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Generalization and Specialization.

1. Definitions.

166. Any question proposed for solution is a Problem. 167. A problem whose given quantities are literal, or general, is a general problem.

168. A problem whose given quantities are numerical, or special, is a special problem, or an example.

169. A number of examples with different given quantities but like conditions and requirements constitute a class.

170. A general problem involves a whole class of examples. It is the type of a class, and its solution the solution of a class.

171. The solution of a general problem gives rise to a formula, which, interpreted, gives a rule for the solution. of every example of a class.

172. The process of converting a special problem into a general one, by substituting literal for numerical quantities, is Generalization.

173. The process of converting a general problem into a special one, by substituting numerical for literal quantities, is Specialization.

2. Examples.

Illustrations.-1. If A can do a piece of work in 4 days and B can do it in 5 days, in what time can they do it working together? Generalize this question and solve it. Solution: Put a for 4 and b for 5. Let x equal the time required for both to do it.

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2. A and B can do a piece of work in a days, A and C in b days, and B and C in c days; in what time can each alone do it? Solve this problem and specialize for a = 10, b=8, and c = 6.

Solution: Let x equal the time required by A, y the time required by B, and z the time required by C; then

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Adding (A), (B), and (C), and subtracting from the sum twice (A), twice (B), and twice (C) respectively, we have

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1. The sum of two numbers is 20, and their difference is 8. Find the numbers.

Suggestion,-Generalize by putting a for 20 and b for 8 in the problem, then 20 for a and 8 for b in the result.

2. A's age is three times B's, but in 12 years it will be only twice B's. Required the age of each.

Suggestion.—Put m for 3, n for 2, and t for 12 in the problem, and 3 for m, 2 for n, and 12 for t in the result.

3. A and B have $170, and % of A's share equals 3/4 of B's. How much has each ?

Suggestion.-Put m for 2/3, n for 3/4, and c for 170, etc.

4. A can do a piece of work in 6 days and B can do it in 8 days. In what time can they do it working together? Generalize and solve.

5. A has $400 more than B, and B has $500 less than C, and they together have $1800. How much has each ? Generalize and solve.

6. If a certain number be increased by 20, the result will be twice as great as when the number is diminished by 10. Required the number. Generalize and solve.

7. What number added to both terms of the fraction / will give the fraction ? Generalize and solve.

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8. B has 40 acres more land than A, but if A buys 60 acres from B, A will have 1 times as much as B. How many acres has each? Generalize and solve.

9. If a man work 5 days and a boy 3 days, they together earn $23, but if the man and boy each work 4 days they together earn $20. Required the daily wages of each. Generalize and solve.

10. The sum of A's and B's ages is c years, and A is d years older than B. Required the age of each. Specialize by making c=36 and d = 8 in the result.

11. Mr. Jones has a coins worth a dollar; some of them are c-cent pieces, and the rest are d-cent pieces. How many of each are there? Specialize by making a = 14, c = 10, and d = 5.

12. The sum of three consecutive numbers is 18. Required the numbers. Generalize and solve.

13. James is a years younger than William; but if m times James's age be subtracted from n times William's, the remainder will be d years. How old is each? Spe

cialize by making a = 4, m = 2, n = 3, and d = 22.

14. If a cows and boxen are worth m dollars, and c cows and d oxen, n dollars, required the value of a cow and of an ox. Specialize by putting 5 for a, 7 for b, 10 for c, 3 for d, 370 for m, and 355 for n.

15. A and B can do a piece of work in d days. After working together c days, B leaves, and A does the balance in a days. In what time could each do it alone? Specialize by putting 30 for d, 18 for c, and 20 for a.

16. If a certain rectangle had been a feet broader and b feet longer, it would have been c square feet larger. But, if it had been b feet wider and a feet longer, it would have been d square feet larger. Required its dimensions. Specialize by making a = 2, b = 3, c= 64, and d = 68.

17. There is a number consisting of two digits whose sum is a, and if b be subtracted from the number, the digits will change places. Required the number. Specialize by putting 13 for a and 27 for b.

18. The wages of a men and b women in one week amount to c dollars, and b men receive d dollars more than e women. What does each receive per week? Put 5 for a, 7 for b, 170 for c, 80 for d, and 6 for e.

19. Three children, taken two at a time, weighed a pounds, b pounds, and c pounds. What was the weight

of each? Put a = 94, b= = 76, and c 90.

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20. A purse holds a crowns and b guineas; c crowns and d guineas fill 17/3 of it. How many will it hold of each? Put 19 for a, 6 for b, 4 for c, and 5 for d. Enunciate the special problem thus formed.

CHAPTER IV.

POWERS AND ROOTS.

Involution of Binomials.

1. Principle.

174. We may learn by actual multiplication that : (a+b)2= a2+2ab+b2

(a - b)2= a2 2ab+b2

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(a + b) = a + 4 a3 b + 6 a2 b2 + 4 a b3 + b2

(a - b)1 = a1 — 4 a3 b + 6 a2 b2 - 4 a b3 + b1

(a+b)5 = a + 5 a b + 10 a3 b2 + 10 a2 b3 + 5 a ba + b5

(a - b) a 5 ab+10 a3 b2-10 a2 b3 +5 a b1 — b5 = — —

By a careful inspection of the above results the following laws will appear:

The Binomial Theorem.

Prin. 72.-1. The number of terms in each result is one greater than the exponent of the binomial.

2. When the binomial is the sum of two quantities, all the terms of the power are positive; when the difference of two quantities, the terms are alternately positive and negative.

3. The first letter occurs in all the terms but the last, and the second letter in all the terms but the first.

4. The exponent of the leading letter in the first term is the same as the exponent of the binomial, and decreases by

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