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QUEST. 2. Divide 100). among A, B, C, so that A may have 201. more than B, and B 101. niore than c.

Let x = A's share, y = B's, and z = c's.
Then x +y + z = 100,

2 Iy + 20,

y=% + 10. In the 1st substit. y + 20 for x, gives 2y + x + 20 = 100; In this substituting % + 10 for y, gives 3z + 40 = 100; By transposing 40, gives

37 = 60; And dividing by 3, gives

z = 20. Hence y = x + 10 = 30, and x = y + 20 = 50.

QUEST. 3. A prize of 5001. is to be divided between two persons, so as their shares may be in proportion as 7 to 8; required the share of each. Put x and y for the two shares; then by the question,

7:8::8:y, or mult. the extremes and the means, 7y

and * 4- 5 = 500;
Transposing y, gives x = 500 - Y;
This substituted in the 1st, gives 7y = 4000 sy;
By transposing 8y, it is 15y = 4000;
By dividing by 15, it gives y = 266};

And hence x = 500- y = 233;. QUEST. 4. What number is that whose 4th part exceeds its 5th part by 10?

Let x denote the number sought.
Then by the question fr - x = 10;
By mult. by 4, it becomes x - x = 40;

By mult. by 5, it gives x = 200, the number sought. QUEST. 5. What fraction is that, to the numerator of which if i be added, the value will be ; but if 1 be added to the denominator, its value will be ?

2

denote the fraction.
y
X + 1

to Then by the quest.

=, and y

y + 1 The 1st mult. by 2 and y, gives 2x + 2 = y; The 2d mult. by 3 and y + 1, is 3x = y +l; The upper taken from the under leaves x

- 2 =l; By transpos. 2, it gives .r = 3. And hence y = 2.x + 2 = 8; and the fraction is

QUEST. 6.

Let

3

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Quest. 6. A labourer engaged to serve for 30 days on these conditions: that for every day he worked, he was to receive 20d. but for every day he played, or was absent, he was to forfeit 10d. Now at the end of the time he had to receive just 20 shillings, or 240 pence. It is required to find how many days he worked, and how many he was idle ?

Let x be the days worked, and y the days idled.
Then 20x is the pence earned, and 10y the forfeits ;
Hence, by the question x + y = 30,

and 20.x Toy = 240;
The 1st, mult. by 10, gives 10.r + 10y = 300;
These two added give

30.r = 540; This div. by 30, gives

= 18, the days worked; Hence

y = 30– X = 12, the days idled. Quest. 7. Out of a cask of wine, which had leaked away, 30 gallons were drawn; and then, being gaged, it appeared to be half full; how much did it hold?

Let it be supposed to have held x gallons,
Then it would have leaked 2x gallons,
Conseq. there had been taken away 1 + 30 gallons.
Hence x = x + 30 by the question.
Then mult. by 4, gives 2.x = x + 120;

And transposing *, gives x = 120 the contents. Quest. 8. To divide 20 into two such parts, that 3 times the one part added to 5 times the other may make 76.

Let x and y denote the two parts.
Then by the question

x + y = 20,

and 3x + 5y = 76. Mult, the 1st by 3, gives

3x + 3y

= 60; Subtr. the latter from the former, gives 24 = 16; And dividing by 2, gives

8. Hence, from the 1st,

x = 20 - y = 12.,

Y =

QUEST. 9. A market woman bought in a certain number of eggs at 2 a penny, and as many more at 3 a penny, and sold them all out again at the rate of 5 for two-pence, and by so doing, contrary to expectation, found she lost 3d.; what number of eggs had she?

Let x = number of eggs of each sort.
Then will zx = cost of the first sort,
And fx = cost of the second sort ;

But

But 5:2 :: 2x (the whole number of eggs) : *;
Hence fx = price of both sorts, at :5 for 2 pence;
Then by the question 1x + fix - x = 3;
Mult. by 2, gives

* + - r = 6;
And mult. by 3, gives 5x – **% = 18;
Also mult. by 5, gives x = 90, the number of eggs of

each sort.

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Quest. 10. . Two persons, A and e, engage at play. Before they begin, a has 80'guineas, and B has 60. After a certain number of games won and lost between them, A rises with three times as many guineas as B. Query, how many guineas did a win of B?

Let x denote the number of guineas A won.
Then A rises with 80 + X,
And B rises with 60 — X;
Theref. by the quest. 80 + x = 180
Transp. 80 and 3x, gives 4x = 100;
And dividing by 4, gives x = 25, the guineas won.

3.2 ;

QUESTIONS FOR PRACTICE.

of the se-,

1. To determine two numbers such, that their difference may be 4, and the difference of their squares 64.

Ans. 6 and 10. 2. To find two numbers with these conditions, viz. that half the first with a 3d part of the second may

make 9, and that a 4th part of the first with a 5th part cond may make 5.

Ans. 8 and 15. 3. To divide the number 20 into two such parts, that a 3d of the one part added to a fifth of the other, may make 6.

Ans. 15 and 5. 4. To find three numbers such, that the sum of the 1st and 2d shall be 7, the sum of the 1st and 3d 8, and the sum of the 2d and 3d 9.

Ans. 3, 4, 5. 5. A father, dying, bequeathed his fortune, which was 28001. to his son and daughter, in this manner; that for every half crown the son might have, the daughter was to have a shilling. What then were their two shares ?

Ans. The son 2000l. and the daughter 8001. 6. Three persons, A, B, C, make a joint contribution, which in the whole amounts to 4001.: of which sun B con

tributes

tributes twice as much as A and 201. more; and c as much as A and B together. What sum did each contribute ?

Ans. A 601. B 1401, and c 2001. 7. A person paid a bill of 1001. with half guineas and crowns, using in all 202 pieces; how many pieces were there of each sort ?

Ans. 180 half guineas, and 22 crowns. 8. Says A to B, if you give me 10 guineas of your money, I shall then have twice as much as you will have left: but says B to A, give me 10 of your guineas, and then I shall have 3 times as many as you.

How
many

had each?

Ans. A 22, B 26. 9. A person goes to a tavern with a certain quantity of money in his pocket, where he spends 2 shillings; he then borrows as much money as he had left, and going to another tavern, he there spends 2 shillings also; then borrowing again as much money as was left, he went to a third tavern,

where likewise he spent 2 shillings; and thus repeating, the. * same at a fourth tavern, he then had nothing remaining. What sum had he at first?

Ans. 35. 9d. 10. A man with his wife and child dine together at an inn. The landlord charged 1 shilling for the child; and for the woman he charged as much as for the child and I as much as for the man; and for the man he charged as much as for the woman and child together. How much was that for each ?

Ans. The woman 200. and the man 32d. 11. A cask, which held 60 gallons, was filled with a mixture of brandy, wine, and cyder, in this manner, viz. the cyder was 6 gallons more than the brandy, and the wine was as much as the cyder and of the brandy. How much was there of each?

Ans. Brandy 15, cyder 21, wine 24. 12. A general, disposing his army into a square form, finds that he has 284 men more than a perfect square ;

but increasing the side by 1 man, he then wants 25 men to be a complete square. Then how many men had he under his command ?

Ans. 24000. 13. What number is that, to which if 3, 5, and 8, be severally added, the three sums shall be in geometrical progression ?

Ans. l. 13. The stock of three traders amounted to 8601. the shares of the first and second exceeded that of the third

by

A

by 240 ; and the sum of the 2d and 3d exceeded the first by 260. What was the share of each ?

Ans. The 1st 200, the 2d 300, the 3d 260. 15. What two numbers are those, which, being in the ratio of 3 to 4, their product is equal to 12 times their sum?

Ans. 21 and 28. 16. A certain company at a tavern, when they came to settle their reckoning, found that had there been 4 more in company, they might have paid a shilling a-piece less than they did; but that if there had been 3 fewer in company. they must have paid a shilling a-piece more than they did, What then was the number of persons in company, what each paid, and what was the whole reckoning?

Ans. 24 persons, each paid 7s. and the whole

reckoning 8 guineas. 17. A jockey has two horses ; and also two saddles, the one valued at 181. the other at 31. Now when he sets the better saddle on the 1st horse, and the worse on the 2d, it makes the first horse worth double the 2d: but when he places the better saddle on the 2d horse, and the worse on the first, it makes the 2d horse worth three times the 1st. What then were the values of the two horses?

Ans. The 1st Gl., and the 2d 91. 18. What two numbers are as 2 to 3, to each of which if 6 be added, the sums will be as 4 to 5 ?

Ans. 6 and 9. 19. What are those two numbers, of which the greater is to the less as their sum is to 20, and as their difference is to 10?

Ans. 15 and 45. * 20. What two numbers are those, whose difference, sum, and product, are to each other, as the three numbers 2, 3,5?

Ans. 2 and 10. 21. To find three numbers in arithmetical progression, of which the first is to the third as 5 to 9, and the sum of all three is 63.

Ans. 15, 21, 27. 22. It is required to divide the number 24 into two such parts, that the quotient of the greater part divided by the less, may be to the quotient of the less part divided by the greater, as 4 to 1.

Ans. 16 and 8. 23. A gentleman being asked the age of his two sons, answered, that if to the sum of their ages 18 be added, the result will be double the age of the elder; but if 6 be

taken

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