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RULE. Q. What, then, is the rule for reducing a mixed or whole number to an improper fraction ?

A. Multiply the whole number by the denominator of the fraction.

Q. What do you add to the product?
A. The numerator.
Q. What is to be written under this result?
A. The denominator.

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More Exercises for the Slate. 2. What improper fraction is equal to 2066? A. 1601. 3. What improper fraction is equal to 7212? A. 4. What improper fraction is equal to 4%? 5. What improper fraction is equal to 12? 6. What improper fraction is equal to 1673? A. 197. 7. What improper fraction is equal to 17141 ? A. 17. 8. What improper fraction is equal to 1441 ? A. 1729. 9. Reduce 302 pounds to 20ths. As zx of a pound=1 s., 2=2 s., the question is the same as if it had been stated thus : In 30£ 5 s. how many shillings? A. 92=605 shillings.

10. In 144 weeks, how many 7ths? A. 192=101 days. 11. In 263 pecks, how many 8ths ? A. 211=211 quarts.

1 XXXVII. TO REDUCE A FRACTION TO ITS LOWEST

TERMS.

Q. When an apple is divided into 4 parts, 2 parts, or ļ, are evidently of of the apple: now, if we take ž, and multiply the 1 and 2 both by 2, we shall have again ; why does not this multiplying alter the value?

A. Because, when the apple is divided into 4 parts, or quarters, it takes 2 times as many parts, or quarters, to make one whole apple, as it will take parts, when the apple is divided into only 2 parts, or halves : hence, multiplying only increases the number of parts of a whole, without altering the value of the fraction.

Q. Now, if we take 4, and multiply both the 2 and 4 by 2, we obtain = 1; what, then, is 3 equal to ?

A. , or

Q. Now it is plain that the reverse of this must be true; for, if we divide both the 4 and 8 in by 2, we obtain, and, dividing the 2 and 4 in Q by 2, we have t; what, then, may be inferred from these remarks respecting multiplying or dividing both the numerator and denominator of the same fraction?

A. That they may both be multiplied, or divided, by the same number, without altering the value of the fraction.

Q. What are the numerator and denominator of the same fraction called ?

A. The terms of the fraction.
Q. What is the process of changing into its equal called ?
A. Reducing the fraction to its lowest terms.

Mental Exercises.

1. Reduce to its lowest terms.
2. Reduce to its lowest terms.
3. Reduce to its lowest terms.
4. Reduce 1 to its lowest terms.
5. Reduce 18 to its lowest terms.
6. Reduce zo to its lowest terms.
7. Reduces to its lowest terms.

Operation by Slate illustrated. 1. One minute is ob of an hour, and 15 minutes are di what part of an hour will make, reduced to its lowest terms? OPERATION.

Q. How do you get the is in this 3,

example ? 15 3

A. By dividing 15 and 60 each by 5. 12 4

Q. How do you get the ?

A. By dividing 3 and 12, each, by 3. Q. How do you know that is reduced to its lowest terms? Ă. Because there is no number greater than 1 that will divide "h the terms of £ without a remainder.

From these illustrations we derive the following

RULE. Q. How do you proceed to reduce a fraction to its lowest terms ?

Ă. Divide both the terms of the fraction by any number that will divide them without a remainder, and the quotients again in the same manner.

Q. When is the fraction said to be reduced to its lowest terms?

A. When there is no number greater than 1 that will divide the terms without a remainder.

More Exercises for the Slate. 2. Reduce 34 of a barrel to its lowest terms. 3. Reduce o of a hogshead to its lowest terms. 4. Reduce o of a tun to its lowest terms. 5. Reduce 1144 of a foot to its lowest terms. 6. Reduce of a gallon to its lowest terms. 7. Reduce 14% of an inch to its lowest terms.

A. 7. A. 4. A. t.

A. 1

1 XXXVIII. TO MULTIPLY A FRACTION BY A

WHOLE NUMBER.

1. If 1 apple cost } of a cent, what will 2 apples cost? How much is 2 times } ?

2. If a horse eat of a bushel of oats in one day, how many bushels will he eat in 2 days? In 3 days? How much is two times & ? 3 times à ?

3. William has of a melon, and Thomas 2 times as much; what is Thomas's part? How much is 2 times ? 2 times ? 2 times ž? 3 times 2? 6 times is?

Q. From these examples, what effect does multiplying the numera for by any number appear to have on the value of the fraction, if the denominator remain the same ?

A. It multiplies the value by that number.

Q. 2 times is =; but, if we divide the denominator 4 (in ď) oy 2, we obtain }; what effect, then, does dividing the denominator by any number have on the value of a fraction, if the numerator remain the same?

A. It multiplies the value by that number.

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Q. What is the reason of this?

Ă. Dividing the denominator makes the parts of a whole so many times larger; and, if as many are taken, as before, 'which will be the case if the numerator remain the same,) the value of the fraction is evidently increased so many times.

Again, as the numerator shows how many parts of a
whole are taken, multiplying the numerator by any number,
if the denominator remain the same, increases the number
of parts taken; consequently, it increases the value of the
fraction.

4. At 1 of a dollar a yard, what will 4 yards of cloth
cost? 4 times it are 1 = of a dollar, Ans. But, by
dividing the denominator of B by 4, as above shown, we
immediately have in its lowest terms.
From these illustrations we derive the following

RULE.
Q. How can you multiply a fraction by a whole number?

A. Multiply the numerator hy it without chang-
ing its denominator.

Q. How can you shorten this process ?

A. Divide the denominator by the whole number, when it can be done without a remainder.

Exercises for the Slate. 1. If a horse consume is of a bushel of oats in one day, how many bushels will he consume in 30 days? A. 19=6 bushels.

2. If 1 pound of butter cost z of a dollar, what will 205 pounds cost? A. 205=3026=304 dollars.

3. Bought 400 yards of calico, at g vf a dollar a yard; what
did it come to ? A. 1200 = $150.

4. How much is 6 times it? A. 1=114
5. How much is 8 times 15 ? A. 420=26=29.
6. How much is 12 times 35? 1.6=6*5=64
7. How much is 13 times it? A. 2742= }$.
8. How much is 314 times ? 4.942=2353 =2351
9. How much is 513 times 11? A. 3511=326 Pr:
10. How much is 530 times ? A. 120 =4833.

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115 Divide the denominator in the following. 11. How much is 42 times 12 ? A. 11. 12. How much is 13 times 2 68? A. 239=1218. 13. How much is 60 times Tão? A. =27.

14. At 2} dollars a yard, what will 9 yards of cloth cost? times 2 are 18, and 9 times $ are j=1}, which, added to 18, makes 19} dollars. A. This process is substantially the same as 1 XXVII., by which the remaining examples in this rule may be performed.

15. Multiply 34 by 367. A. 11921. 16. Multiply 67 by 211. A. 14508.

17. Multiply 36 by 42. A. 12928=1294. 1 XXXIX. TO MULTIPLY A WHOLE NUMBER BY A

FRACTION. Q. When a number is added to itself several times, this repeated addition has been called multiplication; but the term has a more extensive application. It often happens that not a whole number only, but a certain portion of it, is to be repeated several times; as, for instance, If you pay 12 cents for a melon, what will & of one cost ?' 4 of 12 cents is 3 cents; and to get q, it is plain that we must repeat the 3, 3 times, making 9 cents, the answer; when, then, a certain portion of the multiplicand is repeated several times, or as many times as the numerator shows, what is it called ?

A. Multiplying by a fraction. Q. How much is f of 12 ? of 12? 3 of 20? of 20? of 8 ? of 8 ? } of 40? of 40 ? of 40 ? of 40 ?

Q. We found in Multiplication, 1 X., that when two numbers are to be multiplied together, either may be the multiplier; hence, to multiply a whole number by a fraction, is the same as a fraction by a whole number; consequently, the operations of both are the same as that described in || XXVII.; what, then, is the rule for multiplying a whole number by a fraction (For answer, see 1 XXVII.)

Exercises for the Slate. 1. What will 600 bushels of oats cost, at I of a dollar a bushel ? A. $1121.

2. What will 2770 yards of tape cost, at } of a dollar a yard ? A. $3371,

3. Multiply 425 by 53. A. 2210. 4. Multiply 272 by 154. A. 4284. 5. Multiply 999 by 21 . A 21201. 6. Multiply 20 by 57. A. 1017.

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