15.m One of the parts of 20+ 15 is 10+5; what is the c part?

16. A farmer received $2025 for his sugar and $1824 for his cotton. The expense of raising the sugar was $1113, and of the cotton $749; what were his profits? Ans. $1987.

17.m $13 and $8 are two parts of $33; what is the c part?

18. A man paid $5270 for a house and $1835 for improving it. If he sells the house for $7500, what will be his profits? Ans. $395. 19.m If I begin at 40 and count backward, what will be the 12th number?

20. What is the 525th number below 937? Ans. 412. 21.m 50 contains 19, 11, and the c part; what is the latter?

22. A and B are 1529 feet apart; if A goes toward B 375 feet, and B goes toward A 682 feet, how far apart will they then be? Ans. 472 feet.

92. QUESTIONS FOR REVIEW.

What is: 1. Subtraction? 2. The minuend? 3. The subtrahend? 4. The remainder? 5. The sign of Subtraction?

What is denoted by the sign - ?

What is the relation of Subtraction to Addition? What do we find by Subtraction?

What is the: 1. Principle of Subtraction? 2. Rule for Subtraction? How may Subtraction be proved?

When we know the whole and several of its parts, how do we find the c part?

MULTIPLICATION.

INDUCTIVE EXERCISES.

93. 1. How many ones in 4 twos?

Ans. Two taken 4 times, or 2 +2 +2 + 2 = 8.

2. How many are 5 threes?

Ans. Three taken 5 times, or 3+3 +3 +3 +3 = 15. 3. How many are 4 sevens?

Ans. The sum of four sevens, or 28.

4. How many are:

6 (5's)? 3 (9's)? 7 (4's)? 8 (5 dollars)? 5 (10 c.)? 9 (10's)?

5. One boy has two hands; how many hands have 6 boys? Ans. 2 hands taken 6 times, or 12 hands. 6. A horse has 6 nails in each of his 4 shoes; how many nails in all?

Ans. 4 (6 nails), or 6 nails taken as many times as there are shoes.

7. What will 5 hats cost at $4 a piece? Ans. 5 ($4). 8. At 9c. each, what will three melons cost?

9. What will be the cost of 6 pairs of boots at $8 a pair? Ans. 6 ($8)=?

10. If a horse travels 4 miles per hour, how far does he travel in 7 hours?

11. There are 5 trees in the orchard and 20 peaches on each tree; how many peaches are in the orchard? 20 peaches are taken how many times? 5 (20 peaches)=?

DEFINITIONS.

94. Multiplication is a short method of adding equal parts, or the process of taking one number as many times as there are units in another.

95. The number to be taken is called the multiplicand; the number that shows how many times it is to be taken, the multiplier, and the result, the product.

96. The multiplicand and multiplier are called the fac tors of the product.

97. The Sign of Multiplication is X, which is read: times, or multiplied by. When stands between two numbers, it indicates that one of them is to be multiplied by the other.

Thus, 6X5 30 is read: 5 times 6 are 30, or 6 multiplied by 5=30. Here 6 is the multiplicand, 5 the multiplier, and 30 the product. The factors of 30 are 5 and 6.

Since 6X55 × 6, either may be read: 5 times 6 or 6 times 5. If one of the factors is a concrete number, it is regarded as the multiplicand, but may be used abstractly as the multiplier. It is evident that 5 (6) also indicates multiplication. See Art. 69.

98. The Complemental Factors of a number are those factors whose product is equal to that number.

Thus, the complemental factors of 6 are 2 and 3; of 12, 3 and 4, or 2 and 6; of 36, 4 and 9, or 2, 3, and 6, etc., etc.

Point out the multiplicand, multiplier, and product in

ach.

100. Express by signs:

1. 5 times 7 are 35. 2. 8 times 9 are 72.

3. 7 times $4 are $28.

101. PRINCIPLES.

4. 4 taken 3 times equals 12. 5. 9 times 5 are how many ?

6. 6 times 4 pens are 24 pens.

1o. The multiplier is an abstract number.

2o. The multiplicand and product are like numbers. 3°. Multiplication may be effected by Addition.

Let the pupil point out or verify each of these principles in examples 1, 2, 5, 6, and 7 of Art. 99.

Thus, in ex. 5,

1o. 6, the multiplier, is an abstract number.

2o. 2 yards, the multiplicand, and 12 yards, the product, are like numbers.

3o. 6X2 yards = 2 yards+2 yards+2 yards+2 yards+2 yards+2 yards = 12 yards.

102. By Multiplication we find a number when we know its complemental factors.

Thus, if 5 and 7 are the c factors of a number, the number is 5X 7=35; if 6 and 9 are the c factors, the number is 6X 9=54.

SUGGESTIONS TO TEACHERS.-In every example and problem in Multiplication the pupil should be required to point out: 1o. The whole and the c parts.

Thus, in 4X $5=?, the parts are $5, $5, $5, $5, and the whole is the answer required, viz: $20.

2o. The whole and the c factors.

Thus, in 6×5= ?, the c factors are 6 and 5, and the whole is the answer required, 30.

103. The following table contains the products of each two numbers from 0 to 12. It should be thoroughly committed to memory.