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To reduce a mixed number to an improper fraction.

Multiply the whole number by the denominator of the fraction, and add the numerator to the product ; then that sum written above the denominator will form the fraction required. Reduce 1274, to an improper fraction. Ans. 2190 Reduce 514 to an improper fraction. Ans. 8220

To reduce an improper fraction, to an equivalent whole or mixed number. Divide the numerator by the denominator, and the quotient will be the answer. Reduce to an equivalent whole or mixed number. Ans. 2. Reduce to an equivalent whole or mixed number.

Ans. 34. Reduce 143 to an equivalent whole or mixed number.

Ans. 1271 Having now attended to the multiplication of a fraction by a whole number, we will next attend to the multiplication of one fraction by another.

Pup. This looks to me to be very difficult. I can see plainly the reasons given for multiplying a fraction by a whole number, but cannot see how to multiply one fraction by another.

7. Tut. Suppose it were required to multiply i by ; by what has been said respecting the multiplying of whole numbers by a fraction, the operation would consist in dividing { by 5, and multiplying the result by 4; and in order to divide you must multiply its denominator, which you will understand from what I have told you before. After having divided by 5 you must multiply the result by 4, viz. multiply the numerator, 3 by 4. Hence the following rule for multiplying one fraction by another.

Multiply the numerators together for a numerator, and the denominators together for a denominator. Multiply by

Ans. 13.
Multiply ý by it:
Nultiply 5 by is.
Multiply iż by

Ans.
Ans. .

Ans.

When two mixed numbers are to be multiplied together it is best to reduce them to improper fractions, and then proceed as with proper fractions. Thus to multiply 41 by 31 they would be x * = 117.

It is usual in expressing fractions of fractions, to read them thus, į of of of

We have now gone through with multiplication of fractions, and will attend to the division of them.

8. When we say that one number contains another, we speak correctly, provided the numbers are whole ones; but when applied to fractions, it is not strictly correct, for as multiplying a number by a fraction decreases it, so dividing a number by a fraction increases that number. If it were required to divide 48 by 4, the quotient would be 12; but if the divisor had been 2, which is one half of 4, instead of 4, the quotient would have been 24; and if instead of 2 we had divided by 1, the quotient would have been just the dividend, or double the quotient by 2.

Hence it follows that to divide by è, the quotient must be double the dividend. From what has here been said, and from the relation which division has to multiplication, it appears that the division of fractions must be the reverse of the multiplication of them. Hence the following rule.

To divide one fraction by another, invert the divisor, and proceed exactly as in multiplication of fractions. When a whole number is to be divided by a fraction, it may be expressed fractionwise by putting a unit for its denominator; thus 4 is expressed I, and the same may be done in the multiplication of fractions.

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From what has been said respecting the multiplication and division of fractions, it may be proper to give the following table.

By vltiplying the numerator, the fraction is dividiedied.
By multiplying

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By dividing
the denominator, the fraction is

s divided.

multiplied. Pup. You have taught me how to multiply and divide fractions, and have said nothing about adding and subtracting them.

Is it not necessary sometimes to add and subtract them ?

Tut. it is. And it may seem preposterous to you that I should teach you how to multiply and divide before teaching you how to add and subtract. But I have done this because a fraction, being a remainder after division, is more immediately connected with multiplication and division, and is more easily performed on, by them, than by addition and subtraction.

9. When it is required to add or subtract fractions, the reasons of the operations are very plain, if the denominators are all alike. If it be required to add to it is evident that adding together the numerators and placing their sum over the common denominator, will give the true amount.

But when the denominators of the several fractions are different, this will not be the case, for the parts of which they are composed are of different magnitudes. But to avoid this difficulty, the fractions are reduced to a common denominator.

If, for instance, it be required to add to, we must first reduce them to a common denominator in the following manner.

10. If each term of the first fraction be multiplied by 6, the denominator of the second, the first fraction will be changed into ; and if each term of the second fraction be multiplied by 4, the denominator of the first, it will be changed into an each of them equal to their for mer expressions.

Again, if it be required to subtract from the same difficulty presents, as when we wished to add them. But being reduced to a common denominator, we can subtract one from the other and obtain the true difference between them.

Pup. When there are several fractions to be added or subtracted at the same time, how shall I proceed?

Tut. When more than two fractions are given, the process is the same'; for all the denominators must be alike, since each one is the product of all the other denominators. Hence we have the following rule.

“ Any number of fractions are reduced to a common denominator, by multiplying the two terms of each by, the product of the denominators of all the others." Or by multiplying each numerator by all the denominators, except its own, for a numerator, and all the denominators continually together for a common denominator.

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Reduce and to a common denominator. 3 X 9 = 27

new numerators, 4 X 5 = 20

4 X 9 = 36 common denominator. Reduce 1, and to equivelent fractions, having a common denominator.

5 X 8 = 40 X 4 = 160
5 X 4 = 20 X 8 = 160 common denominators.
4 X 8

40 X !
20 X 5 100 new numerators.

32 X

= 160

40

32 X

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Ans. 1960, 160, 160*

100

630

1890 1800 1750

Reduce js, Ķ, , , and } to a common denominator.

Ans. , . Reduce to, , and to a common denominator.

Ans.

3150 3150 31309 315ő. Having now learned how to find a common denominator, you will be able to add and subtract fractions.

To add fractions, first reduce them to a common denominator, then the sum of all the numerators, written over the comnon denominator, will form the fraction required.

When the fractions are mixed numbers, reduce them to improper fractions; and when there are compound fractions, or fractions of fractions, they must be reduced 10 a simple fraction by multiplying all the numerators together for a numerator, and all the denominators together for a denominator. Add and together.

Ans. 41 Add y, and together.

Ans. 33's Add , of and 94 together,

Ans. 1000

To subtract one fraction from another ; prepare them as in addition ; then the difference of the numerators written over the common denominator, will be the difference required. From take.

Ans. .
From 1 take o

Ans. 1
From 5 take 2

Ans. 27.
From 37 take 211

Ans. 83.

From what you have been taught concerning the increase of numbers in a tenfold proportion, and the intricacy of vulgar fractions, it must appear evident to you, that, if fractions were'made to decrease in the same manner that whole numbers increase, or if whole numbers continued their manner of decrease from left to right, to fractions, the labour of operating with them would be much facilitated. We have already a rule for this, which is called

DECIMAL FRACTIONS.

And can you tell me what decimal fractions are ?

Pup. I cannot. I did not know before, that there was more than one kind of fractions. I thought, that when a unit could be broken into any number of parts whatever, and that by these parts any part of a unit coulit be expressed in its true value, and all questions performed by them, that any other kind of fractions would be useless.

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