24. Bought a tun of wine at auction for $250 and sold it at retail for 6 cents a gill, how much was gained on the whole, allowing $37.50, for incidental expenses?

25. Admit a ship's cargo, from London, to be worth 987£. 15s. 6 d., what must the man pay who buys of it? how much in Federal money, allowing 4s. 6d. sterling to the dollar?

26. Two men talking of their ages, one said he was 34 years, 9 months, 20 days old, and the other replied age is exactly of mine, how old was the latter?

of your

of 13 of a dollar,

At the same rate

27. If of of of a pound equals what part of a pound is one dollar worth? what would be the sterling value of 44 dollars?

28. The UNITED STATES government extends over a territory of 2620000 square miles, and the State of New Jersey contains 8320 square miles, how many such States could be formed from the area of the United States' territory?

29. The population of Europe averages 66 persons to every square mile, what would be the number of inhabitants in the UNITED STATES, if the population were as dense as that of Europe, allowing the number of square miles in the former, to be as above?

30. Just sixteen yards of German serge

For 90 dimes had I.

How many yards of that same cloth,

Will 14 Eagles buy?

31. If of a penny equals of a cent, how many dollars are there in 8€. 9s. 6d. 3fars.?

14

32. If of a yard of chintz cost 5 of a dollar what would of an ell English cost at the same rate?

33. A goldsmith sold a tankard for 12£. 6s. 84d. being at the rate of 6s. 9d. per ounce, what was the weight of the tankard.

34. If of a ship is worth of her cargo what would be the worth of both, the cargo being valued at 25000£.?

ARTICLE XVI.

DECIMALS.

WE have already learned (Article XVI., pages 150-152,) that DECIMALS are a kind of FRACTIONS based upon the principle of whole numbers, or integers, which will more fully appear by studying attentively the following

INTEGRAL AND DECIMAL NOTATION TABLE.
Integers.

Decimals.

From which it appears that the higher orders of units in

Units of the 6th order
Units of the 7th order

Units of the 8th order Units of the 9th order

WHOLE NUMBERS have a higher value; but the higher orders of units in DECIMALS have a lower value.

By the laws of NOTATION, therefore, a WHOLE NUMBER commences with the INTEGRAL UNIT, and INCREASES in the ratio of 10 to 1, while the DECIMAL commences at the SAME POINT, and DECREASES in the same manner. Thus, in whole numbers 10 UNITS of the first order make 1 UNIT of the second order, &c., while of an integral UNIT of the first order makes one DECIMAL UNIT of the first order, called onetenth, and of a DECIMAL UNIT, of the first order makes one DECIMAL UNIT of the second order, called one hundredth, &c., in which the INTEGRAL UNIT (unit of the first order,) is regarded as the FOUNDATION of Decimal Notation, as well as of notation in whole numbers. Commencing with the first order of units, therefore, and multiplying by 10 towards the left, and by towards the right, we may proceed either way ad infinitum. We see, also, that the numerator of a decimal is all that appears, the denominator, being hidden from view, is found by NOTATION. In whole numbers we use ciphers to fill the places of those lower orders of units not mentioned in the number to be written, but in decimals, we use ciphers to fill the places of those higher orders of units not mentioned in the quantity to be written. It has been remarked, in NOTATION OF INTEGERS, (pages 17, 18,) that ciphers are used to fill the places left vacant in the number to be represented, this is also true in regard to decimals; and although the ciphers thus used signify nothing in themselves, yet in either of these cases they have a tendency to give value to the significant figures, by assigning them their proper place in the order of notation. Thus, if we wish to represent 8 units of the second order in integers, we set down the 8, and place a cipher on its right, to fill the place of units of the first order, which gives 80, (eighty,) and in the same way, if we wish to represent 8 units of the second order in DECIMALS, we set down the 8, and place a cipher on the left, to fill the place of units of the first order, giving .08, (eight tenths,) i. e., . Again, if we wish to represent 8 units of the third order in DECIMALS, we set down 8, and place ciphers in the places of units of the first and second orders, i. e., in the places of tenths and hundredths; thus, .008. = to 180, &c. Hence, it is plain, that a decimal may be expressed in the

form of a vulgar fraction, by placing the number contained in it, over another number, which is always suggested by the significant figure standing on its right, i. e., by the lowest order of units contained in the decimal; thus, .8= 8 %; .08= 180; .008=1000; 0008=10000; .00008 = 100000; .2689 =208; 5.654-5654, &c.

2689

10000

1000'

8

10

In the same manner any quantity consisting of a decimal, or a whole number, and a decimal (mixed number,) may be reduced to a common fraction by the following

RULE for reducing Decimals to Vulgar Fractions.

First. Commence at the point, and find by notation the RIGHT HAND FIGURE, or LOWEST order of units in the DECIMAL, if THIS is tenths, the denominator will be ten, (10,) or if hundreds, the denominator will be one hundred, (100,) &c.; and,

Second. Having found the denominator, write the quantity to be reduced over it, for a numerator, omitting the decimal point, the result reduced to its lowest terms, will be the frac tion sought, or answer.

NOTE.-If the quantity for reduction is a decimal, without an integer, and has ciphers on the left of a significant figure, omit those ciphers before using the quantity as a numerator; as, .008-10, &c.

5. Reduce 25.0025 to an equivalent vulgar fraction.

Ans. 250025-10801-2530.

=

In the same manner find the value of v. in the following equations where v. represents the vulgar fraction, to which the quantity on the left is equivalent.

Set down the quantities, whether they are fractions or mixed numbers, so that the like orders of units in each may fall exactly under one another, and then add or subtract as in WHOLE NUMBERS, the RESULT will be the ANSWER.

EXAMPLES.

1. What is the value of s. (sum,) in the following equation, 374.6856+98.61402+.84306+.0061+25=s.

Hence, 374.6856+98.61402+.84306+0061+25= 499. 14878=s.

Answer.

2. What is the value of d. (difference,) in 8676.4695— 48.069=d.

Here we subtract, as in WHOLE

numbers, setting down each remain. Thus : 8676.4695 der under the figure subtracted, and

find.

48.069

8628.4005 Ans.

Hence, 8676.4695-48.069-8628.4005=d. Answer.