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

OPERATION.

2X20X12=1=6, Ans.

3. Reduce To of a pound to the fraction of a farthing. Izbo×20=6bX12=3×4=4, Ans.

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

960X12=80×20=X24=1=3, Ans.

5. Reduce of an ell English to the fraction of an inch. 225X5=X44X21=3, Ans.

36

180

6. Reduce TT0's of a mile to the fraction of an inch. 1109508=1360X49=560X161=2472%=‡, Ans. 7. Reduce of a bushel to the fraction of a pint.

320X40X8=×2=3, Ans.

XIII.

TO REDUCE A FRACTION OF A LOWER DENOMINATION

THAT OF A HIGHER.

то

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=1680-210, Ans.

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 higher.

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

1

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

Ans. 225

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.

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

XIV.

Ans. 120 0

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 num bers, and the product will be the least common multiple. 1. What is the common multiple of 3, 5, 8, and 10? 53 5 8 10

23 1 8 2

10×3×1×4X1=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, 10×3=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.

3×5X8=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 lowor 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=22=1, and

X12-24-8 pence.

12

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 of a guinea?

Ans. 21 s. 9 d. 14 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.

XVI.

TO REDUCE A MIXED QUANTITY OF WEIGHTS, MEASURES, ETC., TO THE FRACTION OF A WHOLE NUMBER.

Reduce the given numbers to the lowest denomination mentioned, for a numerator; and then reduce the whole number to the same denomination for the denominator of the required fraction.

1. Reduce 3 d. and 2 qrs. to the fraction of a pound.

2. Reduce 2 gr. 15 lb. 4 oz. 5 dr. to the fraction of a

cwt.

Ans. 3. Reduce 7 oz. 13 dr. to the fraction of a pound.

Ans. .

4. What part of an ell English are 2 qr. 3 na. 0 in. ?

Ans. .

5. What part of a mile are 6 fur. 30 rd. 12 ft. 8 in. 013 br.?

6. What part of a hhd. of beer are 42 gal.?

Ans. 1

Ans. .

XVII.

TO ADD FRACTIONS.

When the denominators are the same, add the numerators; if the denominators are not the same, reduce the fractions to the same denominators.

1. Add 1, 2, .

Ans. 11.

Subtract the smaller numerator from the larger, when the denominators are the same; if the denominators are not the same, reduce the fractions to the same denominators.

2. Reduce,

nator.

83 81

of 14, 1, and to the same denomiAns. 181, 18815, 181ft, 16818.

3. Reduce of of to a simple fraction. Ans. 4. What is the value of 3761 acres of land, at $75 per acre? Ans. $28387,061. 5. What is the cost of 8 cords of wood, at $23 per cord? Ans. $20, or $20,184, or $20,1875. 6. What is the value of 9 pounds of wool, at 384 cents per pound?