find (1) the most probable collective value of the jewels he will draw: (2) the value of his expectation. (3) What would be the value of his expectation, if he were allowed to draw r jewels from each casket? If p=2, and one jewel be drawn from each casket, what is the chance (4) that the value of the jewels drawn is n+r guineas? (5) that the collective value of the jewels drawn is that collective value which is most probable?

n

Ans. (1) (p+1) guineas, if either n be even or p odd ;

n

and 2 (p+1)+ §guineas, n being odd and p even, each of these being equally probable.

72. A person borrows £c on the following terms. It is to be paid off in n years and at the end of each year is to be paid interest at a given rater on the sum remaining unpaid at the beginning of the year, together with such a portion of the principal that the whole sum paid on account of principal and interest together shall be the same for every year. Investigate a formula for the sum to be paid every year.

73. From the gold fields are brought 2n specimens of gold dust, no two of which are of the same degree of fineness. Each specimen is divided into as many equal portions as is necessary to the following operation, viz. to form as many different mixtures as possible by taking a portion of dust from some specimen and mixing it with a portion from some other specimen. Each of these mixtures is now divided into as many equal portions as is necessary to the following operation, viz. to form as many new mixtures as possible by mixing together portions from any n of the former mixtures. Prove that 1.3.5.7...... (2n-1) of the latter mixtures will be of the same degree of fineness as a mixture formed by mixing together all the dust of 2n specimens exactly like the original specimens.

EASY EXERCISES.

EXERCISES. A.

IF a stand for 10, b for 3, and x for 7, what is the value of each of the following quantities?

(16) What is the coefficient of a in each of the quantities 2a, 2ab, abx, Sabx, ma, axx, pax, abxy?

(17) What is the coefficient of 25 in 125?

(18) What is the difference between 3+x, and 3x, when x stands for 7?

(19) What is the difference between 3a+x, and 3a-x, when a stands for 10, and r for 6?

(20) What is the difference between 3a+x, and 3ax, when a stands for 3, and x for 2?

Find the value of each of the following quantities, when a stands for 10, b for 3, and x for 7:—

If a stand for 1, b for 9, and c for 8, find the value of each of the

following quantities:

(31) a2+b2-c3.

(32) 13a2+3b3— 4c2.

(53) 5abc-22b'+3c3.

(34) ab+b'c.

(47) What is the difference between 3a and a3, when a stands for 2? (48) What is the difference between 2 √x and 2+√x, when x is 100? (49) What is the difference between 3 and √x, when x is 64? (50) What is the difference between √a+b and √a+b, when a stands

for 1, and b for 8?

(51) What is the difference between √a, when a stands

for 16, and b for 4?

√

and

Add together (1) a+b, and a+b.

(2) a+b, and a-b.

(3) a-b, and a-b.

(4) a-b+c, and a+b-c.

(5) a-b+c, and a+b+c.

EXERCISES. B.

(6) 1-2m+3n, and 3m-2n+1.

(7) 5m+3, and 2m-4.

(8) 3xy-2x, and xy+6x.

(9) 4p-2q+1, and 7-3p+q. (10) 5ab-2bc, and ab+bc. (11) 2ax+3by, and ax-by. (12) 3a-2b+4c, and 2a-3b+c. (13) xy+x-7, and 3xy-2x+3. (14) p+q-pq, and 2pq−3p+2q. (15) p2+2pq+q3, and p2−2pq+q°. (16) 7ab-5ac+1, and ab+6ac-2.

(17) 7x-6y, -x-3y, -x+y, -2x+3y, and x+8y.

(18) 3-a, -8-a, 7a-1, -a-1, and 9+a.

(19) 2a-5b+3c-d, and a+5b-c+2d.

(20) a2+2ab+b2, and 2a-ab-36'.

(21) 3x2-6x+5, 2x-3-x, and 4-x-2x2.

(22) ac+bd, bd-cd, and ac+cd.

(23) 3x2-4y', x2+y', and '-¿y'.

(24) x3-3x2+3a2x-a3, and 3a3— Qa2x+4ax2— x3.

(25) 8mn+m, and 1-n-7mn.

(26) 9x-8y-7, and 3z-9x+6y+7.
(27) a3-2ab+ab, Sal'-2ab, and b3+c.

(28) a3—§ab3, b3—§a2b, and ab2—§a2b.

(29) 4x2+2xy, şx2-xy+y3, and mx+ny.

(30) ad+2bd-3cd, ad-4bd, and ab+2cd-ac.

(31) I received m shillings from my father, the same from my mother, and ǹ shillings from each of three friends, express the whole sum.

(32) A certain sum is divided between A, B, and C; B receives a pounds more than A, and C receives b pounds more than B; if A receives x pounds, find an expression for the whole sum divided.

EXERCISES. C.

(1) From a take b-x.

(2) From a+b-c-d take a-b+c-d.
(3) From 6a-2b-3c take a-2b+2c.
(4) From a+x-b-5c take x+8b-5c.
(5) From 3x+2y-5z take 2x+3y+4z.
(6) From 2ax+by-c take ax-by+c.
(7) From 3bc-ab+a take 2bc+ab−a.
(8) From xy+x2+y3 take xy-x3+y3.
(9) From 2xy +3x2+4y2 take xy-2x3-y3.
(10) From 2mn+5m−3n take mn+m+n.
(11) From -2xy+mx-py take -3xy-2mx-py.
(12) From 5alc-2ab-3ac take 2abc+ab-ac+1.

(13) From a2-b2+c2 take a2-2b3— 2c2.
(14) From 4ax-3a2+2x2 take 2ax-a2+4x2.
(15) From 3a b+2a c-5c2 take a❜b-a2c-7c2.
(16) From 2xy+3a-a2b+5 take 2a-a2b+6.

(19) The united ages of a father and his son make 60 years, and the father was 30 years old when the son was born, what is the age of each?

(20) Divide 1 into two fractional parts, so that one part shall exceed the other by

EXERCISES. D.

(3) -2xy by 4a.
(4) -2xy by -4a.

EXERCISES. E.

(23) a2-ax-6x2 by a- - 3x. (24) 2ab+6abc-8abcd by 1+3c-4cd.