PROBLEM.

(409.) To find any number of harmonic means between two quantities.

SOLUTION.

Let it be required to find m harmonic means between a and c.

By Prop. 1, (407.), we have

progression of m+2 terms.

1

b

1

an arithmetical

By substituting in the formula la±(n-1)d, we have

Having found the common difference, we are now able to insert m arithmetical means in the arithmetical series.

b,

1

(m+1)bc

(m+1)bc

(m+1)be

(m + 1)bc

mc+b' (m−1)c+2b' (m−2)c+3b' (m−3)c+4b

EXAMPLES.

1. Find an harmonic mean between 3 and 6

2. Find an harmonic mean between x+y and x-y.

4. Find the third of three quantities in harmonical proportion, the

first and second being 3 and 4.

Ans. 6.

5. Find the first of three quantities in harmonical proportion, the second and third being 144 and 104. Ans. 234.

6. Find the fourth of four quantities in harmonic proportion, the first three being 2, 3, and 8. Ans. 16.

7. Find the third of four quantities in harmonical proportion, the first being 10, the second 12, and the fourth 15.

Ans. 12.

8. Find the second of four quantities in harmonical proportion, the first being 5, the third 9, and the fourth 15.

9. Find the first of four quantities in harmonical second being 6, the third 9, and the fourth 15.

10. Find a harmonic mean between 50 and 100. 11. Find a harmonic mean between 25 and 50. 12. Find a harmonic mean between 12 and 25.

Ans. 7.

proportion, the Ans. 44.

Ans. 663,

Ans. 33.

Ans. 164.

13. Find two harmonic means between 1 and 3. Ans. 14 and 2. 14. Find three harmonic means between 10 and 30.

Ans. 12, 15, and 20.

15. Find three harmonic means between 315 and 35.

Ans. 105, 63, and 45.

16. Find fourteen harmonic means between and 4.

Ans. 15, 4, 1, 4, 7, 3, 4, 1, 4, 3, 3, 1, 14, and 2.

17. Find the fifth term of an harmonical progression whose first term is 60 and second term 21. Ans. 7.

18. Find the unknown terms of an harmonical progression consisting of 12 terms, the first being 4 and the fourth 1.

Ans. 2, 1, 4, 3, 4, 1, 4, 3, rt, and J.

19. Find the resulting proportion when a, b, c are in arithmetical progression, and b, c, d in harmonic proportion. Ans. a:b::c: d.

20. Find the nth term of an harmonical progression, a and b being the first two terms.

PROBLEMS IN PROPORTION.

PROBLEM.

(410.) There are two numbers whose product is 24, and the difference of their cubes is to the cube of their difference as 19 is to 1. What are the numbers ?

Let

SOLUTION.

the greater number,

and y the lesser number.

By 2nd condition, x-y': (x-y)':: 19: 1 x2+xy+y2: x3-2xy+y':: 19:1

3xy x + xy + y2 18 19

Prop. 7.
Prop. 11.

(9)

1. There are two numbers whose product is 135, and the differ ence of their squares is to the square of their difference as 4 to 1. What are the numbers? Ans. 15 and 9.

2. There are two numbers which are to each other in the duplicate ratio of 4 to 3, and 24 is a mean proportional between them. What are the numbers ? Ans. 32 and 18.

3. There are two numbers whose sum is 24, and their product is to the sum of their squares as 3 to 10. What are the numbers? Ans. 18 and 6.

4. There are three numbers which are to each other as 3 to 2. If 6 be added to the greater and subtracted from the lesser, the sum will be to the remainder as 3 to 1. What are the numbers?

Ans. 24 and 16.

5. There are two numbers whose sum is 60, and their product is to

the sum of their squares as 2 to 5.

What are the numbers?

Ans. 40 and 20. two parts, which are to each What is the mean proportional Ans. 6,

=4a, show that a+x: 2a :: 26; a—x.

8. If (a+x)2: (a—x)' :: x+y: x−y, show that a:x::V/2a-y:Vỹ.

9. If xy: ab and a:b:: Vc+x: Vd+y show that dr=cy,

:

10. If x2 y2:: 36:25 and 2x+y: x+2 in a ratio compounded of the ratios of 17:2 and 2: 7, what are the values of x and y?

Ans. x 12, and y=10.

11. A person has British wine at 5s. per gallon, with which he wishes to mix spirits at 11s. per gallon, in such proportion that by selling the mixture at 9s. a gallon he may gain 35 per cent. What is the necessary proportion?

Ans. 13 gallons of wine to 5 of spirits.

12. What number is that to which if 3, 8, and 17 be severally added, the first sum shall be to the second as the second to the third? Ans. 31.

13. A merchant having mixed a certain number of gallons of brandy and water, found that if he had mixed 6 gallons more of each there would have been 7 gallons of brandy to every 6 gallons of water, but if he had mixed 6 gallons less of each there would have been 6 gallons of brandy to every 5 gallons of water. How much of each did he mix? Ans. 78 gallons of brandy and 66 of water.

14. A and B speculate in trade with different sums, A gains $150, B loses $50, and now A's stock is to B's as 3 to 2; but, had A lost $50 and B gained $100, then A's stock would have been to B's as 5 to 9. What was the stock of each?

Ans. A's $300 and B's $350.

15. What are the two parts of 14 of which the greater divided by the less is to the less divided by the greater as 16 to 9.

Ans. 8 and 6.

16. In a mixture of rum and brandy, the difference between the quantities of each is to the quantity of brandy as 100 is to the number of gallons of rum; and the same difference is to the quantity or rum as 4 is to the number of gallons of brandy. How many gallons are there of each? Ans. 25 gallons of rum and 5 of brandy.

17. There is a number consisting of three digits, the first of which is to the second as the second is to the third; the number itself is to the sum of its digits as 124 to 7; and if 594 be added to it the digits will be inverted. What is the number! Ans. 248.

18. A corn-factor mixes wheat which cost 10s. a bushel with barley which cost 4s. a bushel, in such proportion as to gain 43 per cent. by selling the mixiure at 11s. a bushel. What is the proportion? Ans. 14 bushels of wheat to 9 of barley.

19. What two numbers are those whose sum, difference, and product, are as the numbers 3, 2, and 5, respectively. Ans. 10 and 2. 20. What two numbers are those whose sum, difference, and product, are as the numbers s, d, and p respectively!

21. There are two numbers in the proportion of to, and such, that if they be increased respectively by 6 and 5, they will be to each other as to. What are the numbers? Ans. 30 and 40.

22. There is a number, the sum of whose digits is to the number itself as 4 to 13; and if the digits be inverted, their difference will be to the number expressed as 2 to 31. What is the number?

Ans. 39.

23. The difference of the cubes of two numbers is to the cube of their difference as 61 is to 1, and the product of the numbers is 320. What are the numbers? Ans. 20 and 16.

24. The sum of the cubes of two numbers is to the difference of their cubes as 559 to 127, and the square of the first multiplied by the second is 294. What are the numbers? Ans. 7 and 6.

25. The difference of the cubes of two numbers is to their product multiplied by their difference as 7 is to 2, and the sum of the numbers is 6. What are the numbers ? Ans. 4 and 2.