DECIMAL FRACTIONS.

PRELIMINARY EXERCISES.

1. If a sheet of paper is divided into 10 equal parts, what part of a sheet is I of these parts?

One of these parts is of a sheet.

2. If one of these tenths is divided into 10 other equal parts, what part of a sheet is I of these parts?

One of these parts is

of, or Too of a sheet.

3. If one of these hundredths is divided into 10 other equa1 parts, what part of a sheet is 1 of these parts?

Each part is

of of, or rooo of a sheet.

4. What is meant by a tenth, a hundredth, a thousandth, etc. A tenth is one of the ten equal parts into which a number or thing may be divided, etc.?

5. How much greater are tens than units; hundreds than tens; thousands than hundreds, etc.?

Tens are 10 times greater than units, and each succeeding order is 10 times greater than the preceding.

6. How much less are tenths than units; hundredths than tenths; thousandths than hundredths, etc.?

Tenths are 10 times less than units; hundredths are 10 times less than tenths; and so on, each succeeding order being 10 times less than the preceding.

7. What places do tens, hundreds, thousands, etc., occupy? Tens occupy the first place on the left of units; hundreds, the second; thousands, the third, etc.

8. Following this analogy, what place should tenths, hundredths, thousandths, etc., occupy?

Tenths in the decreasing scale correspond with tens in the increasing scale; hence they should occupy the first place on the right of units. In like manner, hundredths, which correspond with hundreds, should occupy the second place; thousandths, the third place, etc.

9. How many units make a ten, tens a hundred, etc.? 10. How do the orders of whole numbers increase?

They increase from right to left by the scale of 10.

11. How many tenths make a unit; hundredths a tenth : thousandths a hundredth, etc.

Ten tenths make a unit; ten hundredths make a fenth; ten thousandths make a hundredth, etc.

12. How do the orders of these fractions decrease?

They decrease from left to right by the scale of 10.

ΝΟΤΑΤΙΟΝ

OF DECIMALS.

13. What are Decimal Fractions?

Decimal Fractions are those in which the unit is divided into tenths, hundredths, thousandths, etc.

They arise from dividing a unit into ten equal parts, or tenths; then subdividing one of these tenths into ten other equal parts, or hundredths; and so on, the successive orders decreasing regularly by the scale of 10.

14. How, and upon what principle are they expressed?

By placing a point before the numerator, and assigning to each figure a value according to the place it occupies, as in whole numbers. Thus, is expressed by writing 3 in the first place on the right of units; as, .3; Too by writing 3 in the second place; as, .03; 1000 by writing 3 in the third place; as, .003.

15. What do figures standing in the first, second, third, etc., places on the right of units denote?

When standing in the first place on the right of units, they denote tenths; in the second place, they denote hundredths; in the third place, thousandths, etc.

NOTES.-I. The point used to distinguish decimals from whole numbers, is called the decimal point.

2. These fractions are called decimals from the Latin decem. ten, which indicates their origin and scale of decrease.

16. What is the denominator of a decimal fraction?

It is always 10, 100, 1000, etc., or 1 with as many ciphers annexed as there are decimals in the numerator.

Name the orders of integers, beginning at units. Name the orders of decimals, beginning at units place.

17. What is the effect of prefixing ciphers to decimals? Each cipher prefixed to a decimal, diminishes its value ten times, or divides it by 10.

18. What is the effect of annexing ciphers to decimals? The value is not altered. Thus, .3 .30 .300, etc. 19. How write decimals?

Write the figures of the numerator in their order, assioning to each its proper place below units, and prefix to them the decimal point.

If the numerator has not as many figures as required, supply the deficiency by prefixing ciphers.

NOTE. A decimal and integer written together, are called a mixed number; as, 35.263. (P. 92, Q. 13.)

1. On which side of units are tens? Tenths? Thousands? Hundredths? Hundreds? Thousandths?

2. What is the name of the second place on the right of units? The fourth? The third? The fifth?

3. How many decimal places are required to express tenths? Thousandths? Hundredths? Millionths?

13. Write 6 hundredths. 41 thousandths. 7 thou

sandths.

14. Write 201 ten-thousandths.

sandths.

752 hundred-thou

98 mil

15. Write 5 millionths. 63 millionths.

lionths. 375 millionths.

20. How read decimals?

Read the decimals as whole numbers, and apply to them the name of the lowest order.

REMARK.-The unit's place should always be the starting point

both in reading and writing decimals.

Copy and read the following:

15. .7.

16. .75. 17. .06.

29. 44.0643.

35. 306.46531.

30. 53.21034.

36. 500.00729.

31. 72.05213.

37. 607.329267.

23. 5.078. 18. .121. 24. 6.2356. 19. .065. 25. 7.3062. 20. .008. 26. 8.5602. 32. 84.00605. 38. 730.004308.

**Dictation exercises in reading and writing decimals should be practiced till the class is perfectly familiar with them.

REDUCTION OF DECIMALS.
CASE I.

To Reduce Decimals to Common Fractions.

1. Reduce .27 to a common fraction.

OPERATION.

.27=10, Ans.

ANALYSIS.-Since .27 has two decimal figures, its denominator must be 100. Hence, .27. In the operation we omit the decimal point, and place the denominator 100 under the 27.

21. How reduce decimals to common fractions?

Erase the decimal point, and place the denominator under the numerator. (P. 131, Q. 16.)

NOTE.-After decimals are reduced to common fractions, they may be reduced to lower terms, to a common denominator, etc., and then be treated in all respects like other common fractions. 2. Reduce .35 to a common fraction, and to its lowest terms. Ans. .35, and 35-26. Reduce the following decimals to common fractions in their lowest terms:

To Reduce Common Fractions to Decimals.

1. Reduce to a decimal.

OPERATION.

4) 1.00

.25

ANALYSIS. is equivalent to 1 divided by 4. But I cannot be divided by 4; we therefore reduce it to tenths by annexing a cipher to it, making 10 tenths. (P. 57, Q. 18.) Now of 10 tenths=2 tenths and 2 tenths over. Reducing the 2 tenths to hundredths by annexing a cipher, we have 20 hundredths; and of 20 hundredths 5 hundredths. Therefore, equals .25.