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17. How do you check the work in subtraction ?

18. What do you mean by multiplication? Show that it is a short process of addition.

19. What names are given to the three numbers used in multiplication ? Illustrate.

20. What are factors ? Illustrate.

21. May the multiplicand be abstract ? May it be concrete ? Illustrate both cases.

22. What is the effect of changing the order of the addends ? also of the factors of a number ? Illustrate by the cases of 4 + 7 + 9, and 4 x 7 x 9.

23. What is meant by a power of a number ? by the square of a number ? by the cube of a number ? Illustrate.

24. What is the short way of multiplying an integer by 100 ? of multiplying United States money by 100 ?

25. What do you mean by division ? In what way does division differ from multiplication ?

26. Name the terms used in division and give an example showing the use of each.

27. If the dividend and divisor are concrete, what else may be said of their nature? What is the nature of the quotient ? Illustrate.

28. May the dividend be concrete and the divisor abstract ? Illustrate.

29. What is the short process of dividing an integer by 10 ? Illustrate by dividing 4750 by 10.

30. What is the effect on the quotient of multiplying the dividend or dividing the divisor ? Illustrate each case.

31. What is the effect on the quotient of dividing the dividend or multiplying the divisor ? Illustrate each case.

CHAPTER VI

DECIMAL FRACTIONS

66. Meaning of Decimals. We have already seen that $2.25 means $2 + 25 cents, which is the same as $25 or $24.

In the same way, 2.25 miles means 235 miles, or 24 miles ; 1.50 means 110%, or 1}; 1.75 means 116, or 14; 1.5 means 1%, or 13; 2.125 means 21,240, or 2; and 0.125 means 1260, or k. (See § 65.)

That is, 0.5 or .5 is a fraction whose denominator is not written, it being understood to be 10 from the fact that 5 occupies the first place to the right of the decimal point.

We therefore have the following: 0.5 means i, for the 5 extends to the 10th's place; 0.25 means y, for the 25 extends to the 100th's place; 0.125 means 125, for the 125 extends to the 1000th’s place. The names of the places are, in part, as follows:

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This number is read "one thousand three hundred forty-five and two thousand seven hundred sixty-five ten-thousandths.” The orders beyond ten-thousandths are hundred-thousandths, millionths, tenmillionths, and so on.

67. Decimal Fractions. A fraction whose denominator is not written, but is some power of 10, is called a decimal fraction, or more often simply a decimal.

Thus 0.25, which has the same value as mo, is a decimal fraction.

An integer and a decimal together form a mixed decimal; for example, 2.25.

68. Decimal Point. The period written at the left of the tenths is called the decimal point.

For example, 0.7 = 1o ; 0.08 = 187, or ; 0.005 = Tood, or ado.

It was not necessary to write a zero at the left of the decimal point in the above cases, for 0.5 means the same as .5. The zero is often written there to call attention more quickly to the decimal point.

69. Reading Decimals. We read a decimal precisely as if it were a whole number, and then give it the name of the lowest decimal place.

It is best to pronounce the word and at the decimal point only. Thus 100.023 is read "one hundred and twenty-three thousandths," while 0.123 is read "one hundred twenty-three thousandths."

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Read aloud as dollars and decimals of a dollar: 33. $1.25. 34. $25.05. 35. $26.70. 36. $125.50.

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70. Writing Decimals. Although we write a decimal fraction without its denominator, the denominator is shown by the decimal point and is indicated in reading.

Thus 0.9 has no printed denominator, but when we say tenths" we indicate the fact that the denominator is ten as well as state that the numerator is nine.

Therefore, write only the numerator of the decimal, inserting such zeros as are necessary to indicate the denominator, and place the decimal point before the tenths.

Thus, to write in figures two hundred one hundred-thousandths, we write 201, and prefix two zeros, preceded by a decimal point, so that the last figure is in hundred-thousandths' place. We then have 0.00201.

EXERCISE 28 Write in figures : 1. Seven tenths; one tenth; five tenths; nine tenths.

2. Twelve hundredths; fifty-five hundredths; nine hundredths; one hundredth; twenty hundredths.

3. One hundred twenty-five thousandths; forty-seven thousandths; ten thousandths; five thousandths.

4. One thousand two hundred fifty ten-thousandths; seven hundred sixty-five ten-thousandths.

5. 5 and 63 thousandths; 7 and 70 thousandths. 6. 1352 and 135 thousandths; 60 and 60 thousandths. 7. 1000 and 1 thousandth; 1000 and 10 thousandths. Express in figures as dollars and cents : 8. Two and seventy-five hundredths dollars. 9. Seventy-one and two hundredths dollars. 10. One hundred and eight hundredths dollars. 11. One thousand three and three hundredths dollars. 12. Two hundred four and two hundredths dollars.

71. Reduction of Decimals. Since decimals are only a special form of fractions, we may reduce them as we reduce common fractions.

For example, since po = 8 (8 65), therefore 0.5 = 0.50.

72. Annexing Zeros. Because, as just shown, 0.5 = 0.50, therefore

Annexing zeros to a decimal does not change its value.

73. Similar Decimals. Decimals that have the same number of decimal places are called similar decimals.

Thus 0.75 and 0.25 are similar decimals, and so are 0.150 and 0.275 ; but 0.15 and 0.275 are dissimilar decimals.

74. Reduction of Dissimilar Decimals to Similar Decimals. Since we may give any decimals the same denominators (understood, not written) by annexing or cutting off zeros,

Therefore, to reduce dissimilar decimals to similar decimals, give them the same number of decimal places by annexing or cutting off zeros.

Thus 0.4, 0.72, and 0.125 may all be reduced to thousandths as follows: 0.400, 0.720, and 0.125. Similarly, 0.40 and 0.600 are equal to 0.4 and 0.6.

EXERCISE 29

Reduce to similar decimals :

1. 0.2, 0.17. 2. 0.8, 0.08. 3. 0.7, 0.002. 4. 0.9, 0.130. 5. 0.6, 0.123. 6. 0.5, 0.700. 7. 0.4, 0.050. 8. 0.1, 0.357.

9. 0.12, 0.103. 10. 0.72, 0.008. 11. 0.69, 0.070. 12. 0.30, 0.700. 13. 0.99, 0.999. 14. 0.75, 0.630. 15. 0.50, 0.400. 16. 0.85, 0,392.

17. 0.001, 0.1100. 18. 0.207, 0.2376. 19. 0.412, 0.9320. 20. 0.681, 0.6881. 21. 0.900, 0.7000. 22. 0.620, 0.5700. 23. 0.090, 0.6550. 24. 0.584, 0.5834.

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