The specific forms of these problems are

1. Given the sum of equal quantities, and one of them, to find their number.

2. Given the sum of equal quantities, and their number, to find one of them.

3. Given the effect of two causes, and one of them, to find the other.

4. Given the sum of two or more quantities, and their number, to find their average, or mean.

ILLUSTRATIONS.

1. A man gave to each of his children $720, and to all of them he gave 6480 dollars; how many children had he?

2. A man put 1260 bushels of apples into 20 bags; how many bushels did he put in each?

3. If 1 man can build 725 rods of fence in 29 days, how long will it take 25 men to build the same amount?

4. If a ship sails 1189 miles in 29 days, what is the average rate per day?

Art. 91. The average, or mean, of two or more numbers is that number which, expressed as many times as there are numbers, would amount to their sum. It implies that the sum is considered as distributed into as many equal quantities as there are numbers. It is a popular method of dealing with unequal quantities, when they are considered together.

Art. 92. To find the average, or mean, of numbers.

Rules.-I. Divide their sum by their number. Or

II. Divide the sum of the excesses of the other numbers over the least by the number of numbers, and add the quotient to the least number.

Or

III. Divide the sum of the deficiencies, by which the other numbers are less than the greatest, by the number of numbers, and subtract the quotient from the greatest number.

EXAMPLES FOR PRACTICE.

1. Bought one cow for 36 dollars, and 2 others for 30 dollars

2. Find the average of 4, 8, 12, and 16. 3. Find the average of 9, 12, 15, and 18. 4. Find the average of 26, 37, 48, and 59. 5. A's age is 45, B's 30, C's 35, D's 60, What is the average of their ages?

4

=

) 12

32

Ans. 10. Ans. 13.

Ans. 422.

E's 70 years. Ans. 48 years.

6. The weights of some hogs are as follows: 250, 320, 275, 322, 415, 213, 244, 209, and 195 pounds. What is the average of their weights?

7. A man sold goods in six days to the following amounts: $80, $75, $92, $63, $210, $193. What did his sales average per day?

8. Seven houses are worth, respectively, $10000, $12000, $8500, $7525, $4260, $4180, and $3200. What is their average value?

Art. 93. Given the sum and difference of two numbers, to find the numbers.

Ex. 1. The sum of two numbers is 40, and their difference What are the numbers?

is 14.

ANALYSIS.-First, since 40:

Ans. 13 and 27.

= the less

the greater the less, 40 +14= the greater the less + 14. But the less +14= the greater. Therefore 40+14= twice the greater. Again, since 40 the greater, 40 14: the less the greater greater 14 the less. Therefore 40- 14 twice the less.

=

- 14.

But the

Rules.-I. To the sum add the difference, and divide the result by 2; the quotient will be the greater number. Or

II. From the sum take the difference, and divide the result by 2; the quotient will be the less number.

EXAMPLES FOR PRACTICE.

2. The sum of two numbers is 56, and their difference 16. What are the numbers?

3. The sum of two numbers is 81, and their difference 23. What are the numbers?

4. The sum of two numbers is 843, and their difference is 165. What are the numbers?

Art. 94. Given the sum of more that two numbers, and their differences, to find the numbers.

Rule. From the sum take the difference between the least and every other number, and divide the remainder by the number of numbers; the quotient will be the least number. To the least number add the difference between it and another; the sum is that number.

NOTE. The pupil should show the reason for the rule, in analyzing the solutions of examples.

EXAMPLES FOR PRACTICE.

1. The sum of 3 numbers is 557; the difference between the first, or smallest, and the second is 48, and the difference between the first and third is 113. What are the numbers?

the difference is 15, and the

2. The sum of three numbers is 846; between the first, or smallest, and the second difference between the second and third is 57. What are the numbers?

Subtraction. Simple

Terms.

Principles.

Multiplication. Numbers. Operations. Special.
J Common.

Division.

Rule.
Proof.

Exercises.

CHAPTER VIII.

PROPERTIES AND RELATIONS OF NUMBERS.

Art. 95. The properties of numbers are the qualities which are inseparable from them.

Art. 96. The integral factors of a number are such whole numbers as multiplied together, produce that number. They are often called simply factors of the number. In this

sense

A factor of a whole number is any whole number which when multiplied by another, will produce the given number.

An exact divisor of a number is a divisor which is contained in that number a whole number of times. Thus, 7 is an exact divisor of 7, 14, 21, &c. Such a divisor is often called simply a divisor, or measure of the number divided.

One number is said to be divisible by another, when it contains that other a whole number of times. Thus, 35, 42, 49, &c., are said to be divisible by 7. Such a number is said to be divided, or measured by its divisor.

Division is said to be exact, or without a remainder, when the quotient is a whole number.

NOTE. Strictly speaking, all complete division is exact, and every number is divisible by another equal to, or less than, itself. But convenience restricts the use of the terms exact and divisible to cases in which the quotient is an integer.

Art. 97. A multiple of a given whole number is any whole number of which it is a factor. It is called a multiple because it is considered as produced by multiplying the given number. Thus, 35 is a multiple of 7, because 35 may be produced by multiplying 7 by 5. In like manner 42 is a multiple of 7 by 6.