of two numbers is 6 times their sum; and the sum of their squares is 325; find the numbers.

7. The sum of two numbers is 6, and the difference of their third powers is 56. Find the numbers.

8. There are two numbers such that 3 times the square of the greater added to twice the square of the less is 110; and half their product plus the square of the less is 4. What are the numbers?

9. Two persons A and B leave Cambridge, and walk in the same direction, at a uniform rate; A starts 2 hours before B, and, after travelling 30 miles, B overtakes A; but had each of them travelled half a mile more per hour, B would have travelled 42 miles before overtaking A; at what rate did they travel?

10. A dealer sold 60 bullocks and 80 sheep for £1060; but he sold 42 more of the latter for £90, than he did of the former for £45. Find the price of each bullock and each sheep.

RATIO.

116. The term Ratio is used to express the relation which exists between two quantities of the same kind with respect to magnitude: thus we speak of the ratio of two numbers, of two periods of time, or of any two quantities which may properly be compared with one another in respect of magnitude.

It is evident that this relation can only exist between quantities of the same kind; for a mile or a yard cannot be compared in respect of magnitude with a day or an hour; but a mile can be compared with a yard, or 3 miles with 2 miles; and a day may be compared in respect of magnitude with an hour, or 5 days with 6 days. In order to form a ratio, length must be compared with length, time with time, area with area, and so on.

The comparison between two quantities may be made by either of the two following methods: 1st, by considering "How much one quantity exceeds the other." This relation between the quantities is called their ARITHMETICAL Ratio: 2nd, by considering “How often one is contained in the other." This relation is called their GEOMETRICAL Ratio.