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2. Multiply 54.25 by 2.2802. 3. Multiply 15,278 by .7854. 4. Multiply 79,347 by 23.15. 5. Multiply 350 by .7853.

6. Multiply 20 by .55.

7. Multiply one-tenth by one-tenth.

8. Multiply 25 by twenty-five hundredths,

Ans. 123.70536. Ans. 12, nearly. Ans. 1836.88305. Ans. 274.855. Ans. 11.

9. At 4 mills apiece, what will 400 quills cost?

Ans. 01.

Ans. 6.25.

Ans. $1.80.

To multiply decimals by 10, 100, 1000, &c., remove the decimal point as many places to the right as there are ciphers in the multiplier, agreeably to Article 13. Removing the decimal point one figure to the right increases every figure ten-fold, &c.

10. Multiply 4.56 by 10.

11. Multiply $1.75 by 100. 12. Multiply .006 by 1000.

Ans. 45.6.
Ans. $175.

Ans. 6.

13. A benevolent person, whose income was $6000, gave .12 of it for charitable purposes; what did he give away?

Ans. $720.

14. The capital stock of a bank was of a million of dollars, and 3 of it was owned equally by 4 individuals; how much was owned by each?

25

Ans. $22500.

DIVISION OF DECIMALS.

(ART. 61.) Division being the reverse of multiplication, and it having been shown, in multiplication of decimals, that the product of any two factors must contain as many decimal places as the number of decimals in both factors: now, the divisor and quotient, in division, may be considered as factors of the dividend; hence, the number of decimal places in the quotient must be equal to the difference of the number of places in the divisor, and the number made use of in the dividend.

If the dividend really have no decimals, make a decimal point at the right of the unit, and fill up the blank decimal places with ciphers, as many as you wish. In dividing, the decimal places may run out; we may then annex ciphers and count them as decimals. If the dividend has decimals, but not so many as the divisor, fill up blank places, as before, which will not affect the value of the decimal, (Art. 57). From the above explanations, the reason of the following rule must be evident:

RULE. Divide as in whole numbers, and point off from the right of the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor.

1. Divide 42.42 by 2.113.

EXAMPLES.

Operation.
2.113)42.4200(20.07

42.26

16000
14791

Here we rest, and count off the quotient. There were, originally, 2 decimal places in the dividend, and in the operation we used three more; that is, filled up three blank decimal places with ciphers. This made five, and three, the number in the divisor, taken from it, leaves two decimal places in the quotient.

2. Divide 2.3421 by 21.1.
3. Divide 2.3421 by .211.
4. Divide .8297592 by .153.
5. Divide 8.297592 by .153.
6. Divide 12 by .7854.

Quot. 0.111.

Quot. 11.1. Quot. 5.4232.

Quot. 54.232. Quot. 15.278.

7. Divide 3 by 3; divide 3 by .3; 3 by .03; 30 by .03.

(ART. 62.) The preceding rule for division of decimals is as clear and as explicit as any mere rule can be: but, so careless are pupils in general, about the decimal point, that little or no reliance can be placed on the value of their quotients, until they are taught to use their judgments in each and every case, independent of any rule. To call into exercise this individual reliance, is the object of the following remarks and explanations.

To divide to cut into parts-will not, at all times, give a clear understanding of division, and confusion frequently arises from taking this view of the subject; we better consider it as one number measuring another. For example: How often will .5 of a foot measure 12 feet? In other words, divide 12 by .5, or divide 12 by. Here, if the student should imagine that 12 must be cut into parts, he would make a great error. He must divide 120 tenths into parts; in this case, into 5 parts, because the 5 is .5: or, he may consider that of a foot may be laid down in 12 feet; that is, measure 12 feet 24 times. Or, he may reduce the 12 feet to half feet, and then divide by 1. In all cases, the divisor and dividend must be of the same denomination, before the division can be effected. But, in decimals, these reductions are made so easily that a thoughtless operator rarely perceives them; hence the difficulty in ascertaining the value of the quotient.

We now give a few examples, for the purpose of teaching the pupil how to use his judgment. He will then have learned a rule more valuable than all others.

EXAMPLES.

Divide 15.34 by 2.7. Here, we consider the whole number, 15, is to be divided by less than 3: the quotient must, therefore, be a little over 5. One figure, then, in the quotient, will be whole numbers, the

rest decimals.

Divide 15.34 by .27. Here we perceive that 15 is to be divided, or rather measured, by less than of 1; therefore, the quotient must be more than 3 times 15. Or, we may multiply both dividend and divisor by 100, which will not affect the quotient, and then we shall have 1534 to be divided by 27. Now, no one can mistake how much of the quotient will be whole numbers: the rest, of course, decimals.

Divide 45.30 by .015. Conceive both numbers to be multiplied by 1000; then the requirement will be to divide 45300 by 15, a cornmon example in whole numbers.

By attention to this operation, the student will have no difficulty in any case where the divisor is less than the dividend.

Here is one of the most difficult cases:

Divide .003753 by 625.5. In all such examples as this, we insist upon the formality of placing a cipher in the dividend, to represent the place of whole numbers, thus:

625.5)0.0037530(

We now consider whether the whole number in the divisor will be contained in the whole number in the dividend, and we find it will not: we, therefore, write a cipher in the quotient, to represent the place of whole numbers, and make the decimal on the right, thus: 0.

We now consider that 625 will not go in the 10's, nor in the 100's nor in the 3, nor in the 37, nor in the 375, but it will go in the 3753

We must make a trial at every step, that is, every time we take in view another place; and we must take but one at a time. In this case, then, we shall have 0.000006, the quotient.

Divide 3 by 30. 30 will not go in 3; we, therefore, write 0 for place of whole numbers, and then say, 30 in 30 tenths, 1 tenth times, or 0.1.

Divide .55 by 11.

11)0.55(0.05.

11 in 0, no times; 11 in 5 tenths, no times; 11 in 55 hundredths, 5 hundredths times. It will be observed that we make the decimal point in the quotient as soon as we ascertain it; not wait, and

then find where it should be, by counting, &c.,-a rule against which we have nothing to say; but good judgment, at ready command, is far superior to any rule that can be formed.

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4. If 350 pounds of beef cost $12,25, what is the cost of one pound?

Ans. ,035. 5. At $5,75 per yard, how much cloth can be purIchased with $19,40625? Ans. 3,375 yards. 6. At 7 per cent., how much capital must be invested to yield 602 dollars? Ans. $8600.

7. If a contribution, amounting to $36,72, be made by a congregation consisting of 918 persons, how much is it a-piece? Ans. ,04, or 4 cents.

8. If 275 lemons cost $2,475, how much is it a-piece? Ans. 9 mills.

9. A benevolent individual gave away $600 per annum to charitable objects, which was ,12 of his income; what was his income?

(ART. 63.) By paying some little attention to the relation of numbers, we may abbreviate certain divisions, in a similar manner as we abbreviated multiplication, in Art. 21.

To divide by ,25 is the same as to multiply by 4.

66

66

66

66

by,5 is the same as to multiply by 2. by,75 is the same as to multiply by 4, and divide by 3.

66

66

66

66

To divide by ,125 is the same as to multiply by 8. by,333 is the same as to multiply by 3. by,666 is the same as to multiply by 3, and divide by 2, &c. &c., for any aliquot part of unity. The same is true of aliquot parts of 10, 100, 1000, &c.

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4. Divide 15 by,16%.

5. Divide 4,27 by,333.

Ans. 90. Ans. 12,81.

REDUCTION OF DECIMALS.

(ART. 64.) To reduce a vulgar fraction to its equivalent decimal: the value of any fraction is found by dividing the numerator by the denominator; thus, 2. But in a proper fraction, we cannot divide the numerator without first reducing it. Annexing one cipher, makes it tenths; two ciphers, hundredths; and so on. The quotient, therefore, will be decimal parts. Thus: reduce to a decimal.

8)1,000
,125

We may take this view of the case: Multiply both numerator and denominator, by 10, 100, 1000, &c.; which will not change the value of the fraction, (Art. 35.) We then have 1000. Now divide both numerator and denominator by 8, and we have 125 which is a decimal fraction, by the definition of decimals. whichever view we take, we have the following

10009

Hence,

RULE. Place a decimal point at the right of the numerator, annex ciphers, and divide by the denominator. Compound fractions must first be reduced to simple

ones.

EXAMPLES.

1. Reduce,, and 3, to equivalent decimals.

Ans. ,25,,5,,75. 2. What decimal is equivalent to ? Ans. ,625. 3. What decimal is equivalent to 3 of ?? Ans.,4.

4. Reduce to a decimal.

5. Reduce 25 to a decimal.

12

6. Reduce to a decimal.

Ans. ,9375.

Ans. ,0008.

Ans. ,7058 +

There are many vulgar fractions that cannot be exactly expressed in decimals. The following are some of them: 3,3,9,。,。,。, 7, &c.

1 2 1

4 5

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