43.-Ex. 1. John has 45 cents more than Willie, and 6 times Willie's number equals John's. How many has each?

SOLUTION. Let x equal Willie's number; then 6x will equal John's number, and 6x-x will be the number John has more than Willie, which is 45.

x= Willie's number; 6x= John's number. 6x-x=45 50=45

x= 9, Willie's number. 6x=54, John's number.

If 6x-x, or 5x, is equal to 45, x, or the number Willie has, must be onefifth of 45, or 9, and 6x, or the number John has, must be 6 times 9, or 54.

2. The difference of two numbers is 40, and the larger is 5 times the smaller. What are the numbers? Ans. 10 and 50.

3. The difference between the ages of a mother and her daughter is 24 years, and the mother's age is 3 times that of the daughter. What is the age of each?

4. Five times the amount of my money diminished by three times that amount is equal to $62. How much money have I? Ans. $31.

5. Alice has 36 more books than Susan, and her number is 4 times Susan's number. How many books has each?

44.—Ex. 1. Alfred and Daniel have together a cents, and Daniel has 4 times as many as Alfred. How many cents has each?

2. My horse and carriage cost me a, and the cost of the carriage was twice that of the horse. What was the cost of each?

3. The sum of three numbers is a, the second is 3 times the first, and the third is 4 times the first. What are the numbers?

4. The difference between two numbers is a, and the larger equals 7 times the smaller. What are the numbers?

Ans.

a 7a 6' 6'

5. Divide the number b into such parts that the larger part shall be 4 times the smaller.

6. Albert has a more cents than his brother, and his number is 3 times that of his brother. How many cents has each?

7. If a in the last problem equals 12, what is the value of the results?

8. Divide the number m into three parts that shall be to one another as 1, 2 and 3, and find the value of each, if m equals 60.

45.-1. What is Quantity? The unit of a quantity? A symbol of a quantity? What are known quantities? Unknown quantities? Numerical quantities? Literal quantities? What is a sign? Algebra? 2. What is the Sign of Addition? Of subtraction? Of multiplication? Of division? Of equality? Of aggregation?

3. What is a Factor of a quantity? A coefficient? What coefficient is understood when no coefficient is expressed? What is an exponent? What exponent is understood when no exponent is expressed?

4. What is a Power of a quantity? A root of a quantity? How is a root of a quantity denoted?

5. What are Axioms? Upon what is Algebraic reasoning based?

6. What is an Algebraic Expression? What are the terms of an algebraic expression? What are similar terms? Dissimilar terms? What is a monomial? A polynomial? A binomial? A trinomial? The degree of a term? The numerical value of an algebraic expression?

7. What are Algebraic Processes? What is a problem?

SECTION IV.

ADDITION.

46. Addition is the process of uniting two or more quantities to find their sum.

CASE I.

Terms Similar with like Signs.

47.-Ex. 1. A man earned once two dollars on one day, and twice two dollars on another day. How many times two dollars did he earn in the two days?

2. In one week John saves a certain sum of money, Harry saves twice that sum, and Robert saves three times that sum. How many times the sum do they together save? If x stands for the sum which John saves, what should stand for what they all save?

3. A merchant gained in January a certain amount, in February three times as much, and in March four times as much. How many times the gain in January was the entire gain? Let x represent the gain in January, what will represent the entire gain?

4. Once any quantity, plus three times that quantity, plus four times that quantity, is how many times that quantity?

5. Edward lost a number of cents, George lost four times as many, and Richard five times as many. How many times Edward's loss was the loss of the three boys? Edward's loss, what will represent George's loss? The entire loss?

If

x represent

loss?

Richard's

6. A grocer loses by one customer a sum represented by y, by another a sum five times as large, by another a sum seven times as large, and by another a sum twice as large. What will represent the whole loss? If y stands for ten dollars, what is the grocer's whole loss?

WRITTEN EXERCISES.

48.-Ex. 1. Henry has 2 apples, Alfred 3 apples and Arthur 9 apples. How many apples have they all?

SOLUTION. 2 apples, 3 apples and 9 apples are 14 apples. But let a denote 1 apple, then 2a will denote 2 apples, 3a three apples and 9a nine apples.

2a, 3a and 9a are 14a, the apples they all have.

2a

За

да

14α

2. Henry has lost two apples, Afred 3 apples and Arthur 9 apples. How many apples have they all lost?

SOLUTION. Let a denote 1 apple lost; then, — 2ɑ will denote 2 apples lost; -3a, 3 apples lost, and -9a, 9 apples lost.

- 2a

- 3a

- 9a

-2a, -3a and

9a are 14a, the apples they all lost.

- 14a

3. What is the sum of 7a, 6a and 11a?

4. What is the sum of -6bx, -10bx and - 13bx?

49. Rule for Adding Similar Terms having like Signs.--Add the coefficients, and to their sum prefix the common sign, and annex the common literal part.

10. What is the sum of bz, 5bz and 7bz?

11. What is the sum of 5ac-d, 3ac-7d, 3ac-6d and 8ac-12d?

12. Add -3c(a-x), -4c(a-x), -7c(a-x), -8c(a-x) and 10c(a-x). Ans. 32c(a-x).

CASE II.

Terms Similar with unlike Signs.

50.-Ex. 1. A man gained $100 one day and lost $50 the next day. What was the net result of the two days' transactions? If we give to gains the sign + and to losses the sign what will be the answer?

2. If a man gained $50 one day and lost $100 the next day, what was the result? If we give to gains the sign + and to losses the sign, what will be the answer?

3. If John has 20 cents and owes 15 cents, what is his financial condition? What is it if he has 15 cents and owes 20 cents? If what one has in possession is regarded as positive and what one owes is made negative, what expression will show John's condition in the first case? In the second case?

4. In January a merchant gains a certain sum, in February he loses three times as much, and in March he gains six times as much. Let stand for the gain in January; what will stand for the result of the three months' transactions?

x

5. Twice any quantity, plus four times the quantity, minus three times the quantity, minus five times the quantity, plus seven times the quantity, is how many times the quantity?

WRITTEN EXERCISES.

51.-Ex. 1. John earned in one week 8 dollars and spent 5 dollars, and the next week he earned 6 dollars and spent 7 dollars. How many dollars had he at the end of the second week?