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5. If three persons, A, B, C, can separately perform a work in the times tı, tz, t, respectively; in what time they can do it jointly? and conversely, if they can do it jointly in time t, find in what times they can do it separately. Shew also that (ti+to+tz) -t can never be less than 9, if t be fixed, and when it is equal to 9, t, = t; = tz.

6. A and B can complete a work in a days, A and C in b days, and B can do p times as much as C in a given time; show that B and C (p-1)ab

b-pat can complete the work in

(p+1)(6-0)

days, and that A must do times as much as C. Why cannot p be greater than unity, and a greater than b simultaneously, and consequently this result negative ?

7. The effect m is produced by the action of the agents A and B together in the time a, and by B and C together in the time b, and by C and A together in the time c; in what time can each separately, and all together produce the same effect ?

8. To complete a certain work A requires m times as long a time as B and C together, B requires n times as long as A and C together, and C requires p times as long as A and B together. Explain the nature of the results which the problem admits. Show that 1 1 1 +

=1. m+1'n+1

XXVIII. 1. Determine the relations which must subsist between the coefficients of three equations involving three unknown quantities in order that one of the equations may be derived from the other two. 2. Explain the meaning of the equations

a,x+by+2=2qx+by+c=23c+by+c=1 when the values of x, y, or z take the form of Also shew that they are equivalent to two independent equations when

a, a, b, —Ь. _", —e,

а, а6, —b, c,с. 3. If ax+y+6= 0, 4x+by+a%= 0, batay+z= 0,

shew that aa'+bb,'+cc = abc+2a,bıcı, and (ab-c ')(ca6,4).x? = (bc-a,'(ab—')} = (bc-a,?) (ac—6,4)=?.

4. Find the condition which must exist in order that the equations ax+by=C1, Qgx+bay =C2, a 3x+bay=c;, may be consistent; and shew that under this condition the equations a,x+by+cæ= d, ayx+by+z= d., ayx+by+ez = ds, are inconsistent.

5. Explain the method of indeterminate multipliers for the solution of linear equations, and reduce the equations 4,2+by+oz+d, 2gc+by+c+d2 23x+by+c+d3

to the form mi M2

mg c+a + B +7

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6. If a=ac+ay+a,-, b = bx+buy+b>, a," + a, a, = 0

6, +62 +639 = 0, a,b, +azba+azb, = 0;
Then shall

aa2_ аз

bi ba bg 7. Solve the equations x+y+z=1, ax+by+cz=-(a+b+c),

a*x+by+c9% =(a+b+c)?; and shew that a*x+by+coz=(a*x+by+cʻz)?. 8. Give a solution of the equations 3x + 4y +52 = 10,

4x+5y+62 = 12, 5x + 6y +72 = 14, and point out the meaning of the result.

9. Shew that the following three equations are inconsistent, 27x+3y + 2x=40, x-y+z=3, 2x+4y— 32= 12.

1

1 1 1 1 1 10. Prove that the equations -+-+

b

C +-+

1, y

y c , , c ++ = 0, are equivalent to only two independent equations if

Y a+b+c= 0.

11. The system of equations x(m+n)'+yno+zm' = a,

xno+y(n+1)+zbo=b, xmo+yl+=(1+m)’=c, is incompatible, if mn+ni+Im=0, unless la+mb+nc = 0, and is then indeterminate.

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XXIX. 1. Apply the method of indeterminate multipliers to find the value of x from the equations,

y+=+u = 14, 2x+33+2u = 33, x+y+2u = 18, 3x+2y+2z = 20.

2. Given ax+by+cz+dv=h, a*x +bʼy+c*: + d’v = h', a'r+by+-+d*v = h, a*x+by+oz+div = h , find x, y, z, v.

3. Shew that the five equations, 2(x1 - x.)=(1+75)(xs – x3), 2(x,- x) = (1+5)(+-x), 2(x,-X.)=(1+5)(xs-), 2(x+— X)=(1+5)(x1 – X3), 2(X5 – X)=(1+5)(x,— xs), are equivalent only to two independent relations between X1, X., X3, Xa, and Xz. 4. Eliminate u, x, y, z between the equations

y+z+u = ax, 2+u+x=by, u+x+y=cz, x+y+x=du. 5. If A+B+C=D+ 2d, B+C+D=A+2a, C+D+A=B+2b, D+A+B=C+2c, and A+B+C+D=24,

then shall x = }{a+b+c+d}.

=
2
2

2
6. If y =

find the relation 2

2-4 2 between t and x.

7. What are the conditions necessary in order that an equation, or a series of equations, may give definite values for the unknown quantities contained in them?

8. How many equations must be given in order to eliminate 1, 2, 3, . ..n unknown quantities between them? Explain when and how such elimination is possible.

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9. If there be mtn unknown quantities and only m independent equations; then * of the unknown quantities can be considered as perfectly arbitrary and independent in value.

XXX. Find the solutions in positive integers of the equations :1. 20—3y=1. 2. 73-5y = 1. 3. 35x-19y=1. 4. 193—117y=11. 5. 17x 197 = 42. 6. 13x—177 = 54. 7. 13.x+7y= 141. 8. 9x + 13y = 200. 9. 11x +13y = 400. 10. 9x+13y = 2000. 11. 7+9y = 653. 12. 11c+5y = 100. 13. 30x+43y = 335. 14. 11x +177 = 248. 15. 32x+lly = 182. 16. 17:+23y = 200. 17. 30793+2711y = 37819000.

XXXI. Find the solutions of the following equations in positive integers :1. 3x+7y+172 = 100. 2. 6x-4y+7z=190. 3. 170+23y+3z=200. 4. 17x+19+212= 400. 5. 12x+15y+20% = 1001. 6. 7x+9y+232 = 999. 7. 4x+54 - 14x = 49. 8. 5x+7y+112= 224.

XXXII. Determine the solutions in positive integers of the following equations :

1. x+2y+32 = 20 and 4x+54 +62 = 47.
2. 6x+7y+43 = 122 and 11x+8y-6% = 145.
3. 4x+5y+72 = 560 and 9x+257-492 = 250.
4. 3x+8y+92= 656 and x+5y+43 = 272.

XXXIII. 1. A certain number when divided by 5 and by 4 leaves a remainder of unity in each case, but when divided by 3 it leaves no remainder ; what is the number?

2. What number is that which when divided by 4 leaves a remainder 3, and when divided by 9 leaves a remainder 4 ?

3. Determine the least number which being divided by 3, 5, 7, the remainders shall be 2, 4, 6 respectively.

4. Find a number which when divided by 6, 5, 4, 3, 2, shall leave the remainders 5, 4, 3, 2, 1 respectively.

5. Can an integral number « be found such that when 4, 5, 6 are respectively taken from X, 2x, 3x, and the remainders divided by 7, 8, 9, the quotients shall be integral? If possible, shew that the number of solutions is indefinite; if not possible, explain the reason.

6. Divide 196 into two parts such that the greater divided by 7 gives a remainder 5, and the less divided by 5 gives a remainder 1.

7. Shew that (xa)(y-b)+(y-a)(x-6)= 0 is an indeterminate equation such that for each value of x there is one, and only one value of y, and trace the changes in the value of y, as x changes from a to b.

8. In the equation ax+by = +c, if a and 6 have a common divisor, which is not also a divisor of c, the solution of the equation is impossible in integers.

9. Shew that the number of solutions in positive integers of the equation x+y+z=n+z is jn(n+1).

XXXIV. 1. The sum of two numbers is 78, one of them is divisible by 5 and the other by 3; how many pairs of numbers satisfy these conditions ?

2. Divide 1,000 into two parts, such that one shall be divisible by 7 and the other by 11.

3. Find all the numbers of two digits which exceed six times the left-hand digit by 38.

4. There is a number of two digits which, if its digits be reversed, becomes less by 1 than its half. What is the number? 5. Find two fractions whose denominators are prime to each other,

32 (1) when their sum, (2) when their difference is equal to

45 6. Show that the indeterminate equation 49x+63y = 491, can have no solution in which both x and y are integers.

7. Find what multiples of 355 and of 452 differ from one another by any given number.

XXXV. 1. Find the number of different ways in which a substance of a ton weight may be weighed by weights of 91b. and 141b.

2. Show that £20 can be paid in seven different ways in halfguineas and half-crowns.

3. In how many ways may a bill of £1,000 be paid in half-sovereigns and half-crowns ?

4. A, having only one pound notes, owes 12s. to B, who has only seven shilling pieces. What is the least number which A and B must respectively interchange so that the debt may be discharged ?

5. Supposing a sovereign be worth 25 francs, and a napoleon worth 20 francs, find the simplest way of paying a debt of 28 shillings by giving English and receiving French gold.

6. Find a sum consisting of x pounds and y shillings, the double of which shall be y pounds and a shillings.

7. A has to pay B the sum of 198. 6d., having only 12 half-crowns and 10 shillings, while B has 4 half-crowns and 8 shillings. In how many ways can the payment be effected ?

8. If the 3 per cent. consols are at 64 and the 4 per cents. at 84 in how many ways can £3,480 sterling be invested in these stocks ? Which is most advantageous ? Can there be an equal number of hundreds of each stock ?

9. If capital be invested in the 4 per cents at 96, and in the 5 per cents at 105, how much must be invested in each stock so as to realise 41 per cent. on the capital invested ?

XXXVI. 1. Determine three fractions with prime denominators whose sum is 138

2. Find three fractions whose denominators shall be 6, 9, 18, whose sum shall be 23, and the difference of the first and second equal to that of the second and third.

3. In 150 coins consisting of sovereigns, half-crowns, and shillings, if the sum represented by the half-crowns and shillings be equal to that represented by the sovereigns; find the numbers of the coins.

4. Find the number of ways in which the sum of five pounds can be paid in exactly 50 coins, consisting of half-crowns, florins, and fourpenny pieces.

5. Find in how many different ways the sum of £6 168. can be paid in guineas, crowns, and shillings, so that the number of coins used shall be exactly 16; and in each case determine the number of guineas, crowns, and shillings respectively.

6. In how many ways can a year be made up of 12 months of 28, 30, and 31 days?

7. Four jewellers possessing respectively 8 rubies, 10 sapphires, 100 pearls, and 5 diamonds, presented each from his own stock, one apiece to the rest in token of regard and gratification at meeting; and they thus become owners of stock of precisely equal value; tell what were the prices of their gems respectively. (Vija-Ganita, ch. iv.)

8. The horses belonging to four persons respectively are 5, 3, 6, and 8; the camels appertaining to them are 2, 7, 4, and 1; their mules are 8, 2, 1, and 3; and the oxen owned by them are 7, 1, 2, and 1; all are equally rich ; tell the rates of the horses and the rest. (Pija-Ganita, ch. vi.)

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