same number of boxes of each kind. How many boxes can he fill? Ans. 84. 22. A coal dealer paid $965 for some coal. He sold 160 tons for $5 a ton, when the remainder stood him in but $3 a ton. How many tons did he buy? Ans. 215.

23. A dealer in horses gave $7560 for a certain number, and sold a part of them for $3825, at $85 each, and by so doing, lost $5 a head; for how much a head must he sell the remainder, to gain $945 on the whole? Ans. $120.

24. Bought a Western farm for $22,360, and after expending $1742 in improvements upon it, I sold one half of it for $15480, at $18 per acre. How many acres of land did I purchase, and at 2513

what price per acre

PROBLEMS IN SIMPLE INTEGRAL NUMBERS.

124. The four operations that have now been considered, viz., Addition, Subtraction, Multiplication, and Division, are all the operations that can be performed upon numbers, and hence they are called the Fundamental Rules.

125. In all cases, the numbers operated upon and the results obtained, sustain to each other the relation of a whole to its parts. Thus,

I. In Addition, the numbers added are the parts, and the sum or amount is the whole.

II. In Subtraction, the subtrahend and remainder are the parts, and the minuend is the whole.

III. In Multiplication, the multiplicand denotes the value of one part, the multiplier the number of parts, and the product the total value of the whole number of parts.

IV. In Division, the dividend denotes the total value of the whole number of parts, the divisor the value of one part, and the quotient the number of parts; or the divisor the number of parts, and the quotient the

value of one part.

126. Every example that can possibly occur in Arithmetic, and every business computation requiring an arithmetical opera

tion, can be classed under one or more of the four Fundamental

Rules, as follows:

I. Cases requiring Addition.

There may be given

1. The parts,

2 The less of two numbers and their difference, or the sub

trahend and remainder,

There may be given

To find

the whole, or the sum total.

the greater number or the minuend.

II. Cases requiring Subtraction.

1. The sum of two numbers and

one of them,

2. The greater and the less of two numbers, or the minuend

and subtrahend,

III. Cases requiring Multiplication.

There may be given

1. Two numbers,

2. Any number of factors,

3. The divisor and quotient,

IV. Cases requiring

There may be given

1. The dividend and divisor,
2. The dividend and quotient,
3. The product and one of two
factors,

4. The continued product of

To find

their product.

their continued product.

the dividend.

Division.
To find

the quotient.
the divisor.

}

the other factor.

several factors, and the pro- that one factor.

duct of all but one factor,

127. Let the pupil be required to illustrate the following problems by original examples.

Problem 1. Given, several numbers, to find their sum.

Prob. 2. Given, the sum of several numbers and all of them but one, to find that one.

Prob. 3. Given, the parts, to find the whole.

Prob. 4. Given, the whole and all the parts but one, to find that one.

Prob. 5. Given, two numbers, to find their difference.

Prob. 6. Given, the greater of two numbers and their difference, to find the less number.

Prob. 7. Given, the less of two numbers and their difference, to find the greater number.

Prob. 8. Given, the minuend and subtrahend, to find the remainder.

Prob. 9. Given, the minuend and remainder, to find the subtrahend.

Prob. 10. Given, the subtrahend and remainder, to find the minuend.

Prob. 11. Given, two or more numbers, to find their product. Prob. 12. Given, the product and one of two factors, to find the other factor.

Prob. 13. Given, the continued product of several factors and all the factors but one, to find that factor.

Prob. 14. Given, the factors, to find their product.

Prob. 15 Given, the multiplicand and multiplier, to find the product.

Prob. 16. Given, the product and multiplicand, to find the multiplier.

Prob. 17. Given, the product and multiplier, to find the multiplicand.

Prob. 18. Given, two numbers, to find their quotients.

Prob. 19. Given, the divisor and dividend, to find the quotient. Prob. 20. Given, the divisor and quotient, to find the dividend. Prob. 21. Given, the dividend and quotient, to find the divisor. Prob. 22. Given, the divisor, quotient, and remainder, to find the dividend.

Prob. 23. Given, the dividend, quotient, and remainder, to find the divisor.

Prob. 24. Given, the final quotient of a continued division and the several divisors, to find the dividend.

Prob 25. Given, the final quotient of a continued division, the first dividend, and all the divisors but one, to find that divisor.

Prob. 26. Given, the dividend and several divisors of a continued division, to find the quotient.

Prob. 27. Given, two or more sets of numbers, to find the difference of their sums.

Prob. 28. Given, two or more sets of factors, to find the sum of their products.

Prob. 29. Given, one or more sets of factors and one or more numbers, to find the sum of the products and the given numbers. Prob. 30. Given, two or more sets of factors, to find the difference of their products.

Prob. 31. Given, one or more sets of factors and one or more numbers, to find the sum of the products and the given number or numbers.

Prob 32. Given, two or more sets of factors and two or more other sets of factors, to find the difference of the sums of the products of the former and latter.

Prob 33. Given, the sum and the difference of two numbers, to find the numbers.

ANALYSIS. If the difference of two unequal numbers be added to the less number, the sum will be equal to the greater; and if this sum be added to the greater number, the result will be twice the greater number. But this result is the sum of the two numbers plus

their difference

Again, if the difference of two numbers be subtracted from the greater number, the remainder will be equal to the less number; and if this remainder be added to the less number, the result will be twice the less number. But this result is the sum of the two numbers minus their difference.

Hence,

I. The sum of two numbers plus their difference is equal to twice the greater number.

II. The sum of two numbers minus their difference is equal to twice the less number.

PROPERTIES OF NUMBERS.

EXACT DIVISORS.

128. An Exact Divisor of a number is one that gives an integral number for a quotient. And since division is the reverse of multiplication, it follows that all the exact divisors of a number are factors of that number, and that all its factors are exact divisors.

NOTES.-1. Every number is divisible by itself and unity; but the number itself and unity are not generally considered as factors, or exact divisors of the number.

2. An exact divisor of a number is sometimes called the measure of the number.

129 An Even Number is a number of which 2 is an exact divisor; as 2, 4, 6, or 8.

130. An Odd Number is a number of which 2 is not an exact divisor; as 1, 3, 5, 7, or 9.

131. A Perfect Number is one that is equal to the sum of all its factors plus 1; as 6 = 3 + 2 + 1, or 28 = 14 +7 + 4 + 2 + 1.

NOTE The only perfect numbers known are 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2417851639228158837784576, 9903520314282971830448816128.

132. An Imperfect Number is one that is not equal to the sum of all its factors plus 1, as 12, which is not equal to 6 + 4 + 3 + 2 + 1.

133. An Abundant Number is one which is less than the sum of all its factors plus 1; as 18, which is less than 9 + 6 + 3+2+1.

134. A Defective Number is one which is greater than the sum of all its factors plus 1; as 27, which is greater than 9+3+1.

135. To show the nature of exact division, and furnish tests of divisibility, observe that if we begin with any number, as 4, and take once 4, two times 4, three times 4, four times 4, and so on indefinitely, forming the series 4, 8, 12, 16, etc., we shall have