When fractions occur in the Statements of Compound and Continued Proportions, it is generally most convenient to use the fraction in the third or second terms, in whatever manner may be the best for the multiplication; and for a fraction in a first term, to multiply it and one of the other terms by the denominator of the fraction.

The whole may however be used fractionally in the manner of a Simple Proportion, as in the following operation worked in a fractional form.

EXAMPLE OF A COMPOUND PROPORTION.

If the carriage of 3 cwt. of Goods for 147 miles amounts to 67% s., what will be charged for the carriage of 115 cwt. for 132 miles?

EXAMPLE OF A CONTINUED PROPORTION,

worked with the multiplication of the opposite terms to cancel the fractions in the antecedents.

To find the value of 12000 Hamburg Marks Banco, in Spanish Dollars of Plate, at the exchange of 13 Marks per £ Sterling, and 36 d. Sterling per Dollar of Plate.

DECIMALS.

A Decimal is a number expressing a portion of a unit or an integer in tenths, hundredths, thousandths, &c.

NUMERATION.

The value of a decimal is shown without employing a denominator, by the position which each figure has relative to the place of units: thus, a point being used to separate decimal parts from units, tenths occupy the first place on the right, hundredths the second place, thousandths the third place, &c. as with the figures in the following table.

Which number, supposing it to be applied to Pounds, may be read as 46 Pounds, and 763587 millionths of a pound; but when there are more than two or three places of decimals it is usual to read it thus, forty-six Pounds, decimal, seven, six, three, five, eight, seven.

It is to be observed, that separation of the greater parts into those which are less, in order to be decimally made, must always be into tenths, or the larger part into 10 smaller parts. The integer is also said to be decimally separated when it is done upon the same principle, whatever name may be given to the parts produced: thus, if 10 shillings made 1 pound, 10 pence made one shilling, 10 farthings made 1 penny, and 10 parts made 1 farthing, &c., the pound would be decimally separated.

When this mode of separation is adopted, the reductions are made by merely altering the place of the decimal point: thus, 46 pounds, 7 decimal shillings, 6 decimal pence, and 3 decimal farthings, would be expressed by £ 46.763, and the number of decimal shillings would be 467.63 s., or the decimal pence 4676.3 d., and the decimal farthings 46763 f.

In the same manner as with whole numbers, ciphers are employed to keep the figures in their proper position, by filling up any vacancy that may occur: a cipher is also used when there is not an integer; thus, to express 3 units and 37 ten thousandths, or 3 units, no tenths, no hundredths, 3 thousandths, and 7 ten thousandths, we write 3.0037; and for no units 37 hundredths, or 3 tenths and 7 hundredths, we write 0.37. Ciphers are seldom affixed to the right of the decimal, except when the decimal point is in any operation to be removed beyond the given figure, or when a decimal is to be reduced or expanded into a lower sort of parts; thus, to express 0.37 hundredths as thousandths, we write 0.370, or, as ten thousandths, 0.3700.

When a decimal figure continually repeats,* as 0.4333, &c. the part which repeats may be expressed by placing a dot over the repeating figure; thus,

0.43.

When a decimal number circulates,* as, 0.7925925, &c. the first circulate only is necessary to be expressed, if a dot be placed over the first and last of the circulating figures; as,

0.7925.

The complement of a decimal is the difference between it and unity, as the complement of 0.792563 is 0.207437, the amount of the two being equal to unity.

EXERCISES UPON THE DECIMAL NOTATION.

Ex. 1. Value £ 674.373 as in the preceding table. Ex. 2. Value £ 83.406703 as in the preceding table. Ex. 3. Express by figures, one hundred and twenty-seven pounds and twenty-seven hundredths of a pound.

Ex. 4. Express by figures, fifty-four shillings and thirty-three thousandths of a shilling.

Ex. 5. Express by figures, no pounds and seven millionths of a pound.

Ex. 6. Express by figures, one, decimal, nought nought two

seven.

Ex. 7. Express by figures, no units, decimal, nought nought nought nought nought seven.

For the operations of repeating and circulating decimals, see the end of this Section.

DECIMAL REDUCTION.

CASE 1.

To reduce a decimal into a fraction.

Rule. If the decimal does not repeat or circulate, make the given number the numerator, and 1, with as many ciphers as there are decimal places, the denominator.

If the decimal repeats or circulates, make the given number as before the numerator, but make the part of it that repeats or circulates a fraction, having as many nines for its denominator as there are places of decimals in the part that repeats; the denominator is to be formed of 1, with as many ciphers as there are places of decimals in the part that does not repeat.

EXAMPLE 1.

To express 0.046875 as a fraction.

0.046875-16875 - 1875

1000000

40000 = 1350 = 33Õ

There being 6 places of decimals in the given number, there are 6 ciphers annexed to the I in the denominator; both terms are then successively divided by 25.

Ex. 1. Express 0.45, 0.125, and 0.475 as fractions. 2. Express 4.36, 8.927, and 67.54 as fractions.

Ex. 1.,, 18.

PRODUCTS.

Ex. 2. 4, 83, 6737.

N. B. The performance of the calculations in Decimal Reduction may be postponed until some of the calculations in Addition, &c. have been practised. If they are taken as here placed, some reference will occasionally be necessary to the rules of Decimal Multiplication and Division.

CASE 2.

To reduce a fraction into a decimal.

Rule. Divide the numerator by the denominator, reducing the numerator into tenths, hundredths, &c. by annexing ciphers, as far as may be necessary.

This may be expressed by 0.6. In general, when the division has been carried as far as necessary, if the last remainder is greater than half the divisor, the last quotient figure may be reckoned 1 more; as the above may be called 0.6667. Sometimes the exact value may be required, and then when there is a remainder it must be expressed as a fraction, as 0.66663.

Ex. 2. To reduce 73 into a decimal.

7300) 1.00000000

0.00013698

Or very nearly 0.000137.

EXERCISES.

Ex. 1. Reduce, 4, 1, 4, √, and into decimals. 2. Reduce, 7, V, and 7 into decimals.

3. Reduce,, 16, 1, 25, T15, and

of into decimals.

PRODUCTS.

Ex. 1. 0.5, 0.75, 0.3333, &c., 0.375, 0.4375, 0.35 2. 1.5, 1.75, 1.8333, &c., 26.35

3. 0.25, 0.125, 0.0625, 0.2, 0.04, 0.008, 0.00013698