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SECTION IX.

MULTIPLICATION OF FRACTIONS BY FRACTIONS.

We have seen how we may multiply or divide a fraction by a whole number. We will now inquire how we can multiply or divide one fraction by another. Let us multiply by . First multiply by 2, which gives for the answer. But here we have multiplied by 2, instead of the real multiplier, . Now 2 is 5 times greater than ; the product & then is 5 times greater than it should be. It must therefore be divided by 5. We divide by 5 by multiplying the denominator by 5, giving for the answer.

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In the same way multiply by 1. X. X.

DIVISION OF FRACTIONS BY FRACTIONS.

Let us now divide by . First divide by 3.. This we do by multiplying the denominator by 3, giving for the answer. Here, however, we have divided by 3, instead of the true divisor,. We have used a divisor seven times too large. The quotient, therefore, will be seven times too small; must therefore be multiplied by 7, making the answer 28. In the same way perform the following: 3÷7. 8÷4. 7+1. 5÷4. }÷}.

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The above analysis shows the grounds of the rules usually given in Arithmetics for the multiplication and division of fractions.

For Multiplication, multiply the numerators together for a new numerator, and the denominators for a new denominator. For Division, invert the divisor and proceed as in multiplication.

Sometimes we wish to find the value of a compound fraction, as of; in such cases we may understand the sign of multiplication, X, to stand in the place of the word of, and treat it as a case of multiplication. For in the above example it is plain that one third of is, and two thirds is twice as much, that is, 2.

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What is of of? we have for the answer 108.

Multiplying as we have done above, 24 But this operation may be shortened. We see that 4 appears as a factor both in the

numerator and the denominator. We may then cancel them both, which will have the same effect as dividing both terms of the answer by 4. Again, 3 appears in both the numerator and the denominator, for in the denominator it is a factor of 9. We may therefore cancel 3 in both terms.

2.3.4 The question will then appear thus, X-X substituting 34 93'

24

24

3 in place of the 9. Multiplying together the terms that now remain, we have for the answer. This is the same fraction ass. If you separate the terms of into their prime factors, and cancel what are common to both, the remaining factors will give the fraction 3.

108

Multiply the fractions XXX, writing the terms that are composite in the form of their prime factors, and canceling factors that are common in both, it will stand

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2 2×7.5

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which gives 18.

Multiply XXXX

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Multiply X. X. XX.

TO MULTIPLY OR DIVIDE WHOLE NUMBERS BY FRACTIONS.

The above examples will show how to multiply or divide a whole number by a fraction.

Multiply 7 by . Multiplying 7 by 4 gives 28, which is 5 times too great, because 4 is five times greater than . We must therefore divide the answer by 5, thus 28. As this is more than 1, we can reduce it to a whole number and a fraction. As is equal to 1, 25 will be equal to 5; 28 therefore is equal to 53.

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In this way multiply 6 by 7. 9 by 2. 8 by 4. This operation is in fact the same as multiplying a fraction by a whole number, which has been treated of already. Let us next divide 7 by 2. Dividing 7 by 3 we have }; here, however, we have divided by a number 4 times too great, for 3 is four times greater than 4, If the divisor is 4 times too great, the quotient will be 4 times too small; 3, therefore, must be multiplied by 4, giving 28 for the answer.

Divide 8 by. 9. 11. 10 T

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To reduce an improper fraction, as 13, to a whole number and a proper fraction, we have only to consider how many

whole ones the fraction is equal to, and how much remains. Thus is equal to 3; 13 therefore is equal to 31.

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In like manner, if we have a whole number and a fraction, we may always reduce it to an improper fraction.

ADDITION OF FRACTIONS.

Suppose we wish to add together 3 so that its value shall be expressed in a single expression; we must change 3 to halves, which will be; adding to this we have for the

answer.

In order to unite separate numbers into one expression, they must be of the same kind. We cannot unite 2 bushels and 3 pecks in one expression. It is still 2 bushels and 3 pecks, and we can make nothing else of it; but if we change the bushels to pecks, making 8 pecks, we can then add the 3 pecks, and bring it all into one expression, 11 pecks. So to unite 5 we must change the 5 to thirds, making 15, and add the 3, making. This is called reducing a mixed number to an improper fraction.

Reduce to an improper fraction 73, 81, 47, 51, 61, 91, 32, 52, 151, 162, 133, 202, 21.

Supposing we wish to add to, we must change the to fourths, making; adding these, we have for the answer.

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Let us now add and . This question you perceive has a difficulty which the former ones had not; for is no number of fifths, and therefore we cannot bring the fraction into fifths by any multiplication. We want a number for the denominator which can be divided both by 3 and by 5. Now if you examine, you will find no such number until you come to 15. This, is of course, divisible by 3 and by 5, for these are its factors. We will then take 15 for the denominator. This we call the common denominator. Taking now the fractions and, and changing the denominator 3 to 15, we see that we have made it 5 times as large as it was before; that is, we have multiplied it by 5. We must therefore multiply the numerator by 5, to preserve the value of the fraction. The fraction then becomes without altering its value. Pass

ing now to the second fraction, , we see that, in changing the denominator to 15, we have multiplied it by 3; we must therefore multiply its numerator by 3. This will make the fraction 1. The two fractions will stand, then, 18+, which added together are =178.

TO FIND A COMMON DENOMINATOR.

We can always obtain a common denominator, by multiplying the two denominators together; then, for the numerators, consider, in the case of each fraction, what its denominator has been multiplied by, in order to change it to the common denominator, and multiply the numerator by the same number. Thus each fraction will have had its numerator and its denominator multiplied by the same number, and so its value will not be changed.

What is the value of +? of +? of $+? of +? of +? of +3? of +? of +?

Supposing we wish to add the fractions and . We can proceed as above, and with the common denominator, 24, the fractions will be +2. But we need not employ so large a denominator as 24. We seek the smallest denominator that shall contain both 4 and 6 as a factor. If now we separate 4 and 6 into their prime factors, we shall find the factor 2 belonging both to 4 and to 6; thus, 2×2, 2×3. Now one of these may be cancelled, and we shall still have 2×2 for the number 4, and 2×3 for the number 6. Multiplying the factors which remain, 2×2×3, we have 12 for the smallest common denominator.

From this we see, that, when both the denominators contain the same factor, we may reject it from one of them, and multiply together the factors that remain.

Add to. Here 2X2 is common to both denominators, rejecting it in one, and multiplying, we obtain 24 for the least common denominator.

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Add to. Here 3X3 is common to both denominators, rejecting it in one, and multiplying what remains, we have 54 for the least common denominator.

Add to Add to. Add

to

When more fractions than two are to be added it is often most convenient to add two together first, and then add a third to the sum of these, and so on.

Add ++3

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5 10

First add and, which equal. Next

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16 T2

+;8=19 and 3=1; 19+12=12-17, ans.

Add ++

these add.

Add ++

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First add and fo; then to the sum of

Add 3+3+15. Add +3+3.

We may in the above cases obtain a common divisor by first multiplying all the denominators together, but it is not usually so easily nor so quickly done.

Miscellaneous Examples.

1. A man spends of a dollar in a day; what part of a dollar will he spend in 5 days? How much will he spend in 9 days? How much in 11 days?

2. A man earns of a dollar in a day; how much will he earn in half a day? How much in of a day? How much in of a day?

Here consider whether you can divide the numerator.

3. A man earns earn in half a day? in of a day?

of a dollar in a day; how much can he

How much in † of a day? How much

Consider whether you can divide the numerator; and if you cannot, what you must do.

4. A vessel filled with water leaks so that of its contents will leak out in a week; at this rate, how much will leak out in a day?

What is

of?

5. If a team ploughs of an acre in 6 hours, how much will it plough in one hour? How much in 3 hours?

What is of What is of § ?

6. If a horse runs of a mile in one minute, how far will he run in 2 of a minute?

How far will he run in of a minute?

What is of? What is of ?

7. A man has 7 of a dollar, which he wishes to distribute equally among several persons, giving of a dollar to each; how many can receive this sum, and what will be the remainder?

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