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er fraction? As. Multiply the whole mumber by the denominator and acd in the numerator, under this suin write the denomiirator.

Does tbis alter its value? Ans. It does not.

Why? I have said that the value of a fraction was the quotient arising from dividing the numerator by the denominator, and this operation would bring it back again to the same fraction.

For Example--Reduce 44 to an inproper fraca tion? Ans. ..

What is the value of this? Ans. 2)9

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Hence to find the value of an improper fraction is already done in the above operation.**

Give an example of an improper fraetion reduced to find its value? Ans. 5) 10

2 Ans. How reduce a whole number to a fraction with a given denominator? Ans. Multiply the whole number by the denominator, and put the denominator under the product.

For Example--Reduce 5 to a fraction, whose denominator shall be 6. Ans. 3.

Is the value alterea? Ans. It is not.

How is the value not altered? Ans. According to the definition of the value cf a fraction above, it is not, for 6 in 39, 5 times, the whole number.

What is the meaning of changing a whole number into a fraction with a given denominator? Ans. It means that 8 (or whatever the whole number is) whole appies are to be divided each irrto as many parts, as the given denominator says. # For Ecrumple--Suppose I should tell you to dis wide 3 apples each into 4 parts or quarters, how mamy parts would all the apples make? Ans. 32, i. a 8

the whole number, multiplied by the given denominator 4 (which always shows how many parts 1 whole thing is divided into.)

What wouid be the fractional expression ? Ans.32.
What does the 4 shou?
What is the value of this? Ans. 8.

How reduce whole and mixed numbers in changing compound fractions to simple or 10 proper fractions of equal value? Ans. To improper fractions.

What do then? Ans. Multiply the numerator and denominator together. # How does the rei'uction by the rule not affect the

value? Ans. 1 of 4 is 1 always, whether of an apple or anything else, i.e. 2 times 2, ard so with any other numbers.

Før Example--Reduce of 1 of to a simple fraction? Ans. za

Reduce { of 7? Ans. 36.

How reduce fractions of different denominators to a common denominator? Ans. Multiply each numerator by all the denominators but the one under it, for a new numerator--and all the denominators together for a new denominator.

What d. you mean by a common denominator? Ans. When the denominators are all aliké.

Is the value altered in this reduction? Ans. It is not.

How is the value not altered? Ans. It has leen shown heretofore that multiplying the numerator and denominator by the same number does not a!ter the value, and in this rule they are so multiplied.

Give an example of and reduced to a common denominator? Ans. 4,

What was 1 multiplied by? Ans. 4, as 4 times 1 is 4.

What was its denominator multiplied by? fins. , as 4 times 2 is 8.

ADDITION OF VULGAR FRACTIONS.

What is the rule? Ans. Reduce the fractions to a common denominator, then write the sum of the numerators over the denominator. .

SUBTRACTION OF VULGAR FRACTIONS. What is the rule? Ans. Reduce the fractions as in Addition, than write the difference of the nume. rators over the common denominator."

MULTIPLICATION OF FRACTIONS.

What is the rule? Ans. Reduce all the fractions to single or improper ones, then multiply all the numerators together for a new numerator and all the de. nominators together for a new denominator.

For Example--Multipls } by ? Ans. .
Multiply by f? Ans.
Multiply by ? Ans. j.

DIVISION OF FRACTIONS. What is the rule? Ans. Invert the divisor, and proceed exactly as in multiplication.

For Example--Divide 4 by ?? Ans. 12. low prove this? Ans. šis in, or 2 in 4: 2, non 32 is equal to 2, for the value of a fraction is. &c.

Eramples to crercise the scholar in Fractions You own , B, C 4 and D the rest of a vessel ; D's part was 100 dollars ; how much was each man's part, and what part did Down? Ans. The vessel is divided into 8 parte, all the parts except D's make 7: consequently D's part is d, and is 1 is 100 dollars, is 200 and so on. I own 100 dollars, B 200, C 400, and D

There is a beam, pa is in the ground, in the water, and the rest being 2 feet out of water, how long is the beam? Ans. 16 feet.

How did you do it? Ans. 9 and 5 are 14, there remain 2 parts which are 2 feet, i. e. 1 foot for 1 part, and for 16 parts 16 feet.

You give 5 dollars for 4 of a sled, how much must you give for the other part, and how much is the sled worth?

There are 8 nine-pences in a dollar, now you have

of a dollar, how much money have you? Ans. If dollar.

In the school which you attend, are studying Latin, Greek, } Grammar, and the rest, being 10 in number have just commenced Practical and Mental Arithmetic; how many scholars compose the school? Ans. T =; then 10 are ; and if 10 are , is 50 scholars, the answer..

Note.When the denominators are not alike, they may be reduced to a common denominator, and then they will be very simple.

How can fractions, as a convenient practical method, be reduced to their least common denominator? Ans. Multiply or divide any or each fraction by any number that will make the denominators alike.

How does this not affect the value? Ans. It has been proved that multiplying, or dividing both nu. merator and denominator, by the same number, does not alter the value.

Reduce and to their least common denominator? Ans. Dividing by 2 reduces it: as

EXAMPLES. Suppose there is a mast erected, so that of its length stands in the ground, 12 feet of it in the water, and of its length in the air, or above water; I demand the whole length?

2x} X3

Bott=t Now more makes the whole, and as it is 12 feet, then the whole is 12 times 18=216 Ans.

In an orchard of fruit trees, 4 of them bear apples, £ peaches, & plums, 40 of Them pears, and ten cherries ; how many trees does the orchard contain?

6x} 3x + 2x*

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Pia X D X = then 40peaches and 10 cherries are b, and if 50 are th, then +%, or the whole is 12 times 50–600, Ans.

A gentleman left his son a fortune, 1 of which he spent in 3 months; * of the remainder lasted him 10 months longer, when he had only 2524 dollars left; pray what did his father bequeath him?

.3 : 2524 : non

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Ans. 5389,33} the value of seren parts, or the whole.

T'he third part of an army was killed, the fourth part taken prisoners, and 1000 fled; how many were in this army, how many killed, and how many captives?

}

Ati= now, as more make the whole, or ti, it is evident that must be 1000, and if 5 parts be 1000, what will 12 parts, or the whole be? As 5 : 1000 : 12

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