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A. Continue dividing, as before, till there is no number greater than 1 that will divide two or more numbers without a remainder; then multiplying the divisors and numbers in the last line together, will give the least common multiple required.

More Exercises for the Slate. 3. Find the least common multiple of 4 and 16. A. 16. 4. Find the least common multiple of 10 and 15. A, 30. 5. Find the least common multiple of 30, 35 and 6. A. 210. 6. Find the least common multiple of 27 and 51. A. 459. 7. Find the least common multiple of 3, 12 and 8. A. 24. 8. Find the least common multiple of 4, 12, and 20. A. 60. 9. Find the least common multiple of 2, 7,14 and 49. A. 98.

I XLIII. TO REDUCE FRACTIONS OF DIFFERENT DE

NOMINATORS TO A COMMON DENOMINATOR. Q. When fractions have their denominators alike, they may be added, subtracted, &c. as easily as whole numbers; for example, of and are ; but in the course of calculations by numbers, we shal: meet with fractions whose denominators are unlike; as, for instance, we cannot add, as above, and together: what, then, may be considered the object of reducing fractions of different denominators to a common denominator ?

A. To prepare fractions for the operations of addition, subtraction, &c. of fractions.

Q. What do you mean by a common denominator ?

A. When the denominators are alike. 1. Reduces and to a common denominator. OPERATION

In performing Numer. 2 x 6=12, new numer.

this example, we Denom. 3 X 6=18, com. denom.

take , and multi

ply both its terms Numer. 5 x 3=15, new numer. į by the denominator Denom. 6 X 3 =18, com. denom. of ; also, we mul

| tiply both the terms of by 3, the denominator of; and, as both the terms of each fraction are multiplied by the same number, consequently the value of the fractions is not altered ; | XXXVII.'

From these illustrations we derive the following

RULE. Q. What do you multiply each denominator by for a new denomtrator ?

A. By all the other denominators.
Q. What do you multiply each numerator by for a new numerator?

A. By the same numbers (denominators) that I multiply its denominator by.

Note.-As, by multiplying in this manner, the same denominators are continually multiplied into each other, the process may be shortened; for, having found one denominator, it may be written under each new numerator. This, however, the intelligent pupil will soon discover of himself; and, perhaps, it is best he should.

* More Exercises for the Slate. 2. Reduce j and } to a common denominator. A. I, 39. 3. Reduce ţ and s to a common denominator. A. 13, 18 4 Reduce and 11 to a common denominator. A. 41, 44 á. Reduce g, and I to a common denominator.

A. 14, 198, 19% 6. Reduced, and to a common denominator.

Compound fractions must be reduced to simple fractions be fore finding the common denominator; also the fractional parts nf mixed numbers may first be reduced to a common denominator, and then annexed to the whole numbers. 7. Reduce 1 of f and to a common denominator.

A. 4, 49 8. Reduce 14 and á to a common denominator.

A. 14:11, *. 9 Reduce 102 and 3 of to a common denominator.

A. 1038, 48 10. Reduce 8:41 and 147 to a common denominator.

A. 87477, 147ky Notwithstanding the preceding rule finds a common denominator, it does not always find the least common denominator But, since the common denominator is the product of all tho given denominators into each other, it is plain, that this product ( XLII.) is a common inultiple of all these several denominators, consequently, the least common multiple found by [ XLII will be the least common denominator.

. 11. What is the least common denominator of f, & and } ? OPERATION.

Now, as the denominator of 3 ) 3. 6. 2

each fraction is 6ths, it is evideot

that the numerator must be pra 2 ) 1. 2. 2

portionably increased; that is,

we must find how many 6ths 1.1.1

each fraction is; and, to do this, Ans. 2 x 3=6

we can take , &, and of the

6ths, thus :
of 6=4, the new numerator, written over the 6,=4.
of 6=5, the new numerator, written over the 6,=a.
of 6=3, the new numerator, written over the 6,=8.

Ans. &, &, Hence, to find the least common denominator of several fractions, find the least common multiple of the denominators, for the common denominator, which, multiplied by each fraction, will give the new numerator for said fraction.

12. Reduce and to the least common denominator.

13 Reduce j and It to the least common denominator.

A. poi to 14. Reduce 146 and 134 to the least common denominator.

A. 1419, 1375. Fractions may be reduced to a common, and even to the least common denominator, by a method much shorter than either of the preceding, by multiplying both the terms of a fraction by any number that will make its denominator like the other denominators, for a common denominator; or by dividing both the terms of a fraction by any numbers that will make the denominators alike, for a common denominator. This method oftentimes will be found a very convenient one in practice.

15. Reduce and s to a common, and to a least common denominator.

#X2=f; then f and } = common denominator, A. 2) =]; then f and I = least common denominator, A.

In this example, both the terms of one fraction are multiplied, and both the terms of the other divided, by the same number consequently, (1 XXXVII.) the value is not altered.

16. Reduce is and to the least common denominator.

A. 17. Reduce Boots and to the least common denominator.

A., 18. Reduce za leo and to the least common denominator.

A. * 19. Reduce go and 2 to the least common denominator.

4. zosto

ADDITION OF FRACTIONS.

1 XLIV. 1. A father gave money to his sons as follows; to William of a dollar, to Thomas , and to Rufus s; how much is the amount of the whole? How much are \,, and , added together?

2. A mother divides a pie into 6 equal pieces, or parts, and gives to her son, and to her daughter; how much did she give away in all ? How much are and i added together?

3. How much are $++$? . 4. How much are itttt? 5. How much are igtigt ? 6. How much are ' + % +3%?

When fractions like the above have a common denominator expressing parts of a whole of the same size, or value, it is plain, that their numerators, being like parts of the same whole, may be added as in whole numbers; but sometimes we shall meet with fractions, whose denominators are unlike, as, for example, to add } and à together. These we cannot add as they stand; but, by reducing their denominators to a common denominator, by | XLIII., they make and , which, added cogether as befc. e, make ĝ, Ans.

1. Bought 3 loads of hay, the first weighing 19 cwt., the. second 20 cwt., and the third 22 cwt.; Wuat was the weight of the whole? :

}, }, }, reduced to a common denominator, are equal to 8.

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#8 and 14. these, joined to their respective whole numbers, give the following expressions, viz.

OPERATION. By adding together all the 60ths, Cwt. Cwt. viz. 45, 12 and 40, we have 87=137: 193=1945

then writing the 37 down, and carry} = 2012 ing the whole number, 1, to the = 2248

amount of the column of whole num

bers, makes 62, which, joined with Ans. 6237 cwt. / 37, makes 6237, Ans. 2. How much is of of X, and %, added together? ļoff= ; then and , reduced to a common denominator, give 22 and 3, which, added together as before, give =124, Ans. From these illustrations we derive the following

RULE.
Q. How do you prepare fractions to add them?

A. Reduce compound fractions to simple ones, then all the fractions to a common or least common denominator.

Q. How do you proceed to add ?
A. Add their numerators.

More Exercises for the Slate. 3. What is the amount of 168 yards, 17; yds. and 3} yards A. 3717. 4. Add together and ...

A. 1395 5. Add together 4, f and i1. A. 2475. 6. Add together I's, f and 1. A. 1485. 7. Add together 144 and 15%. A. 3072. 8. Add together of and f of }. A. Zo g. 9. Add together 3}, { of , and ]. A. 412.

SUBTRACTION OF FRACTIONS. 4 XLV. 1. William, having of an orange, gave / to Thomas; how much had he left? How much does á from leave ?

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