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increasing its numerator, without changing its denominator, and that a fraction is diminished by diminishing its numerator, without changing its denominator; also, that a fraction is diininished when its denominator is increased without changing its numerator.

Q. What is meant by the greatest common measure of two numbers?

A. The largest number that will divide them without a remainder, to the lowest terms.

Q. When is a fraction in its lowest terms?

A. When no number but a unit, will measure both its terms.

Q. What is a prime number?

A. A number which can only be divided by itself, or a unit, as 7, 9, 13, 19, 5, 23, &c., as before mentioned in the work.)

Q. What is a composite number?

A. A number which is equal to the products of its factors, as 28 = 7 x 4, 96 6 x 8 x 2 96.

Q. What is an abstract fraction?

A. An abstract fraction is a fraction derived from another, by means of Reduction.

PREPARATORY QUESTIONS.

CASE I.

I. To find the greatest common measure of two given numbers.

RULE.
“The greater by the less divide,
The less, by what remains beside;
The last divisor still again,
By what remains, till nought remains;
And what divides and leaveth nought,

Will be the common measure sought.”
1. What is the common measure of 112 and 120?

Ans, 8.

2. What is the greatest common measure of 379 and 403?

Ans. 31. 3. What is the least common measure of 56 and 108?

Ans. 4. 4. What is the greatest common measure of 57 and 285?

Ans. 57. 5. What is the greatest common measure of 169 and 175?

Ans. 1. 6. What is the greatest common measure of 175 and 455?

Ans. 35. CASE II. II. To find the least common multiple of any given number.

RULE.-If the given numbers are prime to each other, their continued product is their least common multiple.

“If not all prime to all beside,
See what will two or more divide:
Divide the two or more thereby,
The same upon the quotients try;
And if thou canst divide them do;
'Till nought will measure any two,
Then the last quotients multiplied,
And all the numbers which divide
Continually; the product got,

Will be the multiple that's sought.” 1. What is the least common multiple of 3, 5, 8 and 10? Operation, 5)3 5 8 10

2)3 1 8 2

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2. Find the least common multiple between 12, 25, 30, and 42?

Ans. 2100. 3. Find the least common multiple of 12, 16, 20, and 30?

Ans. 240. 4. What is the least common multiple of 25, 35, 60 and 72?

Ans. 12600. 5. What is the least common multiple that will measure 3, 4, 8, and 12?

Ans. 24. 6. What number is the least, that 7, 8, 16, and 28, will measure?

Ans. 112. CASE IV. To reduce fractional parts of a dollar to cents. RULE.—Multiply the numerator by 100 (because 100 cents is a dollar) and divide by the denominator. 1. Bring of a dollar to cents?

100

5

8)500

66

62] or 62cts. Á may be reduced to } because 4 will divide the numerator and denominator without a remainder, thus: 4) =} reduced to its lowest terms. How many cents in } of a dollar? Ans. $.50. 2. Bring j of a dollar to cents?

Ans. $.87}. 3. Bring f of a dollar to cents?

Ans. $.371, 4. Bring § of a dollar to cents?

Ans. $.123. 5. Bring for $

to cents?

Ans. $.75. 6. What number of cents in of a dollar? Ans. $.90. 7. What number of cents in } of a dollar? Ans. $.60. 8. What number of cents in of a dollar? Ans. $.80.

CASE V. To reduce fractions to their lowest denominations, and

also into cents. 1. Reduce $4 to its lowest terms? Ans. $7.1 or 19 3'1 cts, 2. Reduce $}} to its lowest terms? Ans. $or 75 cts.

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3. Reduce $34 to its lowest terms? Ans. $$ or 854 cts. 4. Reduce $to its lowest terms? Ans. $1 or 564 cts. 5. Reduce $7947 to its lowest terms? Ans. $or 40 cts. 6. Reduce $418 to its lowest terms? Ans. $10 or 90 cts. 7. Reduce $114 to its lowest terms? Ans. $16 or 564 cts. 8. Reduce ${} to its lowest terms? Ans. $} or 60 cts. 9. Reduce $14% to its lowest terms? Ans. $or 80 cts.

RATIO OF FRACTIONAL PARTS OF A DOLLAR. 1. What is the ratio between 1 and 1?

Ans. 2. From the preliminary examples it is evident that 2 quarters are = to a half; therefore, the ratio is as 1 to 2 as required. 2. What is the ratio between į and a?

Ans. 3. 3. What is the ratio between į and 3?

Ans. 3. 4. What is the ratio between ţ and ß?

Ans. 3. 5. What is the ratio between į and y?

Ans. 7. 6. What is the ratio between it and 1k?

Ans. 5.

USEFUL THEOREMS IN FRACTIONS. Note to Teachers.--The learner should be required to recite these theorems and to apply them practically.

THEOREM I. TO ADD or SUBTRACT fractions which have the same common denominator, the sum or DIFFERENCE of their numerators must be taken, and the common denominator written under the result.

THEOREM II. To reduce fractions to the same denominator, the two terms of each of them must be multiplied, by the denominator of the other.

THEOREM III. A fraction can be multiplied in two ways; namely, by multiplying its numerator or dividing its denominator. Thus, multiply so by 5 == ' which reduced is į or so + 5 (by dividing the denominator 30 by 5) = 4.

THEOREM IV. A fraction can be divided in two ways, by dividing its numerator or multiplying its denominator: thus, divide

by 4 by dividing the numerator we get ļ and by multiplying the denominator f x 4 = = } which is exactly the same.

THEOREM V. Multiplication alone, according as it is performed on the numerator or denominator, is sufficient for the multiplication and division of fractions; that is, when you multiply the numerator you INCREASE, and when you multiply the denominator you DECREASE.

CASE I,

By dividing the denominator the fraction is multiplied.

CASE II.

By multiplying the denominator the

fraction is } divided.

THEOREM VI. To multiply a whole number by a fraction. Rule.--Multiply the number by the numerator and divide by the denominator: or divide the number by the denominator and multiply the quotient by the numerator. EXAMPLE.—Multiply 20 by first, 20 x 3 = = 15 = 5 X 3 = 15.

COROLLARY. Every common divisor of two numbers must also divide the remainder resulting from the division of the greater of the two by the less.

ADDITION.

CASE VI.

When the numerators are alike and not more than a unit.

Rule.-Multiply the numerator and denominator of the fraction having the least denominator by the common measure of the fractions.

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