EXAMPLE.

Reduce,,,, and, to a common denominator.

Here 5 and 9 may be rejected, their product being equal to 45; and by rejecting 5 and 9, we have their substitute remaining, namely, the 45 to be brought down, and it stands thus:

4) 4 8 45

As there are now no two figures in the

1 2 45 } quotient left that can be divided, the quo tients and the divisor being multiplied together will give the smallest multiple, namely, 45×2×4=360.

And, 78, 8, 35, and, are equal to their original fractions. This method, although not new, is nevertheless well worth the attention of the reader; and will ultimately lead him, by inspection alone, especially when there are but few denominators, to obtain the least multiple of them.

EXAMPLES.

Find the least multiple of 2, 6, 8, 10, 20.

Answer, 120.
What is the least multiple that can be divided by 2, 3,
4, 5, 6, 7, 8, 9?
Answer, 2,520.

What is the least multiple that can be divided by 14,
16, 18, 20?
Answer, 5,040.

CASE THIRD.

To reduce fractions of one denomination to another, retaining the same value.

When the fraction is a great name to be reduced to a smaller one, observe the following,

Rule. Divide the denominator by the parts contained in the integer; but if this cannot at once be done, multiply the terms of the fraction by the least figure that will do it, and then divide. But when the fraction is a small name to be brought to a greater, divide the numerator by the parts contained in the integer, or multiply the terms of the fraction, and then divide as before directed.

EXAMPLE.

Reduce of a £. to the fraction of a penny. Here multiply the terms of the above fraction by 3, and we have.

Now of a penny, Answer.

Reduce of a £. to the fraction of a penny.

Answer, of a penny.

The answer is obtained in a moment by merely dividing the 960 by 240, the pence in a pound.

That dividing the denominator, instead of multiplying

the numerator, may appear more plain by the following; 300240=560÷240.

Reduce of a penny to the fraction of a £.

Answer, of a £.

Here multiply the terms of the above fraction by 48, and we have .

And ÷40. of the £. answer.

That is, there are 384 times of a penny in a £.

Reduce

of a dwt. to the fraction of a pound troy.
Answer, goo.

CASE FOURTH.

To find the proper quantity or value of a fraction in money, weights, or measures.

RULE.

Multiply the numerator of the fraction by the number of times in which the next lower denomination is contained in the first; then divide that product by the denominator; if anything remains, bring it to the next lower name, and divide by the denominator.

Reduce 483 of a £. to its proper value.

When the denominator is an aliquot part of money, weights, or measures, they may very well be answered mentally, by asking this question, "What is the of a £.?" The answer will be one halfpenny; and, consequently, 8 of a £. must be 263 halfpence, which is 131 d., or 10s. 11 d.

The fractional parts of these different denominations are made from the aliquot parts;* thus, for example, 2d. is of

NOTE.

*An aliquot part is also a fractional part; but the word aliquot here signifies that the numerator is an unit, and that the denominator is such as will divide the unit to its lowest name, whether it be money, weights, or measures, without having a remainder.

We say, and that justly, that is an aliquot part of a £, whose value is 34d., because of a pound is 2s. 6d., and again of this 2s. 6d. is 34d. ; and is the same as of or; but we cannot say that of a £. is an aliquot part of a £.,

a £. and, if 2 d. be taken from a £, there remains 19s. 9 d., which is of a £; for 1-=; and if the question is put to divide £95 among 96 men, the answer, if correct, will be 19s. 91⁄2d.; and, on the contrary, to multiply 19s. 9 d. by 96, the answer will be £95; and in the same manner for the rest of the different denominations in weights, measures, &c., so that these fractional parts taken from the integer will always leave an aliquot part of it.

The aliquot and fractional parts will be found to be useful in exchange or interest.

The following examples may be answered without the aid of pen or pencil :

To reduce money, weights, and measures, to fractions.

Rule. Bring the given quantity to the lowest name mentioned for a numerator, under which write the number of those of the same name for a denominator, which reduce to its lowest term.

because its value is 4 d.; now, if there was no fraction after 4 d., then we should say it was an aliquot part.

The knowledge of the aliquot parts of a £. is of the utmost importance, but this may be gained by a little observation.

Again, is an aliquot part of a £., whose value is 1s. 4d.; now, what is the value of of a £? Here the answer must be 7 times 1s. 4d. or 9s. 4d., which is thus obtained with ease mentally.

The reader should always make it a rule to notice the denominator of his fraction when the value is required, to see if it is such as will divide the integer without having a remainder this will imperceptibly lead him to a knowledge of all the aliquot parts, not only in money, but in weights and measures.

If any number be divided by another, and the quotient is an integer, the divisor is an aliquot part of the dividend; if the quotient be not an integer, it is an aliquant or fractional part; thus:

12) 108 (9; 12 is an aliquot part of 108, and so is 9, but take 11) 108 (9.11 is an aliquant part of 108.