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1. If 3 apples cost i of a cent, what will 1 apple cost? How much is :-3?

2. If a horse eat or of a bushel of meal in 2 days, how much will he eat in one day? How much is 4:2

3. A rich man divided f of a barrel of flour among 6 poor men; how much did each receive ? How much is = 6?

4. If 3 yards of calico cost of a dollar, how much is it a yard? How much is $; 3?

5. If 3 yards of cloth cost 16 of a dollar, how much is it a yard?

The foregoing examples have been performed by simply dividing their numerators, and retaining the same denominator, for the following reason, that the numerator tells how many parts any thing is divided into; as, & are 4 parts, and, to divide 4 parts by 2, we have only to say, 2 in 4, 2 times, as in whole numbers. But it will often happen, that the numerator cannot be exactly divided by the whole number, as in the following examples.

6. William divided if of an orange among his 2 little brothers; what was each brother's part?

We have seen, (ST XXXVII.) that the value of the fraction is not altered by multiplying both of its terms by the same number; hence X2= Now, are 6 parts, and William can give 3 parts to each of his two brothers; for 2 in 6, 3 times. A. R of an orange apiece.

Q. In this last example, if (in 1) we multiply the denominator, 4, by 2, (the whole number,) we have }, the same result as before; why is this?

A. Multiplying the denominator makes the parts so many times smaller; and, if the numerator remain the same, no more are taken than before; consequently, the value is lessened so many times. From these illustrations we derive the following

RULE. Q. When the numerator can be divided by the whole number with. out a remainder, how do you proceed ?

A. Divide the numerator by the whole number, writing the denominator under the quotient.

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Q. When the numerator cannot be thus divided, how do you proceed?

A. Multiply the denominator by the whole number, writing the result under the numerator.

Exercises for the Slate. 1. If 8 yards of tape cost of a dollar, how much is it a yard? How much is 18:8.

2. Divideg by 8. 3. Divide its by 6.

A. =*5. 4. Divide zas by 8.

A. Iga 5. Divide 8% by 8. (Divide the numerator.) A. b. 6. Divide 216 by 4.

A. afu=15. Note. When a mixed number occurs, reduce it to an improper fraction ; then divide as before,

7. Divide $6 among five men. A. 6=4+5==127. 8. Divide 214 by 4.

A. 37. 9. Divide 16f by 5.

A. 1301=314. 10. Divide 25*1 by 20.

A. 128=1138 11. Divide 8 by 6.

A. H=112. 12. Divide 114} by 280.

A. 1906.


1. A man, owning of a packet, sells of his part; what part of the whole packet did he sell? How much is of g? 3 x 5 = 15

Ans. The reason of this operation will appear 4 X 8 32 from the following illustration :

Once is g, and 1 of is evidently } divided by 4, which is done (11 XL.) by multiplying the denominator, 8, by the 4, making 32; that is, off = 32.

Again, if I of $ be 32, then of will be 5 times as much, that is, z..

Again, if of ß be , then will be 3 times =}, Ans., as before.

The above process, by close inspection, will be found to

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consist in multiplying together the two numerators for a new numerator, and the two denominators for a new de. nominator.

Should a whole number occur in any example, it may be
reduced to an improper fraction, by placing the figure 1 un-
der it ; thus 7 becomes }; for, since the value of a fraction
(1 XXXIV.) is the numerator divided by the denominator,
the value of 7 is 7; for, 1 in 7, 7 times.
From these illustrations we derive the following

Q. How do you proceed to multiply one fraction by another ?

A. Multiply the numerators together for a new numerator ; and the denominators together for a new denominator.

Note. If the fraction be a mixed number, reduce it to an improper fraction; then proceed as before.

Mentál Exercises. 2. How much is 1 of 1 ? 6. How much is of 1 ? 3. How much is į of 7. How much is of 1? 4. How much is of ? 8. How much is Ty of L? 5. How much is of ģ 9. How much is of ido?


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Q. What are such fractions as these sometimes called ?
A. Compound Fractions.
Q. What does the word of denote?

A. Their continual multiplication into cach other.

Exercises for the Slate. 1. A man, having of a factory, sold of his part; what part of the whole did he sell? How much is į of ? x 8 =IS='s, Ans.

2. At 25 of a dollar a yard, what will of a yard of cloth cost? How much is 25 of s? A. I.

3 X 5 X 3 45 3. Multiply š of by .

8 X7 X7 392 4. Multiply & off by $. A. 14 =s.


A. 1144

3 15

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5. Multiply 5 of 13 by .
6. Multiply 367 by } A. 1868=627=1148.
Note.--If the denominator of any fraction be equal to the numerator of
any other fraction, they may both be dropped on the principle explained in
1 XXXVII. ; thus of 1 of & may be shortened, by dropping the numera-
tor 3, and denominator 3; the remaining terms, being multiplied together,
will produce the fraction required in lower terms ; thus, ^ of 0 of = 4
of 8 == . Ans.
The answers to the following examples express the fraction in its

lowest terms.
7. How much is 1 of of of ? A. 7's
8. How much is of 1 of ?

A. Po
9. How much is 5} times 53 ?

A. 304 10. How much is 161 times 163 ?

A. 2724 11. How much is 20 times ] of ? A. 2=36

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TWO OR MORE NUMBERS. Q. 12 is a number produced by multiplying 2 (a factor) by some other factor; thus, 2X6= 12; what, then, may the 12 be called ?

A. The multiple of 2.

Q. 12 is also produced by multiplying not only 2, but 3 and 6, likewise, each by some other number; thus, 2X6=12; 3x4=12; 6X2= 12; when, then, a number is a multiple of several factors or numbers, what is it called ?

A. The common multiple of these factors.

Q. As the common multiple is a product consisting of two or more factors, it follows that it may be divided by each of these factors without a remainder; how, thén, may it be determined, whether one number is a common multiple of two or more numbers, or not?

A. It is a common multiple of these numbers, when it can be divided by each without a remainder.

Q. What is the common multiple of 2, 3, and 4, then ? A. 24. Q. Why? 4. Because 24 can be divided by 2, 3, and 4, without a remainder.

Q. We can divide 12, also, by 2, 3, and 4, without a remainder; what, then, is the least number, that can be divided by 2 or more numbers, called?

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A. The least common multiple of these numbers.

Q. It sometimes happens, that one number will divide several other numbers, without a remainder; as, for instance, 3 will divide 12, 18, and 24, without a remainder ; when, then, several numbers can be thus divided by one number, what is the number called ?

A. The common divisor of these numbers.

Q. 12, 18, and 24, may be divided, also, each by 6, even ; what, then, is the greatest number called, which will divide 2 or mora num. bers without a remainder ?

A. The greatest common divisor.*

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* In T XXXVII., in reducing fractions to their lowest terms, we were sometimes obliged, in order to do it, to perform several operations in dividing; but, had we only known the greatest common divisor of both terms of the fraction, we might have reduced them by simply dividing once; hence it may sometimes be convenient to have a rule

To find the greatest common divisor of two or more numbers. 1. What is the greatest common divisor of 72 and 84?

OPERATION, In this example, 72 is contained in 84, 1 time, 72 ) 84 (1

and 12 remaining ; 72, then, is not a factor of 84.

Again, if 12 be a factor of 72, it must also be a 72

factor of 84; for 72 +12=84. By dividing 72

by 12, we do find it to be a factor of 72, (for 72
12) 72 (6
*12=6 with no remainder);

therefore, 12 is a

cominon factor or divisor of 72 and 84; and, as

the greatest common divisor of two or more A. 12, common di numbers never exceeds their difference, so 12, visor.

the difference between 84 and 72, must be the

greatest common divisor. Hence, the following RULE :--Divide the greater number by the less, and, if there be no remainder, the less number itself is the common divisor; but, if there be a remainder, divide the divisor by the remainder, always dividing the last divisor by the last remainder, till nothing remain: the last divisor is the divisor sought.

Note.-If there be more numbers than two, of which the greatest common divisor is to be found, find the common divisor of two of them first, and then of that common divisor and one of the other numbers, and so on.

2. Find the greatest common divisor of 144 and 132. A. 12.
3. Find the greatest common divisor of 168 and 84. A. 84.
-4. Find the greatest common divisor of 24, 48, and 96. A. 24.
Let us apply this rule to reducing fractions to their lowest terms.

See T XXXVII. 5. Reduce 184 to In this example, by using the common diviits lowest terms.

sor, 12, found in the answer to sum No. 2, we

have a number that will reduce the fraction to 2) 14 =1, Ans

its lowest terms, by simply dividing both termas

but onca After the same manner perform the following examples :6. Find the common divisor of 750 and 1000; also reduce 108 % to ita lowest terms. A. 250, and



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