4. A bill of 1201. was paid in guineas and moidores, and the number of pieces of both sorts used was just 100; how many were there of each ? Ans. 50 of each.

5. Two travellers set out at the same time from London and York, whose distance is 150 miles; one of them goes 8 miles a day, and the other 7; in what time will they meet ? Ans. 10 days.

6. At a certain election 375 persons voted, and the candidate chosen had a majority of 91; how many voted for each? Ans. 233 for one and 142 for the other.

7. There is a fish, whose tail weighs olb. his head weighs as much as his tail and half his body, and his body weighs as much as his head and his tail; what is the whole weight of the fish ? Ans. 72lb.

3

8. What number is that, from which if 5 be subtracted, of the remainder will be 40 ?

Ans. 65.

9. A post is one fourth in the mud, one third in the water, and 10 feet above the water; what is its whole length? Ans. 24 feet.

10. After paying away one fourth and one fifth of my money, I found 66 guineas left in my bag; what was in it at first ? Ans. 120 guineas.

11. A's age is double that of B, and B's is triple that of C, and the sum of all their ages is 140; what is the age of each ? Ans. A's = 84, B's = 42 and C's = 14.

12. Two persons, A and B, lay out equal sums of money in trade; A gains 1261. and B loses 871. and A's money is now double that of B; what did each lay out ?

Ans. 3001.

13. A person bought a chaise, horse and harness, for

60l.; the horse came to twice the price of the harness,

and

and the chaise to twice the price of the horse and the harness; what did he give for each ?

Ans. 131. 6s. 8d. for the horse, 61. 138. 4d. for the har ness, and 401. for the chaise.

14. Two persons, A and B, have both the same income; A saves one fifth of his yearly, but B, by spending 50l. per annum more than A, at the end of 4 years finds himself rool. in debt; what is their income ?

Ans. 1251.

15. A gentleman has two horses, and a saddle worth pol. Now if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the first; what is the value of each horse ? Ans. One 30l. and the other 40l.

16. To divide the number 36 into three such parts, that of the first, of the second, and of the third, may be all equal to each other.

Ans. The parts are 8, 12 and 16.

17. A footman agreed to serve his master for 81. a year and a livery, but was turned away at the end of 7 months, and received only 21. 13s. 4d. and his livery; what was its value? Ans. 41. 16s.

18. A gentleman was desirous of giving 3d. a piece to some poor beggars, but found, that he had not money enough in his pocket by 8d.; he therefore gave them each 2d. and had then 3d. remaining; required the number of beggars. Ans. 11.

19. A hare is 50 leaps before a grey hound, and takes 4 leaps to the grey hound's 3; but 2 of the grey hound's leaps are as much as 3 of the hare's; how many leaps must the grey hound take to catch the hare ? Ans. 300.

20. A person at play lost of his money, and then won 3 shillings; after which he lost of what he then had, and

Ww

and then won 2 shillings; lastly, he lost then had, and, this done, found he had but ing; what had he at first ?

of what he 12s. remainAns. 20s.

21. To divide the number 90 into 4 such parts, that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, difference, product, and quotient, shall be all equal to each other.

Ans. The parts are 18, 22, 10, and 40, respectively. 22. The hour and minute hands of a clock are exactly together at 12 o'clock; when are they next together?

Ans. 1 hour, 5 min.

23. There is an island 73 miles in circumference, and 3 footmen all start together to travel the same way about it; A goes 5 miles a day, B 8, and C 10; when will they all come together again ? Ans. 73 days.

24. If A can do a piece of work alone in 10 days, and A and B together in 7 days, in what time can B do it alone? Ans. 23 days.

25. If three agents, A, B and C, can produce the effects a, b, c, in the times e, f, g, respectively; in what time would they jointly produce the effect d?

26. If A and B together can perform a piece of work in 8 days, A and C together in 9 days, and B and Cin 10 days; how many days will it take each person to perform the same work alone ?

Ans. A 14 days, B 174, and C

49

QUADRATIC EQUATIONS.

2337

A SIMPLE QUADRATIC EQUATION is that, which involves the square of the unknown quantity only.

An

An AFFECTED QUADRATIC EQUATION is that, which involves the square of the unknown quantity, together with the product, that arises from multiplying it by some known quantity.

Thus, ax=b is a simple quadratic equation,

And ax2+bx=c is an affected quadratic equation.

The rule for a simple quadratic equation has been given already.

All affected quadratic equations fall under the three following forms.

1. x2+ax=b

2. x2-ax=b

3. x-ax=-b.

The rule for finding the value of x, in each of these equations, is as follows:

RULE.*

1. Transpose all the terms, that involve the unknown quantity, to one side of the equation, and the known terms to the other side, and let them be ranged according to their dimensions.

2. When

* The square root of any quantity may be either + or -, and therefore all quadratic equations admit of two solutions. Thus, the square root of n2 is +2. or -n; for either +n x +n, ornx-n is equal to +n2. So in the first form, where

a

2

is found =+, the root may be either +<+

a

2

4

or -√b+, since either of them being multiplied by itself

4

a2

will produce 6+, And thisambiguity is expressed by writing the

4

2. When the square of the unknown quantity has any coefficient prefixed to it, let all the rest of the terms be divided by that coefficient.

3. Add

In the first form, where x=±√6+, the first value

2

a

4

2

is always affirmative; for

the greatest square must necessa

The second value, viz. x=-6+, will always

2

a

4

2

be negative, because it is composed of two negative terms,

n x2+ax=b, we shall have x=+b+

Therefore, when x

a for the affirmative value of x, and x=6+

2

for the negative value of x,

In the second form, where

وست

2

a

the first

4

2

it is composed of two affirmative terms. The second value, viz