respectively 30.37, 31, 33.6756, and 28.6 gallons. How many gallons in all ?

8. A man bought 5 lots, containing, respectively, 26.62, 220.2007, 56.9, 5.8%, and 150.682 acres. How many acres in all ?

Ans. 460.31945. 9. Add 360.00025, 3.75, 567.893, 60,000.637, 200.050006, .0003625, 20.05.

10. Find the sum of 27, .625, 6, 3.6, 26.3125, 5.6, .813.

SUBTRACTION OF DECIMALS.

ART. 48. Ex. 1. From 60.025 take 3.0825.

60.0250

3.0825

56.9425 Ans.

Explanation.-Same as in addition.

RULE.

Write the numbers as in addition of decimals, subtract as in whole numbers, and point off as in addition of decimals.

5. From 362 ten-thousandths take 1056 millionths.

Ans. .035144.

6. From 875 thousandths take 62 ten-millionths.

7. From 100.001 take 93.00075.

Ans. 7.00105.

8. A man bought 8.75 yards of linen at one time and 29.0056 at another. He afterwards sold 25 yards. How many has he left?

9. From 7 tenths take 7 ten-millionths.

10. From 10001 ten-thousandths take 10001 ten-millionths.

MULTIPLICATION OF DECIMALS.

ART. 49. Ex. 1. Multiply 2.5 by .25.

2.5

.25

.625 Ans.

Explanation.-2.5=2%, .25=2, and hence 25x25=?%X ==.625.

00

RULE.

Multiply as in whole numbers, and point off as many figures in the product as there are decimal places in the multiplicand and multiplier.

Note. If there are not enough figures in the product, prefix ciphers. Thus: 1.6 x .016-.0256; .01 x .003=.00003.

Examples.

2. Multiply 37.5 by 4.5.

3. Multiply $16.37 by 3 hundredths.

4. What is 12 hundredths of $100.15 ?
5. What is 7 tenths of .201 thousandths?
6. Multiply .0015 by .125.

DIVISION OF DECIMALS.

ART. 50. All the examples in Division of Decimals fall under one of three cases, viz. :

1. When the decimal places in the dividend equal those of the divisor.

2. When the decimal places of the dividend exceed those of the divisor.

3. When the decimal places of the dividend are less than those of the divisor.

These three cases are illustrated in the following examples: Ex. 1. Divide 6.25 by .25.

.25)6.25

25. Ans.

Explanation.-Since the quotient arising from dividing one number by another of the same denomination is a whole number, 625 hundredths divided by 25 hundredths must give 25 units.

Ex. 2. Divide .864 by 3.6.

3.6).864.(.24 Ans.

72

144.
144

Explanation.-36 tenths (3.6) is contained in 8 tenths (the same denomination) 0 times; hence there are no units. in the quotient. 36 tenths is contained in 86 hundredths 2 tenths of a time and

14 hundredths remaining. 36 tenths is contained in 144 thousandths 4 hundredths of a time. Hence .864÷3.6=.24. Ex. 3. Divide 13.2 by .033.

Explanation.-13.213.200=13200 thousandths, which divided by 33 thousandths must 400. give 400, a whole number.

.033)13.200

RULE.

FIRST CASE.-Divide as in whole numbers; the quotient will be in units.

SECOND CASE.-Divide as in whole numbers, and point out as many places in the quotient as the decimal places of the dividend exceed those of the divisor.

THIRD CASE.-Make the decimal places of the dividend equal to those of the divisor by annexing ciphers, and then ceed as in whole numbers. The quotient will be in units.

pro

Note.-In either case, if there is a remainder, the division may be continued by annexing ciphers; but each cipher thus annexed will give one decimal figure in the quotient.

Proof. It is well for the student to test the correctness of his answer by multiplying the divisor by the quotient. If the quotient is correct, the product will be the dividend..

12. Divide 4.2 by 311.

13. Divide $16 by $0.25.

14. Divide 3 by 1.25.

Ans. .024.

Note. In this example, we annex two ciphers to make the division possible; this gives two decimal places in the dividend. We add another cipher to obtain the quotient figure 4; thus making in all three decimal places.

15. Divide 5 by 400.

16. Divide 9 by 1500.

17. Divide 6.4 by 80.

18. Divide.1 by .121.

Ans. .08.

19. Divide 6 by .08.

20. Divide 16 by .033].

CONTRACTIONS.

ART. 51. To divide a decimal by 10, 100, 1000, etc., remove the decimal point as many places to the left as there are ciphers in the divisor.

Note. If there are not figures enough in the number, prefix ciphers.

Examples.

1. Divide 6.25 by 100.

2. Divide .25 by 10.

3. Divide .45 by 1000.

4. Divide .01 by 100.

Ans. .0625.

ART. 52. To multiply a decimal by 10, 100, 1000, etc., remove the decimal point as many places to the right as there are ciphers in the multiplier. Thus 62.5 x 100=6250;

REDUCTION OF DENOMINATE NUMBERS.

ART. 53. A denominate number is composed of concrete units of different weights, measures, etc.

Denominate numbers are of two kinds, simple and com

pound.

A simple denominate number is composed of units of a single denomination, as 10 pounds; 12 hours.

A compound denominate number, or simply a compound number, is composed of units of several denominations of the same weight, measure, etc., as 5 days 16 hours 20 minutes.

Reduction is the process of changing the form of a denominate number without altering its value.

Remark. In treating of Denominate Numbers, we omit both tables and rules. The student is supposed to be familiar with the tables in common use.

ART. 54. To reduce a denominate number of a higher denomination to a simple denominate number of a lower.

Examples.

1. Reduce 5 lb. 6 oz. 10 pwt. 18 gr. of silver to grains.

36000 s.

Ans. 36000 s.