Rule. Write the figures at the right of the decimal point, each in its proper order, noting vacant orders, if any, by ciphers.

2.1. Ans. .7. 6. 10. Ans. .106. | 10. 71.

100

Ans. 7.16. 3. 18. Ans. .03. 7. 1. Ans. .007. 11. 191. Ans. 19.009.

Ans. 101.65.

15. Twenty-three ten-thousandths.

16. Thirty-six units, and five tenths.

17. One hundred one units, and sixty-five hundredths.

18. Sixteen units, and sixteen thousandths.

Ans. 16.016.

19. Three hundred twenty-five ten-thousandths. 20. Five units, and three hundred thousandths. 21. Nineteen units, and six hundred thirty-one tenthousandths.

REDUCTION.

138. Annexing a cipher to a decimal does not alter the value of the decimal.

For, the order of the significant figures of the decimal is not changed. Thus, .3, or .30, is the same as

139. Hence, to change decimals having different denominators, to equivalent decimals having a common denominator,

Recite the Rule. What effect has the annexing of a cipher to a decimal? How do you change decimals having different denominators to equivalent decimals having a common denominator?

Make each decimal have the same number of decimal orders, by annexing ciphers.

EXERCISES.

Change to equivalent fractions having a common denominator,

A Decimal to a Common Fraction.

140. 1. Change .35 to a common fraction.

OPERATION.

.35% = o, Ans.

Removing the decimal point and writing the denominator, we have, which, reduced,

is. Therefore, .35 is equal to.

Rule. Write the denominator to the decimal, omit the decimal point, and reduce the common fraction to its lowest terms.

A Common Fraction to a Decimal.

141. 1. Change & to a decimal fraction.

Recite the Rule for reducing a decimal to a common fraction.

OPERATION.

4)3.00

75, Ans.

equals 3÷4. Since we can not divide 3 by 4, we reduce 3 to tenths, and have 30 tenths; of 30 tenths is 7 tenths, with a remainder of 2 tenths; we write the 7 tenths.

The 2 tenths we reduce to hundredths, and have 20 hundredths; of 20 hundredths is 5 hundredths, which we write. Therefore, equals .75.

Rule. - Annex ciphers to the numerator, divide by the denominator, and point off as many orders for decimals as there were ciphers annexed

Reduce to decimals:

Examples.

2. .

3.2.

4. .

142.

When a common fraction can not be exactly expressed in decimals, the division may be carried to a sufficient degree of exactness, and the sign + annexed to the result to indicate its incompleteness.

8. Reduce to a decimal of three places.

9. Reduce to a decimal of four places.

Ans. .444+.

Ans. .8571+.

10. Reduce to a decimal of four places.

Ans. .4166+.

143.

ADDITION.

Since ten of any order of decimals make one of the order next higher, decimals may be added in the same manner as simple integers. Hence,

Recite the Rule for reducing a common fraction to a decimal. Why may decimals be added the same as simple integers?

Rule. Write the numbers so that figures of the same order shall stand in the same column, add as in integers, and place the decimal point in the amount under those in the numbers added.

4. Add 6.051, .095, 31.6, and 1.0075.

5 Add 13.015, .9031, 12.186, and .09.

6. Add 120.16, .003, 1401.7, 1.5, and 19.8.

7. Add 7 units and 8 thousandths, 28 units and 16 hundredths, 56 units and 7 tenths. Ans. 91.868.

8. A merchant bought four lots of goods; for the first he paid $69.125; for the second, $193.3; for the third, $1008.56; and for the fourth, $752.375; how much did he pay for the whole? Ans. $2023.36.

SUBTRACTION.

144. Since ten units of any order of decimals make one of the order next higher, decimals may be subtracted in the same manner as simple integers. Hence,

Rule. Write the less number under the greater, so that figures of the same order shall stand in the same column; subtract as in integers, and place the

Recite the Rule. Why may decimals be suotracted as in whole numbers? Recite the Rule.

Here, in Example 3, as there are no thousandths in the minuend to subtract from, we consider that order in the minuend as filled by 0, since annexing a cipher to a decimal does not alter its value.

4. Subtract 5.91 from 13.675.

5. Subtract 18.16 from 19.5.

6. From 37 dollars take 31.375 dollars.

Ans. 7.765.

Ans. 1.34.

7. Bought 160 acres of land, and sold from it 141.125 acres; how much then had I left?

Ans. 14.85.

8. From 15 take 15 hundredths. 9. A farmer owning 78 hundredths of a farm, sold 725 thousandths of it; how much of it had he left?

10. From 10 take 1 ten-thousandth.

11. Subtract 139.216 from 400.95.

Ans. .055.

Ans. 9.9999.

Ans. 261.734.

12. What is the difference between 64.075 and .195326? 13. From 107 take .0007.

Ans. 106.9993.

14. From forty-three units and seventy-five thousandths, take thirty-five units and sixty-seven hundredthousandths. Ans. 8.07433.

Is the value of a decimal altered by annexing a cipher?