X.

TO DIVIDE ONE FRACTION BY ANOTHER.

Invert the divisor, and proceed as in multiplication.

1. Divide by 3.

OPERATION.

x=1=3, Ans.

This process deserves the careful attention of the pupil. To make the illustration of the rule more intelligible, let us suppose the above sum to be given thus: If a man pays of a dollar for of a yard of cloth, what will a yard cost? If of a yard cost of a dollar, one third will cost one half of, or. If of a yard cost, one yard will cos three times, or 1-3.

6. Divide 17 by 6417.

7. Divide 11543}}}}} by 3344221J††††. 8. Divide 1374228 by 28898 t t 88 77.

66

255669

9. Divide 333287788 by 1,999.

10. Divide 61 by 81.

82344

TO REDUCE FRACTIONS TO THE SAME DENOMINATOR.

Multiply the numerator and denominator of each fraction

by the denominators of all the other fractions.

1. Reduce †, 7, and to the same denominator.

1848 1760

Ans. 1, 18, 1792, 1778.

It will be seen by inspecting the above process, that each fraction retains the same value after it is reduced, as it had before. And this is evident, because both terms of each fraction are multiplied by the same numbers.

2. Reduce,, to the same denominator.

Ans.,, and .

3. Reduce and to the same denominator.

TO REDUCE A FRACTION

XII.

Ans. 4 and 8.

OF A HIGHER DENOMINATION TO THAT OF A LOWER.

Multiply the given fraction by that number of the next lower denomination which makes a unit of the given denomination. Proceed in this manner until the fraction be reduced to the required denomination.

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

OPERATION.

ToX12=1=1, Ans.

As the value of shillings is 12 times greater than that of pence, therefore to reduce a fraction of a shilling to the fraction of a penny, the number of parts taken in the fraction of a shilling must be increased 12 times, or the value of the parts must be increased 12 times. Multiplying the numerator of a fraction multiplies the number of parts in a given fraction; dividing the denominator of a fraction increases the value of the number of parts in the given fraction. Hence, either of these ways may be adopted, according to the character of the sum.

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

OPERATION.

×20=4×12=14=4, Ans.

3. Reduce 12 of a pound to the fraction of a farthing. 1200×20=65×12=4×4=t, Ans.

4. Reduce of a pound troy to the fraction of a grain.

9600×12=goo×20=&×24=24=3, Ans. 5. Reduce 25 of an ell English to the fraction of an inch. 25X5=4=45×24=3%=}, Ans.

36

6. Reduce TTo'ggo of a mile to the fraction of an inch. 110880X8=13960X40=13860×164=2472%=4, Ans.

7. Reduce of a bushel to the fraction of a pint.

320X4=X8X2-,

XIII.

Ans.

TO REDUCE A FRACTION OF A LOWER DENOMINATION TO THAT OF A HIGHER.

Divide the given fraction by that number of the same denomination which makes a unit of the next higher denomination. Proceed in this manner until the fraction be reduced to that of the required denomination.

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

÷÷÷12=÷÷20=1630=zko, Ans.

84

As 12 in the denomination of pence are equal to only 1 in the denomination of shillings, therefore the number of parts in the fraction of a penny must be diminished 12 times or the value of the parts must be diminished 12 times, to express, in the denomination of shillings, the value of the fraction of a penny. Either mode of reduction may be adopted according to the nature of the sum.

The same principle applies in every step of the process, in reducing a fraction of a lower denomination to that of a higher.

2. Reduce of a grain to the fraction of a pound.

Ans.

44

3. Reduce of an inch to the fraction of an ell English.

Ans. 25.

4. Reduce of an inch to the fraction of a mile.

Ans. 110880

5. Reduce of a pint to the fraction of a bushel.

Ans. 3 ਹ•

6. Reduce of a farthing to the fraction of a pound.

XIV.

Ans. Too

TO FIND THE LEAST COMMON MULTIPLE OF TWO OR MORE

NUMBERS.

Divide by some number that will divide two or more of the given numbers without a remainder, and place the several quotients and the undivided numbers under the given numbers, and so continue to divide, until no number, exceeding unity, will divide two or more of them without a remainder. Multiply together all the divisors, quotients, and undivided numbers, and the product will be the least common multiple. 1. What is the common multiple of 3, 5, 8, and 10? 5|3 5 8 10 23 1 8 2

10X3X1X4X1=120, Ans.

To discover the propriety of this process, a close inspection of the example given, is necessary.

The first divisor (5) multiplied by the second divisor (2) is equal to 10, which are a multiple of 5. As 3 are not divided by 2 and 5, they remain 3, and enter into the process as a factor, 10x3=30, which are a multiple of 3, 5, and 10. But 30 are not a multiple of 8. Let 8 enter into the process as a factor, and the product then obtained will be a multiple of 8. Multiply 5 by 2 and 4, and we have 8 as a factor, and the product 40 is a multiple of 8. Then if 30 are a multiple of 3, 5, and 10, any number of times 30 are a multiple of the same numbers. And if 40 are a multiple of 8, so any number of times 40 are a multiple of 8. Therefore 40x3=120, a multiple of 3, 5, 8, and 10.

Again. The product of 3X5 is a multiple of 3 and 5; that product multiplied by 8 is a multiple of 3, 5, 8 and 10, because all these numbers are component parts of that product.

3X5=15, a multiple of 3 and 5.

3X5X8=120, a multiple of 3, 5, 8, and 10.

2. What is the least common multiple of 3, 4, 8, and 12?

Ans. 24.

3. What is the least common multiple of 4 and 6?

Ans. 12.

4. How large must that vessel be, that may be filled by each one of the following measures, viz. 4 quarts, 6 quarts, 10 quarts, 12 quarts? Ans. 60 quarts.

XV.

TO FIND THE VALUE OF A FRACTION IN THE KNOWN PARTS OF A WHOLE NUMBER.

Multiply the given fraction by that number of the lower denomination which makes a unit of the denomination of the given fraction. If the product be an improper fraction, reduce it to a whole or mixed number. If there be a fraction over, proceed with it as before.

1. What is the value of of a £1?

×20=29=1, and

X12-28 pence.

or of a shilling.

Ans. 1 s. and 8 d.

For the demonstration of this rule, see sect. XII. Vulgar Fractions.

2. What is the value of 3 of a guinea?

Ans. 21 s. 9 d. 1 qr.

3. What is the value of & of a pound troy?

Ans. 10 oz. 13 pwts. 8 grs.

4. What is the value of of a rod?

Ans. 144 ft. 19 in.