Find the rates of travelling, and the distance between the extreme mile-stones.

or

Now at first A walks 4 and rides 3 miles) while B walks 3 and rides 4 miles)'

A walks 4 while B walks 3 and rides 1;

that is (since horse's rate is double of B's), while B walks 3 miles; .. A's and B's rate at first may be represented by 8y and 7y respectively.

Again,

while

A walks x- 3 and rides x

- 4,

B walks x 4 and rides x-3;

.. A walks x-3 while B walks x-4 and rides 1,

that is, while B walks x − 4 and A walks 1;

.. A walks x

7

while B walks x - 4,

2

1

1

but A walks 8y +

while B walks 7y+

2

2

... distance = 16 miles; rates of travelling at first = 4 and 34 miles per hour respectively.

6. A and B set out to walk together in the same direction round a field, which is a mile in circumference, A walking faster than B. Twelve minutes after ▲ has passed B for the third time, A turns and walks in the opposite direction until six minutes after he has met him for the third time, when he returns to his original direction and overtakes B four times more. The whole time since they started is three hours, and A has walked eight miles more than B. A and B diminish their rate of walking by one mile an hour, at the end of one and two hours respectively. Determine the velocities with which they began to walk.

Let x = number of miles per hour of A at the first,

In 3 hours A has gone x + 2 (x − 1) = 3x – 2 miles,

.. by the question 3x-2-(3y-1)=8; .. x - y = 3,

that is, the relative speed of A and B is 3 miles per hour; therefore A will gain a circumference on B in § of an hour, and will therefore be passing B for the third time at the end of the first hour.

Also since the relative speed of A and B is the same in the last hour as in the first, and since A passes B for the fourth time at the end of the third hour, therefore he will pass him all the four times within the last hour; the first time being exactly at the commencement of the third hour.

Now in 12 minutes after the first hour the distance between

1

2

A and B is c-y-1)= miles; .. time of first meeting

5

(x+y-1); and time of meeting twice more = 2÷ (x + y − 1).

In 6 minutes the distance between them=

now turns, the time of overtaking B

749. The equations in the preceding Chapter and their solutions, and the solutions in the present Chapter, are due to the Rev. A. Bower, late Fellow of St John's College. Should any student desire more exercises of this kind, he is referred to the collection of algebraical equations and problems edited by Mr W. Rotherham of St John's College.

MISCELLANEOUS EXAMPLES.

1. Exhibit {n√(a2 + b3) − a √(m2 + n3)}3+ b3m3 as a square.

2.

Extract the square root of 6 +6 + √14+ √21.

3. Find the scale of notation in which the number 16640 of the common scale appears as 40400.

5. At a contested election the number of candidates was one more than twice the number of persons to be elected, and each elector by voting for one, or two, or three, or as many persons as were to be elected, could dispose of his vote in 15 ways; required the number of candidates.

T. A.

...

32.

6. In how many ways may the sum of £24. 158. be paid in shillings and francs, supposing 26 francs to be equal to 21 shillings? 7. Find the sum of n terms of the series

+

(1 + z) (1 + 22) (1 + 2′) (1 + z®)

8. Shew that 1 + 2x* is never less than x2 + 2x3.

+

9. If an equal number of arithmetic and geometric means be inserted between any two quantities, shew that the arithmetic mean is always greater than the corresponding geometric mean.

10. If x be any prime number, except 2, the integral part of (2+√3)-2x+1+1 is divisible by 12x.

11. Shew that if n=pq, where p and q are positive integers,

13. If p be the probability à priori that a theory is true, q the probability that an experiment would turn out as indicated by the theory even if the theory were false, shew that after the experiment has been performed, supposing it to have turned out as expected, the probability of the truth of the theory becomes Р

P+q-pq

14. Of two bags one (it is not known which) is known to contain two sovereigns and a shilling, and the other to contain one sovereign and a shilling; a person draws a coin from one of the bags, and it is a sovereign, which is not replaced. Shew that the chance of now drawing a sovereign from the same bag is half the chance of doing so from the other. Supposing the drawer might keep the coin he draws, what is the value of the expectation ?

15. All that is known of two bags, one white and one red, is that one of them, but it is not known which, contains one sovereign and four shilling pieces, and that the other contains two sovereigns and three shilling pieces; but a coin being drawn from each the event is a sovereign out of the white bag and a shilling from the other. These coins are now put back, one into one bag, and the other into the other, but it is not known into which bag the sovereign was put. Shew that the probability of now drawing a sovereign is in favour of the red bag as compared with the white in the ratio of 13 to 9.

16. If n be the number of years which any individual wants of 86, find the value of an annuity of £1 to be paid during his life; adopting De Moivre's supposition that out of 86 persons born, one dies every year, until they are all extinct.

LVI. MISCELLANEOUS THEOREMS.

750. The present Chapter consists of some miscellaneous theorems on the following subjects; abbreviation of algebraical multiplication and division, vanishing fractions, permutations and combinations, convergency and divergency of series, continued fractions, and probability.

751. In multiplying together two algebraical expressions it is sometimes convenient to abridge the written work by expressing only the coefficients. For example, suppose it required to multiply 2x + x2 - 3x + 1 by x2+3x-2; we may proceed thus:

A similar abridgement of the written work may be made in division.

This mode of operation has been sometimes called the method of detached coefficients.