Here, the imaginary expression which enters into the value of x, shows an impossibility in the required conditions. Indeed it is not possible to divide 20 into two parts, the product of which shall be 125, for the greatest product that can be formed of the parts of 20, is 100, the product of 10 by 10.

5. The sum of two numbers is 8, and the sum of their 4th powers 1312; what are those numbers?

Let x half the difference of those numbers, and for the sake of brevity, put 82a, and 1312=b.

The greater number will be a+x, and the lesser a- —x (17); then we shall have the equation,

a1+4a3x+6a2x2 + 6ax3 + 202

+ a1 — 4a3x+6a2x2· 40x3 + x + 3 = b

and reducing,

2a+12a2x2+2x1=b.
a1+6a2x2+x1 = }} b ;
x2+6a2x2=b — a1.

Dividing by 2,

transposing,

x=±√3a2±√1⁄2b+8a1.

Putting in the place of a and b, their values, we have,

x=1V-3x(4)3± √1312+8x(4) V-48/656+2048.

2

=

V-4852=√4=2.

Half the difference of the numbers is, therefore, 2, and since half the sum is 4, the greater number is 6, and the lesser 2.

6. Find two numbers, such that the sum of their squares being subtracted from three times their product, 11 will remain; and the difference of their squares being subtracted from twice their product, the remainder will be 14.

Let

then,

and,

the greater number, and y = the less;

3xy-(x2+ y2)= 11;

Qxy —(x2 — y2)=14;

or performing the subtraction indicated,

Making azy, and substituting it in the place of y, the equations become

Dividing the first of these equations by the second, and casting out the factor 2 from both numerator and denominator of the first member of the resulting equation, we have,

42z-14-142222z-11 +11z2.

Transposing and reducing,

5 5

5'

Substituting this value of z in the equation

3x2z -x2 x2x2 = 11,

The numbers are, therefore, 5 and 3.

7. A person bought cloth for £33. 15s. which he sold again at £2. 8s. per piece, and gained by the bargain as much as one piece cost him. Required the number of pieces?

Ans. 15.

8. A and B set off at the same time, to a place at the distance of 150 miles. A travels 3 miles an hour faster than B, and arrives at his journey's end 8 hours and 20 minutes before him. At what rate did each person travel per hour?

Ans. A 9 miles and B 6.

9. There are two numbers whose product is 120. If 2 be added to the lesser, and 3 subtracted from the greater, the product of the sum and remainder will also be 120. What are the numbers? Ans. 15 and 8.

10. What number is that, which being divided by the product of its digits, the quotient is 2, and if 27 be added to the number itself, the digits will be inverted? Ans. 36.

11. The sum of two numbers multiplied by the lesser is equal to 40, and their difference multiplied by the greater is 12; what are the numbers? Ans. 4 and 6.

12. Divide the number 13 into three such parts, that their squares may have equal differences, and the sum of those squares may be 75. Ans. 7, 5, and 1.

13. The sum of two numbers is 9, and the sum of their cubes 243. Required the numbers. Ans. 3 and 6.

14. The sum of two numbers is 10, and 4th powers 1552. Required the numbers.

15. The sum of two numbers is 7, and 5th powers 3157. Required the numbers.

the sum of their. Ans. 4 and 6.

the sum of their Ans. 5 and 2.

16. Find two numbers, such that their sum, their product, and the difference of their squares, may all be equal to one another. Ans.V5, and ± √5.

17. Divide the number 100 into two such parts, that the sum of their square roots may be 14. Ans. 64 and 36.

18. Divide the number 40 into two such parts, that twice their product shall be equal to the cube of the less number. Ans. 8 and 32.

19. A and B hired a pasture, into which A put 4 horses, and B as many as cost him 18 shillings a week. Afterwards B put in two additional horses, and found that he must pay 20 shillings a week; at what rate was the pasture hired?

Ans. B had 6 horses in the pasture at first, and the price of the whole pasture was 50 shillings per week.

20. A mercer bought a piece of silk for £16. 4s., and the number of shillings he paid per yard, was to the number of yards as 4 to 9. How many yards did he buy, and what was the price per yard?

Ans. 27 yards at 12 shillings per yard. 21. There are two numbers in the proportion of 4 to 5, and the difference of their squares is 81. What are the numbers? Ans. 12 and 15.

In

22. A person bought for a dollar, as many pounds of sugar as were equal to half the number of dollars he laid out. selling the sugar, he received for every 100lbs. as many dollars as the whole had cost him, and he received on the whole $20,48. How many dollars did he lay out, and what did he give for a pound?

Ans. He laid out $16, and gave 12 cents per pound. 23. What two numbers are those, whose difference multiplied by the greater, produces 40, and by the less, 15? Ans. 8 and 3.

24. What two numbers are those, which being both multiplied by 27, the first product is a square, and the second, the root of that square; but being both multiplied by 3, the first product is a cube, and the second the root of that cube? Ans. 243 and 3. 25. Find two numbers, such that the less may be to the greater as the greater is to 12, and the sum of their squares 45. Ans. 3 and 6.

26. What two numbers are those whose product is 20, and the difference of their cubes 61? Ans. 4 and 5.

27. What number is that, which being divided by the product of its two digits, the quotient is 5; and 9 being subtracted from the number itself, the digits will be inverted? Ans. 32.

23. Divide 20 into three such parts, that the continual product of these parts may be 270; and that the difference of the first and second may be 2 less than the difference of the second and third. Ans. 5, 6, and 9.

such parts, that the
Ans. 4 and 7.

29. Divide the number 11 into two product of their squares may be 784. 30. Find two numbers, such that the sum of their squares may be 89, and their sum multiplied by the greater may produce 104.

Ans. 5 and 8.

CHAPTER V.

PROGRESSIONS AND LOGARITHMS.

ARITHMETICAL PROGRESSION.

(78.) An arithmetical progression, or progression by dif ferences, is a series of numbers, in which the terms continually increase or decrease by a constant number, which is called the ratio or common difference of the progression, (Arith. 60).

Thus, 1, 3, 5, 7, 9, 11, &c. is an increasing progression ; and, 11, 9, 7, 5, 3, 1, is a decreasing progression.

The ratio or common difference in these progressions is 2. In order to investigate the properties of arithmetical progressions, we will take the general progression,

the first term of which is a, the last term l, the common difference q, and the number of terms n.