and x(x+5)=the whole number of men ; also (x+845) X5 the whole number of men; ... x2+5x=5x+4225, and x2=4225; .. x=±65, And, consequently, 5x+4225-325+4225-4550, the number of men. Here although (Art. 262), the negative value of x, will not answer the conditions of the problem, yet it will satisfy the above equation; for, if we substitute -65 for x, we shall have (-65)2 +5(-65)=5(-65)+4225; that is, or 4225-325 =-3254225; .. 4225=4225, or 4225-4225 =o, that is, o=0.

--

Prob. 9. It is required to divide the number a into two such parts, that the squares of those parts may be in the proportion of

in to n.

Let x=one of these parts; then a x=the other; and according to the enunciation of the problem, we shall have the equation,

By resolving separately the two equations of the first degree comprised in the above formula, namely, x=+(a−x)√m', and x=-(ax)√m',

we shall have, from the first,

x=

a✓m'

'1+ √m"

By the first solution, the second part of the pro

posed number is a

a√m'

α

and the two

a✓m'

parts,

1+√m' 1+√m"

α

and

',

1 + √m' 1+m" are, as was required in the enunciation of the question, both less than the number proposed.

By the second solution, we have

Their signs being contrary, the number a is not properly speaking, their sum, but their difference.

400. When we make m=n, that is, if we suppose that the squares of the two required parts are equal, we have m'=1; the first solution gives two equal a result which is evident, whilst the second solution gives two infinite results (Art. 166),

parts

a and a,

1-1

a

0

These are proper results, according to the above enunciation, since that the quantities required must be infinitely great, with respect to their difference a, if we can suppose the ratio of their squares equal to unity.

Now if a=18, m=25, and n=16; then substitut

ing these values in the formula

and

α

a√m' 1+√m' 1+ √m' we shall find 10 and 8 equal to the two parts required, the same as in Ex. 2., which is a particular case of this general problem.

Prob. 10. What two numbers are those, whose sum is to the greater as 10 to 7; and whose sum, multiplied by the less, produces 270?

Ans. 121 and ±9. Prob. 11. What two numbers are those, whose difference is to the greater as 2 to 9, and the difference of whose squares is 128? Ans. 18 and ±14.

Prob. 12. A mercer bought a piece of silk for 167. 4s.; and the number of shillings which he paid for a yard was to the number of yards as 4: 9. How many yards did he buy, and what was the price of a yard? Ans 27 yards, at 12s. per yard. Prob. 13. Find three numbers in the proportion of,, and ; the sum of whose squares is 724.

Ans. 12, 16, and ±18.

Prob. 14. It is required to divide the number 14 into two such parts, that the quotient of the greater

part, divided by the less, may be to the quotient of the less divided by the greater as 16:9.

Ans. The parts are 8 and 16. Prob. 15. What two numbers are those whose difference is to the less, as 4 to 3; and their product, multiplied by the less, is equal to 504?

360.

Ans. 14 and 6. Prob. 16. Find two numbers, which are in the proportion of 8 to 5, and whose product is equal to Ans. 24, and ±15. Prob. 17. A person bought two pieces of linen, which, together, measured 36 yards. Each of them cost as many shillings per yard, as there were yards in the piece; and their whole prices were in the proportion of 4 to 1. What were the lengths of the pieces? Ans. 24, and 12 yards.

Prob. 18. There is a number consisting of two digits, which being multiplied by the digit on the left hand, the product is 46; but if the sum of the digits be multiplied by the same digit, the product is only 10. Required the number. Ans. 23. Prob. 19. From two towns, C and D, which were at the distance of 396 miles, two persons, A and B, set out at the same time, and met each other, after travelling as many days as are equal to the difference of the number of miles they travelled per day; when it appears that A has travelled 216 miles. How many miles did each travel per day?

Ans. A went 36, and B 30. Prob. 20. There are two numbers, whose sum is to the greater as 40 is to the less, and whose sum is to the less as 90 is to the greater. What are the numbers? Ans. 36, and 24.

Prob. 21. There are two numbers, whose sum is to the less as 5 to 2; and whose difference, multiplied by the difference of their squares, is 135. Required the numbers. Ans. 9, and 6.

Prob. 22. There are two numbers, which are in the proportion of 3 to 2; the difference of whose

fourth powers is to the sum of their cubes as 26 to 7. Required the numbers. Ans. 6, and 4.

Prob. 23. A number of boys set out to rob an orchard, each carrying as many bags as there were boys in all, and each bag capable of containing 4 times as many apples as there were boys. They filled their bags, and found the number of apples was 2916. How many boys were there? Ans. 9 boys.

Prob. 24. It is required to find two numbers such, that the product of the greater, and square of the less, may be equal to 36; and the product of the less, and square of the greater, may be 48.

Ans. 4, and 3. Prob. 25. There are two numbers, which are in the proportion of 3 to 2; the difference of whose fourth powers is to the sum of their cubes as 26 to 7. Required the numbers. Ans. 6, and 4.

Prob. 26. Some gentlemen made an excursion. and every one took the same sum. Each gentleman had as many servants attending him as there were gentlemen; and the number of dollars which each had was double the number of all the servants; and the whole sum of money taken out was $3456. How many gentlemen were there? Ans. 12.

Prob. 27. A detachment of soldiers from a regiment, being ordered to march on a particular service, each company furnished four times as many men as there were companies in the regiment; but those becoming insufficient, each company furnished 3 more men; when their number was found to be increased in the ratio of 17 to 16. How many companies were there in the regiment?

Ans. 12.

Prob. 28. A charitable person distributed a certain sum among some poor men and women, the numbers of whom were in the proportion of 4 to 5. Each man received one-third of as many shillings as there were women more than men. Now the men received all together 18s. more than the women. How many were there of each?

Ans. 12 men, and 15 women.

Prob. 29. Bought two square carpets for 621. 1s. for each of which I paid as many shillings per yard as there were yards in its side. Now had each of them cost as many shillings per yard as there were yards in the side of the other, I should have paid 17s. less. What was the size of each?

Ans. One contained 31, and the other 64 square yards.

had

Prob. 30. A and B carried 100 eggs between them to market, and each received the same sum. If A had carried as many as B, he would have received 18 pence for them; and if B had only taken as many as A, he would have received 8 pence. How many each? Ans. A 40, and B 60. Prob. 31. The sum of two numbers is 5 (s), and their product 6(p): What is the sum of their 5th powers? Ans. 275 (s5-5ps3+5p2s).

CHAPTER X.

ON

QUADRATIC EQUATIONS.

401. Quadratic equations, as has been already observed, (Art. 388), are divided into pure and adfected. All pure equations of the second degree are comprehended in the formula x2-n, where n may be any number whatever, positive or negative, integral or fractional. And the value of x is obtained by extracting the square root of the number n; this value is double, for we have, (Art. 295), x=±n, and in fact, (n)2=n. This may be otherwise explained, by observing, (Art. 106), that x2-n=(x+ √n).(x — √n)=0, and that any product consisting of two factors becomes nought, when there is no restriction in