CASE VII. To find the value of a fraction in the inferiour denominations of the integer.

RULE.-Multiply the numerator by the parts in the next inferiour denomination. and divide the product by the denominator. Then if any thing remains, multiply it by the parts in the next inferiour denomination, and divide by the denominator as before, and so on, as far as necessary; then the quotients, placed in order, will be the value of the fraction required.

EXAMPLES.

1. Find the value of of a pound sterling. Ans. 3s. 6d.

7 20

40)140(3s, 6d. Ans.

120

20

12

240

240
0

U

2. What is the value of 3. What is the value of

4. Find the value of
5. What is the value of

DEM.-The numerator of a fraction being considered as a remainder, in Division, and the denominator as the divisor, it is plain, that the operation here, is only continuing the division in the inferiour denominations of the integer.

6. Find the value of of a dollar.
7. Reduce of a mile to its proper quantity.

Ans. 60 cents.

Ans. 4fur. 22rds. 4yds. 2ft. lin. 24b. c. CASE VIII. To reduce a fraction from one denomination to another, retaining the same value.

RULE. Consider how many of the less denomination make one of the greater; then multiply the numerator by that number, if the reduction be to an inferiour denomination, but multiply the denominator, if to a superiour denomination.

EXAMPLES.

1. Reduce of a pound to the fraction of a penny.

12)

-2x2x12=729-50, the answer.

Ans..

DEM.-The reason of this rule is obvious, for it is the same as the rule of Reduction in whole numbers from one denomination to another.

2. Reduce of a penny to the fraction of a pound. Ans. 13.

3. Reduce of a pound to the fraction of a penny.

4. Reduce 2s. 6d. to the fraction of a pound.
5. Reduce of a cwt. to the fraction of a pound.
126

Ans.
Ans..

Ans. §.

6. Reduce of a yard to the fraction of an Ell English.

Ans..

It is often better to set the sum down as a compound fraction, and obtain the answer by comparing, thus:

of

the answer.

of an Ell English to the fraction of a yard. of of yd. the answer.

of

7. Reduce

8. Reduce

of a day to the fraction of a year.

of 85145 Ans. Ans.

9. Reduce 4 furlongs to the fraction of a mile.

ADDITION OF VULGAR FRACTIONS,

Is collecting in one sum, broken numbers. RULE-Reduce compound fractions to single ones, mixed numbers to improper fractions; fractions of different integers to those of the same; and all of them to a common denominator; then the sum of the numerators written over the common denominator will be the sum of the fractions required.

EXAMPLE.

1. Add and together. 5X6-30, numerator of §. 5x9=45, numerator of 5.

75, sum of the numerators. 9x654, common, denominator. Then +35.

Ans. 1 DEM. Before fractions are reduced to a common denominator, they are quite dissimilar, because in the firs fraction, 5, a unit is di

vided into 9 parts, and in the second fraction, §, a unit is di vided into 6 parts; consequently the parts are unequal until reduced to a common denominator, then we have 34 and 5, which make 4. That we do not alter the value of the fraction by the operation is evident, because we multiply the numerator and denominator by the same number; thus,

5X6 30 5X9 45 and

9X6 54 6X9 54

Now it is plain, that a unit in each fraction is divided into

54 parts, and consequently the numerators may be addeo, because they are parts of one common integer.

2. Add and together.

3. Add, and together:

4: Add 4 and 9 together.

5. What is the value of of a shilling and

Ans. 13.

Ans. 24.

Ans. 14.

of a pound? Ans. 18s: 3d.

of a shilling? 13s. 10d. 2 qr. of an hour together. Ars. 2d. 14h. 30min. and of a penny. Ans. 3s. 1d. 1gr.

Ans, 6 furlongs 28 poles.

10 Add 2 of a shilling to

of a penny.

Ans. 9d. 1qr.

་

SUBTRACTION OF VULGAR FRACTIONS,

Is taking one broken number from another.

RULE.—Prepare the fractions the same as in Addition; then subtract one numerator from the other, and set the remainder over the common denominator, for the difference of the fractions sought.

NOTE. When the given fractions have a common denominator, they may be added and subtracted without any further reducing. EXAMPLES.

1. What is the difference between 5 and 2?

Here, =2, the answer.

DEM.It is evident, that the difference between and isor, because the fractions have a common denominator, that is, a unit in each fraction is divided into six equal parts, therefore the difference of the numerators placed over the common denominator, must express the difference of the

fractions.

2. What is the difference between 7 and ?

7X3=21

2X8=16

8X3-24 com. denom.

11, and 15, therefore, 21-15, the answer. DEM. The same reason for

the reduction of the fractions, might be offered here, as in Addition, hence the operation of the work is manifest.

Ans. 1m. 2fur. 16pol.
Ans. 15d. 4h. 48min.

9. From 35 days, take 19 days. 10. From of a pound, take of a shilling.

Ans. 4s. 1d.

MULTIPLICATION OF VULGAR FRACTIONS,

Is repeating a whole or broken number, by a part or the parts of an integer.

RULE.-Reduce mixed numbers, if there be any, to equivalent fractions; then multiply all the numerators together for a new numerator, and all the denominators together for a new denominator, which will give the product required.

1. Multiply by 2.

EXAMPLES.

1x3, the Answer.

DEM. It is evident, that as many times as the numerator of a fraction is increased, so many times the value of the fraction is increased; then when we multiply the numerator of the fraction,, by 3, the fraction is increased 3 times, but we do not wish to increase the value of the fraction 3 times, but as much less as the denominator 4, indicates; then when we multiply the denominator of the fraction by 4, it makes the value of the fraction 4 times less, because it takes 4 times the number of parts to make a unit. Multiplying the denominator of a fraction by any number, is the same as dividing the numerator by the same number.

2. What is the product of and?

3. Required the product of and.

4. Required the product of 12 and 16.

5. What is the product of 40 multiplied by +?

6. Multiply 6 by of 5.

7. Multiply 12 by 4 of 4.

8. Required the product of,, and 3. 9. What is the product of 34 and 4?

10. Required the product of 123 multiplied

W

Ans. 1.

Ans. 4.

Ans. 1991.

Ans. 10.

Ans. 20.

Ans. 7.

Ans. 1.

Ans. 1411.

by 7.

Ans. 88.

DIVISION OF VULGAR FRACTIONS,

Is finding how often a part or the parts of an integer is contained in a given sum.

RULE.-Prepare the fractions as in Multiplication; then invert the divisor, and proceed exactly as in Multiplication; the products will be -the quotient required.

1. Divide 20 by 2.

EXAMPLES.

Ans. 40.

20. Invert the divisor, and it stands taus, 2x2=4o—40, the Answer.

DEM.-It is evident, that division of fractions might be performed by dividing the numerator of the dividend by the numerator of the divisor, and the denominator of the dividend by the denominator of the divisor; but as it would often give fractional results for a new numerator and denominator, it is not employed, since we can adopt an easier method, going on the principle, that Division is exactly the reverse of Multiplication; therefore we invert the terms of the divisor and proceed exactly as in Multiplication; and to prove the truth of the rule, we need only invert the reasoning in Multiplication, because in Multiplication the work is nothing more than multiplying the multiplicand by the numerator of the multiplier, and dividing the product by the denominator of the multiplier; and in Division, the work is nothing more than multiplying the dividend by the denominator of the divisor, and dividing the product by the numerator of the divisor.

10. 20

10. Divide of 4 by 45.

Rule of Three Direct in Vulgar Fractions.

Ans. 41.

RULE.-Prepare the given terms, as in Multiplication, and state the question the same as in whole numbers; then multiply the second and third terms together, and divide the product by the first: Or, invert the first term, and multiply the three together, as in Multiplication; the last product will be the answer, in the same name of the second term,

EXAMPLES.

1. If of a yard cost of a pound; what will of a yard cost? lyd.; £: to the Answer. The better way, however, is to invert the first term, thus fyd. X:3×:: 7 := . ££1 38. 4d., the Answer.