ORAL EXERCISE

1. The subtrahend is 7, the minuend 21; what is the difference? How much is 21 minus 7?

2. The difference is 11, the minuend 16; what is the subtrahend? What number minus 16 equals 11?

3. The subtrahend is 27, the remainder 14; what is the minuend? 27 minus what number equals 14?

4. The minuend is 25 more than the subtrahend; what is the difference?

5. The subtrahend is 47 less than the minuend; what is the difference?

6. The subtrahend and difference together equal 39; what is the minuend?

7. Mr. Roberts owed Mr. Lambert $75 yesterday. To-day he has paid him $25. What balance is due?

WRITTEN EXERCISE

Perform these subtractions, and check every answer by adding the remainder and subtrahend. See how many you can perform in five minutes.

17. $48,002.75-$31,609.50. 18. $29,875.25-$20,975.26.

MULTIPLICATION

ORAL EXERCISE

1. Read the products by 2: 4, 7, 1, 9, 6, 2, 8, 3, 11, 5, 10. This or a similar set of numbers should be written on the board.

2. In the same way, read rapidly the products by the various numbers from 3 to 10. Also by 20, 30, 100.

3. Count by 2's from 2 to 20, and give the multiplication table of 2's. Give the multiplication tables from 3 to 10.

4. How much is 4+4 +4 +4 +4? 5 times 4? Multiplication is a short form of what other process?

21. Abstract numbers. A number that does not refer to any particular kind of object or measure is called an abstract number.

For example, 10, 42, 31, are abstract numbers.

22. Concrete numbers. A number that refers to some particular kind of object or measure is called a concrete number. For example, $10, 32 ft., 6 mo., are concrete numbers.

23. Multiplication. The process of taking one number as many times as there are units in another is called multiplication.

This means multiplication by an abstract integer. Thus, to multiply $3 by 2 means that $3 is taken 2 times.

24. Multiplicand. The number multiplied is called the multiplicand.

25. Multiplier. The number by which we multiply is called the multiplier.

$325 multiplicand

3 multiplier

$975 product

26. Product. The result of multiplying is called the product.

27. Nature of the numbers. It is therefore seen that 1. The multiplier must be thought of as abstract.

2. The product is like the multiplicand.

That is, if we have 3 times 17 ft., the 3 is abstract, and the product is feet, like the multiplicand.

28. Multiple. The product of two abstract integers is called a multiple of either.

For example, 35 is a multiple of 7 and of 5.

29. Factors. The numbers which multiplied together make another number are called its factors.

For example, the factors of 30 are 2, 3, and 5.

tions.

Teachers are advised not to require much memorizing of definiThe important thing is that the words shall be used correctly. Those just defined are rarely heard in business. In reading 2 × $6 and $6 × 2, teachers should follow the custom of the school. The former is preferably read "two times $6," and the latter "$6 multiplied by 2." In order not to confuse children who have learned other ways of reading, this book uses the word times where there might be any misunderstanding.

30. Power. The result of taking a number any number of times as a factor is called a power of the number.

For example, 2 × 2 = 4, and 4 is called the second power, or square, of 2. 2 x 2 is written 22. So 2 × 2 × 2, or 23, is the third power of 2. It is also called the cube of 2.

ORAL EXERCISE

1. State four multiples of 7; of 3; of 9; of 11; of 15. 2. State the factors of 35; of 77; of 21; of 49; of 121. 3. State the squares of 7, 6, 9, 8, 10, 11, 20, 100, 1000. 4. What is the square of 5? the cube of 3? the fourth power of 2? the cube of 5? of 2? of 10?

31. The process. You have already learned that in multiplying, for example, $635.50 by 215,

This is the complete operation:

But we write only this:

$635.50

215

3177 50

6355 0

127100

$136632.50

32. Zero in the multiplier. If a zero appears in the multiplier,

This long process might be taken: But we need only this:

MULTIPLYING BY POWERS OF 10

ORAL EXERCISE

1. Multiply by 10: 10 ct., 50 ct., $1.20, $3.50. 2. Multiply by 100: 3, 42, 565, $1.50, $21.75.

3. Multiply by 1000: 4, 7, 10, 40, 52, 100, 250.

4. State a short way of multiplying an integer by 10; by 100; by 1000.

5. State a short way of multiplying a decimal by 10; by 100; by 1000.

33. Multiplying by powers of 10. To multiply an integer by 10, annex a 0; by 100, two 0's; by 1000, three O's.

34. To multiply a decimal by 10, move the decimal point 1 place to the right; by 100, 2 places; by 1000, 3 places.

Hence to multiply 27 by 40, multiply by 4 and annex a 0.

Also to multiply $25.16 by 300, multiply by 3 and move the decimal point 2 places to the right.

27

40

1080

$25.16

300

$7548

‹ 11. If a train goes 48.2 mi. an hour, how far will it go in 20 hr., at the same rate?

12. If it costs $14.75 a year to educate you, how much does it cost to educate 300 children, at the same rate?