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When the price is more than a shilling and less than two.
Take parts for the excess above a shilling, and add the quotients to the given quantity, and the amount is the answer in shillings.
Example.- What is the price of 217 lb. of coffee, at 1s. 10d. per pound ?
217 6d. =
108 6 at 6d.
72 4 at 4d.
191. 178. 10d.
What is the price of 7506 lb. at 1s. 9d.?
Ans. £680. 4s. 7d. At 1s. 11d. per pound, what will 5000 lb. cost?
Ans. £479. 33. 4d. What will 2705 lb. cost, at 1s. 113d.?
Ans. £267. 13s. 7åd.
When the price is an even nu mber of shillings.
Multiply the given number by half the given price, doubling the product of the unit figure for shillings.
279 lb. at 6s. per pound.
Here 3 times 9 are 27, twice 27 are 54s., put down and carry 2
; 3 times 7 are 21 and 2 are 23, 3 times 2 are 6 and 2 are 8.
Or thus, taking the aliquot parts of a pound:
£279 58. =
1 69 15s. 1s.
13 19s. Anz. 83 14s.
What is the price of 3254 lb. at 4s. per pound?
Ans. £650. 16s. What is the price of 3123 lb. at 16s. per pound?
Ans. £2498. 8s.
When the quantity consists of several denominations. Multiply the price by the highest, and take aliquot parts for the remaining denominations.
At £4. 13s. 4d. per cwt., what is the price of 25 cwt. 2 qrs. and 14 lbs. of
2 6 8 the price of 2 qrs. 14 lbs. = 0 11 8 the price of 14 lbs.
£119 11 8 Ans.
The given price multiplied by 25 gives the price of 25 cwt. ; then 2 qrs. being the } of a cwt. the į of the price per cwt. is taken; and 14 lbs. being the 1 of 2 qrs., the 4 of £2. 6s. 8d., the price of 2 qrs., is taken ; and these two sums added to the price of twenty-five hundred, give the answer to the question.
Sold 5 cwt. 1 qr. and 8 lbs. of sugar, at £3. 15s. 8d. per cwt.; how much did I get for the whole?
Ans. £20. 2s. 7 d.
Sold 56 cwt. 1 qr. and 17 lb. of sugar. at £2. 15s. 9d . per cwt. ; how much am I to receive for the whole ?
Ans. £157. 4s. 4 d. Bought sugar at £3. 14s. 6d. the cwt. ; what did I give for 15 cwt. 1 qr. 10 lb. ?
Ans. £57. 2s. 9 d. What is the value of 27 cwt. 2 qrs. 15 lb. of logwood at £1. 1s. 4d. per cwt. ?
Ans. £29. 98. 64d. Bought 72 cwt. 2 qrs. 14 lbs. of tobacco, at £4. 168. 8d. per cwt. ; what did the whole cost? Ans. £351. Os. 5d.
THESE are also fractions; and before closing this little Treatise on Arithmetic, think it necessary to say a few words as to their nature and use; as it is impossible to carry on those more nice and extensive processes of calculation resorted to by the engineer, the astronomer, the chemist, or surveyor, without them. The difference between these and the fractions treated of in the beginning of the work, consists in this; that Vulgar Fractions described quantities by 1, 3, 4, &c.; those of which I am now about to treat, always, as their name imports, describe quantities in tenths, hundredths, thousandthsindeed always in tenths or some submultiples of tenths.
Why they are more applicable to these nicer calculations spoken of above is, that they admit of greater simplicity of statement, and that they enable us, even in Recurring Decimals, to come within a millionth part of what we wish to express.
In the Fractions already treated of, we required two lines of figures in their statement, one expressive of a numerator, and another of a denominator. In these fractions the denominator being always understood to be ten, or some multiple of ten, is never written.
Thus the decimal of į is •5, five being the half of ten ; the decimal of 4 is .25, twenty-five being the quarter of a hundred; the decimal of .is .75, having just three times as much for its numerator as į has.
Decimals are written after whole numbers, thus 79.5, 176.25, 279.75. The whole number being written, a point is made on its right, and we then write the decimal; and however many figures the decimal so written
may have to express it, we must never forget that it is less than a unit of any quantity or thing we may intend to express by the whole nnmber. Thus the value of a decimal is determined by its approach to the place of units, and the further we drive the expressive figure from that place, the more we lower the value of the decimal; thus 79.5 represents 79% but 79.05 represents 7916. We thus see the importance of the cipher in whole numbers and decimals; the pupil being already aware that if written before a whole number it does not alter the value ; and now that, although an important figure when written before the expressive figure of a decimal, it is entirely useless when written after it; thus, 5 is half, is the same, we gain nothing by writing it, and it is therefore never used but to decrease the value of the decimal by being placed before it.
REDUCTION OF DECIMALS. Vulgar Fractions may be reduced to decimals, by adding ciphers to the numerator, and dividing by the denominator until evenly contained, thus :
Or Decimals may be reduced to Vulgar Fractions, thus :
But the following figures, that cannot be thus evenly divided, are called recurring decimals, }, }, , }, although these parts cannot be correctly expressed by decimals, the denominators not dividing evenly ten or any multiplier of ten; we can always come sufficiently near for our purpose;
and the value of the article under consideration must always determine the pupil as to the number of places; for instance, I would think it necessary to have more to express the one-third of a pound than I would the one-third of a penny; id. or •33, would bring me within the hundredth part of a penny; but a pound being 240 times as much as a penny, it of course would have more decimal places; thus,
which would be within a millionth part of the third of a pound.
Having now, I think, said sufficient to instruct the pupil in the value and use of decimals, our next step must be their Addition, Subtraction, Multiplication, and Division, which operations are all performed the same as in whole numbers, except that in stating lines of figures for Addition and Subtraction, we are not to arrange them with the unit or right figure under each other, but be guided by the decimal point, which is the index of value.