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11. What part of 1 cwt. is 3 qrs. 15 lb. 14 oz. ?

Ans. 11.

12. What part of 1 bu. is 3 pks. 7 qts. 1 pt. Ans. §‡. 13. What part of 15 pipes is 25 gals.? Ans. 395• NOTE. In this example 15 pipes must be reduced to gals.; or, which is better, divide both 15 and 25 by 5, and then reduce the quotient 3 pipes to gals Dividing both by 5 will not affect the answer, (§ XLI.) since, both numerator and denominator are equally divided.

14. What part of 2 miles is 7 fur. 11 in. 2 b. c,?

76032

43200

Ans. 33211 15. What part of 1 mo. is 22 d. 15 h. 1 m.? A. 32581 16. What part of the whole duration of the world have you lived?

17. The earth's diameter, is 7,9113 miles in diameter nearly, and the highest peak of the Himmaleh mountains, is estimated to be 27,677 ft. above the level of the ocean. What part of the whole diameter of the earth is the height of this mountain?

MENTAL EXERCISES.

LV. 1. What part of a penny is of a farthing? of a farthing?

2. What part of a shilling is of a penny? of a penny?

3. What part of a pound is shilling?

4. What part of a penny is 5. What part of a gallon is

of a shilling? of a

of a farthing?

of a quart?

It is evident, that as fractions are not in their nature different from other numbers, but only in their form, we need no other rules for reducing them from one denomination to another, than those, given for whole numbers. Hence, to reduce a Fraction from a lower to a higher denomination,

DIVIDE AS IN REDUCTION ASCENDING OF WHOLE NUMBERS.

6. Reduce of a shilling to the Fraction of a £.

Ans. To=80.. NOTE. The results should always be reduced to their lowest terms.

7. Reduce

8. Reduce

a cwt.

of a dwt. to the Fraction of a lb. Troy.
Ans. 1830=336•
of a lb. Avoirdupois to the Fraction of

Ans. T1⁄2T·

9. What part of a week is 2 of an hour? Ans. 717• 10. Reduce 51 furlongs to the Fraction of a mile.

Ans.

11. What part of a mile is 7 of a b. c. ?
12. Reduce 273 gals. to the Fraction of a butt.
13. Reduce 655 qts. to the Fraction of a bu.

MENTAL EXERCISES.

§ LVI. 1. 1 of a penny is how many farthings? ¦ of a penny?

2. of a shilling is how many pence? of a shilling? 3. of a shilling is how many pence? how many pence and farthings?

4. of a lb. Troy is how many oz.? ↓ of a lb. ?

The rule, of course, for the reason assigned in the last section, is the same as in whole numbers. Hence, to reduce a Fraction from a higher denomination to a lower,

MULTIPLY AS-IN REDUCTION DESCENDING OF WHOLE NUMBERS. of a £ to the Fraction of a penny.

5. Reduce

6. Reduce

7. Reduce

40

A. 349.

of a hhd. to the Fraction of a qt.
of a cwt. to the Fraction of an oz.
A. =88.

8. What part of a pt. is of a bar. ?

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1440

1441

9. What part of a minute is 14 of a day? A. 1441 10. What part of an E. E. is of a yd? A. First reduce it to the Fraction of a quarter by (S LV.) and then to the Frac tion of an E. E.

52

11. What part of a hhd. is of a puncheon? A. 4-1. 12. What part of a guinea is 5 of a £? A. 299. 13. Reduce 1624,6, pun. to tier. A. 3249. 14. Reduce 19421 tier. to pun. A. 971013.

MENTAL EXERCISES.

§ LVII. 1. In of a shilling how many pence? In?

From these latter examples it will be observed, that the value of a decimal is not altered by annexing cyphers; for 1 tenth is the same as 10 hundredths and 2 tenths is the same as 200 thousandths, the significant figure, being just as far from the units' place in one case as in the other. But, as has been observed above, cyphers prefixed, (between the unit's place and the significant figures,) materially affect the value, since every cypher, thus prefixed, removes each significant figure, one place farther from the units' place, and, of course diminishes its value ten fold. As these principles are important, we state them concisely as follows,

1. CYPHERS ON THE RIGHT OF A DECIMAL DO NOT ALTER THE VALUE. But

II. EACH CYPHER ON THE LEFT DECREASES THE VALUE TEN

FOLD.

Thus. .5, .50, .500, .5000, and 50000, are all equal in value, being each equal, as is evident to. But .5, .05, .005, .0005, .00005, and ,000005 decrease in ten fold proportion.

The pupil will now see the use of the point or period, which we have employed to separate decimals from whole numbers. If he were required to read this decimal, .15, he would call it 15 hundredths, but if there were no point, as here, 15, he would call it 15 (units or whole numbers,) simply. The point, then, enables him to determine what name and what value belongs to a decimal, by showing him its distance from the units' place. As it separates decimals from whole numbers, it is usually called a SEPARATRIX. The following examples contain whole numbers and decimals mixed, and are therefore called MIXED NUMBERS.

Three, and five tenths

Four, and seventy four hundredths
8.93 7.625 9.842 7:6324 5.55555.
82.594 90.006 825.0304 9004.010203

3.5

4.74

827.34359.

NOTE. When speaking of Federal Money, we remarked that, dollars and dimes might all be read as dimes; dollars, dimes, and cents might all be read as cents; and dollars, dimes, cents, and mills might all be read as mills. The reason of this is that Federal Money increases in ten fold proportion, in the same manner as decimals. For the same reason whole numbers and tenths may all be read together as tenths; whole numbers, tenths and hundredths may all be read together as hundredths; and so on to any distance. Thus 2.7 may be read. two, and seven tenths, or twenty seven tenths; 3.25 may be read three and twenty five hundredths, or three hundred and twenty five hundredths, &c.

The pupil will perhaps be ready to demand, what are decimals but Fractions. Fractions (§ xxvI.) are ex

pressions for parts of numbers, and so are decimals. We reply that decimals ARE Fractions, and are often called DECIMAL FRACTIONS. The only differerence between them and any other Fractions is, that they increase and decrease just like whole numbers, by tens, and therefore we are not obliged to write down their denominators. If we please, however, we may write the denominators, and this we shall easily do, from observing the name of the decimal. Thus, 2.5=25.75=70.

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It will be seen by these examples that,

10000

27.93485=

THE DENOMINATOR OF A DECIMAL ALWAYS CONSISTS OF A UNIT, OR 1, WITH CYPHERS ANNEXED.

Fractions having other denominators, are called VULGAR FRACIt will be further seen by the above, that,

TIONS.

THE NUMBER OF CYPHERS IN THE DENOMINATOR OF A DECIMAL IS THE SAME AS THE NUMBER OF PLACES BELONGING TO THE DECIMAL.

Or, the distance of the lowest figure of the decimal from the separatrix, determines the number of cyphers in the denominator. When the denominator is actually written, cyphers on the left of the decimal may be neglected; but when it is written, and afterwards removed, the cyphers must of course be carefully restored. In the following, let the denominators be removed, and the numerators writen decimally.

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When decimals are written with their denominators, like Vulgar Fractions, it often happens that they can be reduced to lower terms. Let the following be thus written and reduced as low as possible.

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6162000

12. 39.00002

18. 345.858

NOTE. We have mentioned Federal Money as being similar to decimals. In fact, the denominations of Federal Money are decimals. The dollar is consider. ed the unit or whole number; des stand in the tenths' place, cents in the hundredths', and mills in the thousandths'. The French weights, measures, &c., we have likewise seen are on the principles of decimals. (§ XXXIX.) The advantage in the use of decimals consists in the uniformity of their denominators, and in their regular increase by 10 like whole numbers. Hence, we have always to carry for 10; and the operations of the four ground rules are as simple as in whole numbers.

SLIX. 1. Change a dollar to a decimal.

Here we wish to find how many tenths, hundredths, &c. of a dollar, there are in a dollar. Now a tenth of a dollar is a dime; hence, we have to inquire how many dimes there are in a dollar. a dollar is 10 times a dime, because 10 dimes make a dollar. Then a dollar is of a dime=5 dimes=$0.5 Ans. 2. Change to a decimal.

2

This question is the same as the last, except that it is not Federal Money. We wish here to find how many tenths, hundredths, &c. there are in a unit.

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2

a unit

A. $0.125.

is 10 times a tenth, of a tenth,=.5 Ans.
3. Change of a dollar to a decimal.
4. Change to a decimal. A. .125
5. Change to a decimal.

A. .0625

NOTE. If we multiply by 10, to reduce the Fraction to tenths, it becomes 18 of a tenth, which not being a whole tenth, we are obliged to place a cypher in the tenths' place, and proceed as before.

6. Change to a decimal. A. .2

NOTE. It is best to reduce the vulgar Fraction, first, to its lowest terms. 7. Change to a decimal. A. .2

8. Change to a decimal.

30

A. .5

9. Change,,, and to decimals.

A. .25, .625, .2 and .1875

16

10. Change, and to decimals.

15

1.6

11. Change, 70, 300, and

A. .025, .05 and .01 to decimals.

12. Change, and to decimals.

Another mode of explaining the above operation may be given. 13. Change to a decimal.

If both numerator and denominator of a Fraction be multiplied by the same number, the value will not be

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